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0321693620_INS_p1-4a.qxp 0321693620_INS_p1-4a.qxp 12/1/10 11:09 AMAM Page 1 11 0321693620_INS_p1-4a.qxp12/1/10 12/1/1011:09 11:09 AMPage Page Key Key Formulas KeyFormulas Formulas Key Key Formulas KeyFormulas Formulas Key Key Formulas KeyFormulas Formulas Key Key Formulas KeyFormulas Formulas From From Larson/Farber Larson/Farber Elementary Elementary Statistics: Statistics: Picturing Picturing thethe World, World, Fifth Fifth Edition Edition From Larson/Farber Elementary Statistics: Picturing the World, Fifth Edition ©© 2012 Prentice Prentice Hall Hall © 2012 2012 Prentice Hall From From Larson/Farber Larson/Farber Elementary Elementary Statistics: Statistics: Picturing Picturing thethe World, World, Fifth Fifth Edition Edition From Larson/Farber Elementary Statistics: Picturing the World, Fifth Edition ©© 2012 Prentice Prentice Hall Hall © 2012 2012 Prentice Hall From From Larson/Farber Larson/Farber Elementary Elementary Statistics: Statistics: Picturing Picturing thethe World, World, Fifth Fifth Edition Edition From Larson/Farber Elementary Statistics: Picturing the World, Fifth Edition ©© 2012 Prentice Prentice Hall Hall © 2012 2012 Prentice Hall From From Larson/Farber Larson/Farber Elementary Elementary Statistics: Statistics: Picturing Picturing thethe World, World, Fifth Fifth Edition Edition From Larson/Farber Elementary Statistics: Picturing the World, Fifth Edition ©© 2012 Prentice Prentice Hall Hall © 2012 2012 Prentice Hall CC HAPTER 222 CHAPTER HAPTER Class Class Width Width Class Width= == Sample Sample Standard Standard Deviation Deviation ofof a Frequency aa Frequency Distribution: Distribution: Sample Standard Deviation of Frequency Distribution: Range Range ofof data data Range of data Number Number ofof classes classes Number of classes s s=s == CC C n n-n -1- 11 1 round 11round 2 22 upup toto next next convenient convenient number number round up to next convenient number 1Lower 1Lower class class limit2 limit2 + +1Upper class class limit2 limit2 1Lower class limit2 + 1Upper 1Upper class limit2 Midpoint Midpoint = == Midpoint 2 22 Class Class frequency frequency Class frequency f ff Relative Relative Frequency Frequency = == = == Relative Frequency Sample Sample size size Sample size n nn gg xgxx m mm = == Population Population Mean: Mean: Population Mean: NN N x -m - mm Value Value - -Mean Value - Mean Mean x xStandard Standard Score: Score: z z=z == = == Standard Score: s ss Standard Standard deviation deviation Standard deviation CC HAPTER 333 CHAPTER HAPTER Classical Classical (or(or Theoretical) Theoretical) Probability: Probability: Classical (or Theoretical) Probability: Number Number ofof outcomes outcomes inin event event EE Number of outcomes in event E P1E2 P1E2 = == P1E2 Total Total number number ofof outcomes outcomes Total number of outcomes inin sample sample space space in sample space gg xgxx x x= Sample Sample Mean: Mean: x == Sample Mean: n nn # w2# #w2 g g1x 1x g 1x w2 x x= Weighted Weighted Mean: Mean: Weighted Mean: x == gg wgww Empirical Empirical (or(or Statistical) Statistical) Probability: Probability: Empirical (or Statistical) Probability: Frequency Frequency ofof event event EE Frequency of event E f ff P1E2 P1E2 = == = == P1E2 n nn Total Total frequency frequency Total frequency # f2# #f2 g1x g1x g1x f2 Mean Mean ofof a Frequency aa Frequency Distribution: Distribution: x x= x == Mean of Frequency Distribution: n nn = =1Maximum entry2 entry2 - -1Minimum entry2 entry2 Range Range = 1Maximum 1Maximum entry2 - 1Minimum 1Minimum entry2 Range 2 22 gg 1x - -m2 g1x 1x - m2 m2 Population Population Variance: Variance: Population Variance: s == NN N 2 22 gg 1x - -m2 g1x 1x - m2 m2 2 22 gg 1x - -x2 g1x 1x - x2 x2 Sample Sample Variance: Variance: Sample Variance: s == n nn -1- 11 2 22 s s=s == 2s 2s = == Sample Sample Standard Standard Deviation: Deviation: 2s Sample Standard Deviation: g1x g1x - - x2 g1x 2 22 -x2x2 CC C n n-n -1- 11 Empirical Empirical Rule Rule 68-95-99.7 68-95-99.7 Rule) Rule) For For data data with with a aa Empirical Rule(or(or (or 68-95-99.7 Rule) For data with (symmetric) (symmetric) bell-shaped bell-shaped distribution: distribution: (symmetric) bell-shaped distribution: 1. 1. About About 68% 68% ofof thethe data data lies lies between between and m mm - -s m mm + +s. 1. About 68% of the data lies between and -s s and + s. s. m mm - -2s 2. 2. About About 95% 95% ofof the the data data lies lies between between and and 2. About 95% of the data lies between and - 2s 2s m mm + +2s. + 2s. 2s. m mm - -3s 3. 3. About About 99.7% 99.7% ofof the the data data lies lies between between and and 3. About 99.7% of the data lies between and - 3s 3s m mm + +3s. + 3s. 3s. Chebychev’s Chebychev’s Theorem Theorem The portion portion ofof any any data data setset lying lying Chebychev’s TheoremThe The portion of any data set lying within within standard deviations deviations ofof the the mean mean is is at kk 1k1k 7 77 1212 within standard deviations of the mean is at at k standard 1k 12 1 11 least least 1 1-1 -- 2 . 22.. least k kk 22 2 22 s2ss= == gg 1x - -m2 P1x2 P1x2 g1x 1x - m2 m2 P1x2 Standard Standard Deviation Deviation ofof a Discrete aa Discrete Random Random Variable: Variable: Standard Deviation of Discrete Random Variable: 2 22 2 22 s ss = == 2s 2s g1x P1x2 P1x2 = == 22 g1x - -m2 2s 2g 1x - m2 m2 P1x2 Expected Expected Value: Value: E1x2 E1x2 = =m = == gg xP1x2 Expected Value: E1x2 = mm gxP1x2 xP1x2 Binomial Binomial Probability Probability ofof successes inin trials: xx nn Binomial Probability of successes in trials: x successes n trials: n!n! n! xx P1x2 pxxxpqpxnxq-qnxn--= == pxpqpxnxq-qnxn--xx P1x2 = =n=CnnxCC P1x2 1n1n - -x2!x! 1n - x2!x! x2!x! Population Population Parameters Parameters ofof a Binomial aa Binomial Distribution: Distribution: Population Parameters of Binomial Distribution: 22 Variance: Variance: s2ss= =npq Variance: = npq npq Standard Standard Deviation: Deviation: 1npq s ss = == 1npq Standard Deviation: 1npq P1A P1A oror B2B2 = =P1A2 + +P1B2 - -P1A and and B2B2 P1A or B2 = P1A2 P1A2 + P1B2 P1B2 - P1A P1A and B2 22 s2 s= Variance Variance ofof a Discrete aa Discrete Random Random Variable: Variable: Variance of Discrete Random Variable: AA B:B: Probability Probability ofof occurrence occurrence ofof both both events events and and A B: Probability of occurrence of both events and Probability Probability ofof occurrence occurrence ofof either either oror or both: both: AA BB Probability of occurrence of either or or both: A B or CC C NN N Mean Mean ofof a Discrete aa Discrete Random Random Variable: Variable: m mm = == g gxP1x2 xP1x2 gxP1x2 Mean of Discrete Random Variable: Mean: Mean: m mm = =np Mean: = np np # P1B2 # #P1B2 P1A P1A and and B2B2 = =P1A2 A BB if if and and are P1B2 if A and are P1A and B2 = P1A2 P1A2 A B are independent independent independent Population Population Standard Standard Deviation: Deviation: Population Standard Deviation: CC HAPTER 444 CHAPTER HAPTER P1E¿2 P1E¿2 = =1= 1-1 -P1E2 Probability Probability ofof a Complement: aa Complement: P1E¿2 - P1E2 P1E2 Probability of Complement: # P1B # #P1B P1A P1A and and B2B2 = =P1A2 P1B P1A and B2 = P1A2 P1A2 ƒ Aƒ2A ƒ A22 22 s2s= 2 22 s ss = == 2s 2s = == 2s g1x g1x - - x2 f g1x 2 22 -x2x2 ff if if and and are P1A P1A oror B2B2 = =P1A2 + +P1B2 A BB if A and are P1A or B2 = P1A2 P1A2 + P1B2 P1B2 A B are mutually mutually exclusive exclusive mutually exclusive Permutations Permutations ofof objects taken taken a time: aa time: nn r at rr at Permutations of objects taken at time: n objects P=rr nPnnrP n!n! n! == , where ,,where r r…r …… n nn where 1n1n 1n r2! - r2! r2! Distinguishable Distinguishable Permutations: Permutations: , ,, n1nnalike, n 2nnalike, ÁÁ Distinguishable Permutations: alike, alike, Á 11 alike, 22 alike, alike: nknnalike: alike: kk n!n! n! , , Á Á !nkk!! , n1n!n1#1!n!#2#n!n2#2!n!#2#n!n2Á 2!! nkn Á Á where where n1nn+ n n + +n where +2nn+ +3nn+ +Á +knnk=k =n = nn 11 + 22 + 33 + Combination Combination ofof objects taken taken a time: aa time: nn r at rr at Combination of objects taken at time: n objects n!n! n! C=rr == nCnn rC 1n1n - -r2!r! 1n - r2!r! r2!r! Copyright © 2012 Pearson Education, Inc. Geometric Geometric Distribution: Distribution: The The probability probability that that the the first first Geometric Distribution: The probability that the first x -x1x--11 success success will will occur occur onon trial trial number number isx is xx , ,, P1x2 = =p1q2 success will occur on trial number is P1x2 P1x2 = p1q2 p1q2 where where q q= where q =1= 1-1 -p. - p. p. Poisson Poisson Distribution: Distribution: The The probability probability ofof exactly exactly x xx Poisson Distribution: The probability of exactly xe-m mxmemx-m e-m occurrences occurrences inin anan interval interval is is , where ,, where P1x2 = == occurrences in an interval is P1x2 where P1x2 x!x! x! and and is the mean mean number number ofof occurences occurences e eLe L2.71828 mm and is the the mean number of occurences L 2.71828 2.71828 m is per per interval interval unit. unit. per interval unit. CC HAPTER 555 CHAPTER HAPTER s ss sxss= = xx = 1n 1n 1n x -m - xmmxx x xx -m - mm Value - -Mean Value Value - Mean Mean x xz-Score = == z-Score = == = == z-Score sxssxx Standard Error Standard Error Standard Error 1n s>s> 1n s> 1n CC HAPTER 666 CHAPTER HAPTER Interval Interval forfor :: xc- Confidence cm :mmx 6 66 m mm 6 66 x x+ Confidence Interval for c-Confidence x -E -E E x +E, + E, E, s ss EE = =z=c zzcc if if s where where is known and and the the population population is isis E s is where if s is known known and the population 1n 1n 1n s ss n nÚ 30,30, EE = =t=c ttcc if if normally normally distributed distributed oror oror the n ÚÚ 30, E normally distributed or or if the the 1n 1n 1n population population is is normally oror approximately approximately normally normally population is normally normally or approximately normally ss n n6 3030 distributed, distributed, is unknown, and and s is n 66 30 distributed, is unknown, unknown, and zcs zzccss2 22 a aa b bb m :mmn Minimum Minimum Sample Sample Size Size toto Estimate Estimate :: n= n == Minimum Sample Size to Estimate EE E p,p, Point Point Estimate Estimate forfor the the population population proportion proportion ofof p, Point Estimate for the population proportion of x xx n= np n == p successes: successes: p successes: n nn cc- Confidence pp Interval Interval forfor Population Population Proportion Proportion (when c-Confidence p (when Confidence Interval for Population Proportion (when np nnp n+ n -E n +E, npnp Ú ÚÚ 5 and 55 and nqnq Ú ÚÚ 52:52: p 6 66 p p6 p where where np nq 52: p -E E p 66 p + E, E, and where x -m - mm Value Value - -Mean Value - Mean Mean x x= == z z=z == s ss Standard Standard deviation deviation Standard deviation Central Limit Theorem population 3030 Central Limit Theorem (n((nÚ oror population is isis n ÚÚ 30 Central Limit Theorem or population normally distributed): normally distributed): normally distributed): mxmm= m = mm xx = s 22 s s Variance the Sampling Distribution: Variance ofof the Sampling Distribution: = Variance of the Sampling Distribution: s2xss= xx = n nn x xx -m - mm forfor a Mean aa Mean unknown, t- Test t-t-Test m :mmt:: = ,, for ss tt == , for Test for Mean for unknown, s unknown, s> s> 1n s>1n 1n Test forfor a Proportion aa Proportion (when :52:: z- zpp npnp Ú ÚÚ 5 and 55 and nqnq Ú ÚÚ 5252 Test for Proportion (when and z-Test p (when np nq np np n -p mpm nmpnpn p np np n -p pp z z= = == z == sps 1pq>n 1pq>n nspnpn 1pq>n 2 22 2 22 1n1n - -12s 1n1n - -12s 1n - 12s 12s 1n - 12s 12s 6 6 s s 6 6 6 s 6 CC CC C xR2xxR2R2 C xL2xxL2L2 and and d.f. = =n d.f. and d.f. = nn -1- 11 n1p nnn 2p n2p n222-2 -1p 1p p p - 1p -2p2p2222 x x1xx+ +2xx22 11p 11 11p11 11 + z z=z == ,, where p p= , where where p == n1nn+ n +2nn22 11 + 1 11 1 11 p qppaqa q a + ++ b bb BB B n1nn11 n2nn22 and and q q= and q =1= 1-1 -p. - p. p. 22 Chi-Square Chi-Square Test Test forfor a Variance aaVariance or Standard Standard Deviation Deviation s2ssor s:s: Chi-Square Test for Variance or Standard Deviation s: 2 22 1n1n - -12s 1n - 12s 12s 1d.f. 1d.f. = =n x == 1d.f. = nn -12 - 12 12 2 22 s ss ng xy - -1-gx21gy2 nngxy gxy 1g x21g y2 1 gx21gy2 CC HAPTER 888 CHAPTER HAPTER z-Test zTwo-Sample Two-Sample forfor the the Difference Difference Between Between Means Means z-Test Two-Sample Test for the Difference Between Means (Independent (Independent samples; samples; oror normally normally n1nnand n2nnÚ 3030 (Independent samples; and or normally Ú 30 11 and 22 Ú distributed distributed populations): populations): distributed populations): r r=r == 2 22 2 22 22 2 22 2n gxgx 2ng x- -1-gx2 11gx2 2ng 2ng y2yy-1-gy2 11gy2 2n gx2 2ng gy2 forfor the the Correlation Correlation Coefficient: Coefficient: t- Test t-t-Test Test for the Correlation Coefficient: t = tt == r rr -r2 r22 1 1-1 - r BB n nB n -2- 22 (d.f. (d.f. = =n (d.f. = nn -2-)22)) Equation Equation ofof a Regression aa Regression Line: Line: yn yn= + +b, Equation of Regression Line: yn =mx = mx mx + b, b, ng ng xy- -1-gx21gy2 1g x21g y2 nxy gxy 1 gx21gy2 where where and and mm = == where and m 2 22 2 22 n gx ng x- -1-gx2 11gx2 n gx gx2 gg ygyy gg xgxx b b=b =y = == - -m = yy -mx - mx mx -m m n nn n nn Two-Sample Two-Sample forfor the the Difference Difference Between Between Means Means t-Test t-t-Test Two-Sample Test for the Difference Between Means (Independent (Independent samples samples from from normally normally distributed distributed (Independent samples from normally distributed populations, populations, ): n1nnor n2nn6 3030 populations, or 6 30):): 11 or 22 6 Coefficient Coefficient ofof Determination: Determination: Coefficient of Determination: 2 22 n1y g y2 g1y - y2 y2 Explained Explained variation variation Explained variation g1y i nnii 22 r2 rr= == = == 2 Total Total variation variation Total variation gg 1y -ii -y2 g1y - y2 y222 i1y 1x1x -11 -x 1m m -2x2x22-22 - 1m 1m - 2m2m2222 11x 1 11 t =tt == sxs1s-xxx11-2-xx22 If If population variances variances are are equal, equal, d.f. = =n n 2- 22 d.f. If population population variances are equal, d.f. =1nn+ +2nn11 + 22 n i2ynyn2i2i222 gg 1y -ii -y g1y i1y and and and Standard Standard Error Error ofof Estimate: Estimate: se ss= = Standard Error of Estimate: = ee CC C n n-n -2- 22 2 22 2 22 1n1n 12s 1n 12s - 12s 12s + 1n - 12s 12s 1 + 11 + 2 22 11n11 21n22 1 1 1 1 1 1 # # # + ++ . .. sxs1s-xxx11-2-xx=22 == Interval Interval forfor c- Prediction cy: y: 6 66 y y6 yn yn+ n1nn11 n2nn22 Prediction Interval for c-Prediction y:yn ynyn -E -E E y 66 yn +E, + E, E, CC n1nn+ n 2- 22 BB C +2nnB 11 + 22 where where where If If population variances variances are are not not equal, equal, d.f.d.f. is is the If population population variances are not equal, d.f. is the the 2 22 n1x n1x x2 n1x - x2 x2 0 00 1 11 s21 ss2121 s22 ss2222 E = t 1d.f. E = t s s 1 1 + + + + 1d.f. = =n E = t s 1 + + 1d.f. = nn -22 - 22 22 c e c e c e 2 2 2 22 smaller smaller ofof n1nn1- or 11 or n2nn1- and 11 and sxs1s-xxx11-2-xx=22 == + ++ . .. 2 smaller of or and 11 22 CC C n nn ng ng x xx- -1-gx2 11gx2 ng gx2 n CC C1nn1 n2nn2 1 2 CC HAPTER 10 10 CHAPTER HAPTER 10 Test Test Statistic Statistic forfor the the Kruskal-Wallis Kruskal-Wallis Test: Test: Test Statistic for the Kruskal-Wallis Test: Given Given three three oror more more independent independent samples, samples, the the test test Given three or more independent samples, the test statistic statistic forfor the the Kruskal-Wallis Kruskal-Wallis test test is isis statistic for the Kruskal-Wallis test 2 22 1O1O - -E2 1O - E2 E2 gg g Chi-Square: xx == EE E 22 Chi-Square: Chi-Square:x 2 = R21R R2121 R22R R2222 Á R2kR R2k2k 1212 12 HH = == a aa + ++ + +Á + ++ b bb H + Á nknnkk N1N + +12 N1N N1N + 12 12n1nn11 n2nn22 Goodness-of-Fit Goodness-of-Fit Test: Test: d.f. = =k d.f. Goodness-of-Fit Test: d.f. = kk -1- 11 Test Test ofof Independence: Independence: Test of Independence: - -31N + +12. = =k - 31N 31N + 12. 121d.f. . 1d.f. 1d.f. = kk -12 - 12 12 d.f. = =1no. ofof rows - -121no. ofof columns - -12 d.f. rows columns d.f. = 1no. 1no. of rows - 121no. 121no. of columns - 12 12 Spearman Spearman Rank Rank Correlation Correlation Coefficient: Coefficient: Spearman Rank Correlation Coefficient: 22 2 22 and and s21 ssÚ s2,ss2d.f. = n 1, d.f. = n 1- 11 d.f. and Ú d.f. =1nn- 1, 1, d.f. =2nn11 Ú N N 11 DD 22 2,, d.f. N= D= Test Test Statistic Statistic forfor the the Runs Runs Test: Test: Test Statistic for the Runs Test: Two-Sample Two-Sample Test forfor Variances: Variances: F-FFF = Two-Sample Test for Variances: F-Test F 2 22 6 gd 66gd gd rs rr=ss =1= 1-1 -- 2 22 n1n n1n n1n- -12 - 12 12 s21 ss2121 ,, where ==2 , 2where where s2 ss222 gn gn x BxxBB gn x-ii -iAx iiAiAx MS MS SSSS MS SS B BB B BB FF = == , where ,,where MS MS =B == = == where F MS B B MS MS k kk kk - -1 11 MS k -1- 11 WW W One-Way One-Way Analysis Analysis ofof Variance Variance Test: Test: One-Way Analysis of Variance Test: 2 22 CC HAPTER 999 CHAPTER HAPTER Correlation Correlation Coefficient: Coefficient: Correlation Coefficient: 22 x 2x = Interval Interval forfor Population Population Variance Variance c- Confidence cs2s:s22:: Confidence Interval for Population Variance c-Confidence 2 22 d dgg 1d - -d2d2 d -m - dmmdd g1d 1d d2 gg dgdd tt == t = , where ,, where d d= where d == , s,,dssd= = d= n nn sd>ssd1n AA 1n A n nn- --1 11 d>>1n Two-Sample Two-Sample forfor the the Difference Difference Between Between z-Test zTwo-Sample Test for the Difference Between z-Test Proportions Proportions ( n(1(npn1,1p, must bebe atat least least 5):5): n2nqn22q Proportions and must be at least 5): pn,1nqn1,1qqn,,2npn2,2p, pand , and q must n n6 3030 population population is is normal oror nearly nearly normal, normal, and and .30.. n 66 population is normal normal or nearly normal, and 1d.f. = =n 1d.f. 1d.f. = nn -12 - 12 12 s21ss2121 s22ss2222 where where sxs1s-xxx11-2-xx= + ++ = where = 22 n1nn11 n2nn22 CC C zc zzc2 22 np n qnp naqnqnaab cbb Minimum Minimum Sample Sample Size Size toto Estimate Estimate :: n= p :ppn Minimum Sample Size to Estimate n =p = EE E forfor the the Difference Difference Between Between Means Means (Dependent (Dependent t-t-Test t- Test Test for the Difference Between Means (Dependent samples): samples): samples): x xx -m - mm z- zm :mmz:: z=z == ,, for ss Test forfor a Mean aa Mean known with with a aa , for z-Test s known Test for Mean for known with 1n s>s> 1n s> 1n normal normal population, population, oror forfor n nÚ 3030 normal population, or for n ÚÚ 30 1x1x x m -2x2x222-2 -1m - 1m 1m - 2m2m2222 11x11 1 11 z z=z == , ,, sxs1s-xxx11-2-xx22 Interval Interval forfor Population Population Standard Standard cc- Confidence Confidence Interval for Population Standard c-Confidence Deviation Deviation s:s: Deviation s: Transforming Transforming a zaa zScore toto anan Value: x-xx x= + +zs Transforming Score to an Value: z-Score x-Value: x =m = mm + zs zs CC HAPTER 777 CHAPTER HAPTER n qnp np nqnqn p EE = =z=c zzcc E n n BB B n 2 22 2 22 1n1n - -12s 1n1n - -12s 1n - 12s 12s 1n - 12s 12s 2 22 6 6 s s 6 6 6 s 6 xR2xxR2R2 xL2xxL2L2 Standard Standard Score, Score, oror Score: z- zz-Score: Standard Score, or Score: Mean the Sampling Distribution: Mean ofof the Sampling Distribution: Mean of the Sampling Distribution: Standard Deviation the Sampling Standard Deviation ofof the Sampling Standard Deviation of the Sampling Distribution (Standard Error): Distribution (Standard Error): Distribution (Standard Error): 2 22 gg 1n -ii -12s g1n - 12s 12s SSSS SS i ii i1n WW W and and = == MS = == MS and MS WW W NN - -k NN - -k N - kk N - kk ( d.f. ((d.f. d.f. = k d.f. = N - -k d.f. = kk -1, - 1, 1, d.f. =N N -)kk)) N N DD N= D= CC HAPTER 11 11 CHAPTER HAPTER 11 Test Test Statistic Statistic forfor Sign Sign Test: Test: Test Statistic for Sign Test: When When the the test test statistic statistic is is the smaller smaller number number n n… 25,25, When the test statistic is the the smaller number n …… 25, ofof + +or - -signs. of or signs. + or - signs. 1x1x + +0.52 - -0.5n 1x + 0.52 0.52 - 0.5n 0.5n , where When When isx is the ,, where n n7 25,25, z z=z == xx When where is the the n 77 25, 2n 2n 2n 2 22 smaller smaller number number ofof and and isn is the total total + +or - -signs nn smaller number of or signs and is the the total + or - signs number number ofof + +and - -signs. number of and signs. + and - signs. Test Test Statistic Statistic forfor Wilcoxon Wilcoxon Rank Rank Sum Sum Test: Test: Test Statistic for Wilcoxon Rank Sum Test: RR - -m R - RmmRR sum sum ofof the the ranks ranks forfor the the z z=z == , where ,, where RR = == where sum of the ranks for the R sRssRR n1n1n n 12 n111n +2nn+ + 12 12 11n+ 11 + 22 + smaller smaller sample, sample, mRmmR=R == , ,, smaller sample, 2 22 n1nn121n1n n 12 n221n +2nn+ + 12 12 11n+ 11 + 22 + sRssR=R == ,, and n1nn… n2nn22 , and and … 11 … 12 BB 12 B 12 When When and and the the test test statistic statistic is is n1nn… 2020 n2nn… 20,20, G, When and the test statistic is G, … 20 … 20, G, 11 … 22 … the the number number ofof runs. runs. the number of runs. When When oror the the test test statistic statistic is isis n1nn7 2020 n2nn7 20,20, When or the test statistic 7 20 7 20, 11 7 22 7 GG - -m G - GmmGG ofof runs, runs, z z=z == , where ,, where GG = =number where number of runs, G = number sGssGG n121nn22 2n2n 12n and mGmmG= + +1, and = + 1, 1, and G= n1nn+ n +2nn22 11 + 2n2n n121n12n n n n2212n 12n -1nn-2n2n2222 12n 1n121nn22 11 . .. sGssG= = G= 2 2 2 BB B 1n1n n 22 1n n 12 +2n2n221n +2nn- 12 12 11n+ 11 + 11n+ 11 + 22 - 0321693620_INS_p1-4a.qxp 0321693620_INS_p1-4a.qxp 12/1/10 11:09 AMAM Page 1 11 0321693620_INS_p1-4a.qxp12/1/10 12/1/1011:09 11:09 AMPage Page Key Key Formulas KeyFormulas Formulas Key Key Formulas KeyFormulas Formulas Key Key Formulas KeyFormulas Formulas Key Key Formulas KeyFormulas Formulas From From Larson/Farber Larson/Farber Elementary Elementary Statistics: Statistics: Picturing Picturing thethe World, World, Fifth Fifth Edition Edition From Larson/Farber Elementary Statistics: Picturing the World, Fifth Edition ©© 2012 Prentice Prentice Hall Hall © 2012 2012 Prentice Hall From From Larson/Farber Larson/Farber Elementary Elementary Statistics: Statistics: Picturing Picturing thethe World, World, Fifth Fifth Edition Edition From Larson/Farber Elementary Statistics: Picturing the World, Fifth Edition ©© 2012 Prentice Prentice Hall Hall © 2012 2012 Prentice Hall From From Larson/Farber Larson/Farber Elementary Elementary Statistics: Statistics: Picturing Picturing thethe World, World, Fifth Fifth Edition Edition From Larson/Farber Elementary Statistics: Picturing the World, Fifth Edition ©© 2012 Prentice Prentice Hall Hall © 2012 2012 Prentice Hall From From Larson/Farber Larson/Farber Elementary Elementary Statistics: Statistics: Picturing Picturing thethe World, World, Fifth Fifth Edition Edition From Larson/Farber Elementary Statistics: Picturing the World, Fifth Edition ©© 2012 Prentice Prentice Hall Hall © 2012 2012 Prentice Hall CC HAPTER 222 CHAPTER HAPTER Class Class Width Width Class Width= == Sample Sample Standard Standard Deviation Deviation ofof a Frequency aa Frequency Distribution: Distribution: Sample Standard Deviation of Frequency Distribution: Range Range ofof data data Range of data Number Number ofof classes classes Number of classes s s=s == CC C n n-n -1- 11 1 round 11round 2 22 upup toto next next convenient convenient number number round up to next convenient number 1Lower 1Lower class class limit2 limit2 + +1Upper class class limit2 limit2 1Lower class limit2 + 1Upper 1Upper class limit2 Midpoint Midpoint = == Midpoint 2 22 Class Class frequency frequency Class frequency f ff Relative Relative Frequency Frequency = == = == Relative Frequency Sample Sample size size Sample size n nn gg xgxx m mm = == Population Population Mean: Mean: Population Mean: NN N x -m - mm Value Value - -Mean Value - Mean Mean x xStandard Standard Score: Score: z z=z == = == Standard Score: s ss Standard Standard deviation deviation Standard deviation CC HAPTER 333 CHAPTER HAPTER Classical Classical (or(or Theoretical) Theoretical) Probability: Probability: Classical (or Theoretical) Probability: Number Number ofof outcomes outcomes inin event event EE Number of outcomes in event E P1E2 P1E2 = == P1E2 Total Total number number ofof outcomes outcomes Total number of outcomes inin sample sample space space in sample space gg xgxx x x= Sample Sample Mean: Mean: x == Sample Mean: n nn # w2# #w2 g g1x 1x g 1x w2 x x= Weighted Weighted Mean: Mean: Weighted Mean: x == gg wgww Empirical Empirical (or(or Statistical) Statistical) Probability: Probability: Empirical (or Statistical) Probability: Frequency Frequency ofof event event EE Frequency of event E f ff P1E2 P1E2 = == = == P1E2 n nn Total Total frequency frequency Total frequency # f2# #f2 g1x g1x g1x f2 Mean Mean ofof a Frequency aa Frequency Distribution: Distribution: x x= x == Mean of Frequency Distribution: n nn = =1Maximum entry2 entry2 - -1Minimum entry2 entry2 Range Range = 1Maximum 1Maximum entry2 - 1Minimum 1Minimum entry2 Range 2 22 gg 1x - -m2 g1x 1x - m2 m2 Population Population Variance: Variance: Population Variance: s == NN N 2 22 gg 1x - -m2 g1x 1x - m2 m2 2 22 gg 1x - -x2 g1x 1x - x2 x2 Sample Sample Variance: Variance: Sample Variance: s == n nn -1- 11 2 22 s s=s == 2s 2s = == Sample Sample Standard Standard Deviation: Deviation: 2s Sample Standard Deviation: g1x g1x - - x2 g1x 2 22 -x2x2 CC C n n-n -1- 11 Empirical Empirical Rule Rule 68-95-99.7 68-95-99.7 Rule) Rule) For For data data with with a aa Empirical Rule(or(or (or 68-95-99.7 Rule) For data with (symmetric) (symmetric) bell-shaped bell-shaped distribution: distribution: (symmetric) bell-shaped distribution: 1. 1. About About 68% 68% ofof thethe data data lies lies between between and m mm - -s m mm + +s. 1. About 68% of the data lies between and -s s and + s. s. m mm - -2s 2. 2. About About 95% 95% ofof the the data data lies lies between between and and 2. About 95% of the data lies between and - 2s 2s m mm + +2s. + 2s. 2s. m mm - -3s 3. 3. About About 99.7% 99.7% ofof the the data data lies lies between between and and 3. About 99.7% of the data lies between and - 3s 3s m mm + +3s. + 3s. 3s. Chebychev’s Chebychev’s Theorem Theorem The portion portion ofof any any data data setset lying lying Chebychev’s TheoremThe The portion of any data set lying within within standard deviations deviations ofof the the mean mean is is at kk 1k1k 7 77 1212 within standard deviations of the mean is at at k standard 1k 12 1 11 least least 1 1-1 -- 2 . 22.. least k kk 22 2 22 s2ss= == gg 1x - -m2 P1x2 P1x2 g1x 1x - m2 m2 P1x2 Standard Standard Deviation Deviation ofof a Discrete aa Discrete Random Random Variable: Variable: Standard Deviation of Discrete Random Variable: 2 22 2 22 s ss = == 2s 2s g1x P1x2 P1x2 = == 22 g1x - -m2 2s 2g 1x - m2 m2 P1x2 Expected Expected Value: Value: E1x2 E1x2 = =m = == gg xP1x2 Expected Value: E1x2 = mm gxP1x2 xP1x2 Binomial Binomial Probability Probability ofof successes inin trials: xx nn Binomial Probability of successes in trials: x successes n trials: n!n! n! xx P1x2 pxxxpqpxnxq-qnxn--= == pxpqpxnxq-qnxn--xx P1x2 = =n=CnnxCC P1x2 1n1n - -x2!x! 1n - x2!x! x2!x! Population Population Parameters Parameters ofof a Binomial aa Binomial Distribution: Distribution: Population Parameters of Binomial Distribution: 22 Variance: Variance: s2ss= =npq Variance: = npq npq Standard Standard Deviation: Deviation: 1npq s ss = == 1npq Standard Deviation: 1npq P1A P1A oror B2B2 = =P1A2 + +P1B2 - -P1A and and B2B2 P1A or B2 = P1A2 P1A2 + P1B2 P1B2 - P1A P1A and B2 22 s2 s= Variance Variance ofof a Discrete aa Discrete Random Random Variable: Variable: Variance of Discrete Random Variable: AA B:B: Probability Probability ofof occurrence occurrence ofof both both events events and and A B: Probability of occurrence of both events and Probability Probability ofof occurrence occurrence ofof either either oror or both: both: AA BB Probability of occurrence of either or or both: A B or CC C NN N Mean Mean ofof a Discrete aa Discrete Random Random Variable: Variable: m mm = == g gxP1x2 xP1x2 gxP1x2 Mean of Discrete Random Variable: Mean: Mean: m mm = =np Mean: = np np # P1B2 # #P1B2 P1A P1A and and B2B2 = =P1A2 A BB if if and and are P1B2 if A and are P1A and B2 = P1A2 P1A2 A B are independent independent independent Population Population Standard Standard Deviation: Deviation: Population Standard Deviation: CC HAPTER 444 CHAPTER HAPTER P1E¿2 P1E¿2 = =1= 1-1 -P1E2 Probability Probability ofof a Complement: aa Complement: P1E¿2 - P1E2 P1E2 Probability of Complement: # P1B # #P1B P1A P1A and and B2B2 = =P1A2 P1B P1A and B2 = P1A2 P1A2 ƒ Aƒ2A ƒ A22 22 s2s= 2 22 s ss = == 2s 2s = == 2s g1x g1x - - x2 f g1x 2 22 -x2x2 ff if if and and are P1A P1A oror B2B2 = =P1A2 + +P1B2 A BB if A and are P1A or B2 = P1A2 P1A2 + P1B2 P1B2 A B are mutually mutually exclusive exclusive mutually exclusive Permutations Permutations ofof objects taken taken a time: aa time: nn r at rr at Permutations of objects taken at time: n objects P=rr nPnnrP n!n! n! == , where ,,where r r…r …… n nn where 1n1n 1n r2! - r2! r2! Distinguishable Distinguishable Permutations: Permutations: , ,, n1nnalike, n 2nnalike, ÁÁ Distinguishable Permutations: alike, alike, Á 11 alike, 22 alike, alike: nknnalike: alike: kk n!n! n! , , Á Á !nkk!! , n1n!n1#1!n!#2#n!n2#2!n!#2#n!n2Á 2!! nkn Á Á where where n1nn+ n n + +n where +2nn+ +3nn+ +Á +knnk=k =n = nn 11 + 22 + 33 + Combination Combination ofof objects taken taken a time: aa time: nn r at rr at Combination of objects taken at time: n objects n!n! n! C=rr == nCnn rC 1n1n - -r2!r! 1n - r2!r! r2!r! Geometric Geometric Distribution: Distribution: The The probability probability that that the the first first Geometric Distribution: The probability that the first x -x1x--11 success success will will occur occur onon trial trial number number isx is xx , ,, P1x2 = =p1q2 success will occur on trial number is P1x2 P1x2 = p1q2 p1q2 where where q q= where q =1= 1-1 -p. - p. p. Poisson Poisson Distribution: Distribution: The The probability probability ofof exactly exactly x xx Poisson Distribution: The probability of exactly xe-m mxmemx-m e-m occurrences occurrences inin anan interval interval is is , where ,, where P1x2 = == occurrences in an interval is P1x2 where P1x2 x!x! x! and and is the mean mean number number ofof occurences occurences e eLe L2.71828 mm and is the the mean number of occurences L 2.71828 2.71828 m is per per interval interval unit. unit. per interval unit. CC HAPTER 555 CHAPTER HAPTER Standard Deviation the Sampling Standard Deviation ofof the Sampling Standard Deviation of the Sampling Distribution (Standard Error): Distribution (Standard Error): Distribution (Standard Error): s ss sxss= = xx = 1n 1n 1n x -m - xmmxx x xx -m - mm Value - -Mean Value Value - Mean Mean x xz-Score = == z-Score = == = == z-Score sxssxx Standard Error Standard Error Standard Error 1n s>s> 1n s> 1n CC HAPTER 666 CHAPTER HAPTER Interval Interval forfor :: xc- Confidence cm :mmx 6 66 m mm 6 66 x x+ Confidence Interval for c-Confidence x -E -E E x +E, + E, E, s ss EE = =z=c zzcc if if s where where is known and and the the population population is isis E s is where if s is known known and the population 1n 1n 1n s ss n nÚ 30,30, EE = =t=c ttcc if if normally normally distributed distributed oror oror the n ÚÚ 30, E normally distributed or or if the the 1n 1n 1n population population is is normally oror approximately approximately normally normally population is normally normally or approximately normally ss n n6 3030 distributed, distributed, is unknown, and and s is n 66 30 distributed, is unknown, unknown, and zcs zzccss2 22 a aa b bb m :mmn Minimum Minimum Sample Sample Size Size toto Estimate Estimate :: n= n == Minimum Sample Size to Estimate EE E p,p, Point Point Estimate Estimate forfor the the population population proportion proportion ofof p, Point Estimate for the population proportion of x xx n= np n == p successes: successes: p successes: n nn cc- Confidence pp Interval Interval forfor Population Population Proportion Proportion (when c-Confidence p (when Confidence Interval for Population Proportion (when np nnp n+ n -E n +E, npnp Ú ÚÚ 5 and 55 and nqnq Ú ÚÚ 52:52: p 6 66 p p6 p where where np nq 52: p -E E p 66 p + E, E, and where x -m - mm Value Value - -Mean Value - Mean Mean x x= == z z=z == s ss Standard Standard deviation deviation Standard deviation Central Limit Theorem population 3030 Central Limit Theorem (n((nÚ oror population is isis n ÚÚ 30 Central Limit Theorem or population normally distributed): normally distributed): normally distributed): Mean the Sampling Distribution: Mean ofof the Sampling Distribution: Mean of the Sampling Distribution: Test forfor a Proportion aa Proportion (when :52:: z- zpp npnp Ú ÚÚ 5 and 55 and nqnq Ú ÚÚ 5252 Test for Proportion (when and z-Test p (when np nq np np n -p mpm nmpnpn p np np n -p pp z z= = == z == sps 1pq>n 1pq>n nspnpn 1pq>n 2 22 2 22 1n1n - -12s 1n1n - -12s 1n - 12s 12s 1n - 12s 12s 6 6 s s 6 6 6 s 6 CC CC C xR2xxR2R2 C xL2xxL2L2 mxmm= m = mm xx = s 22 s s Variance the Sampling Distribution: Variance ofof the Sampling Distribution: = Variance of the Sampling Distribution: s2xss= xx = n nn 2 22 1n1n - -12s 1n - 12s 12s 1d.f. 1d.f. = =n x == 1d.f. = nn -12 - 12 12 2 22 s ss n1p nnn 2p n2p n222-2 -1p 1p p p - 1p -2p2p2222 x x1xx+ +2xx22 11p 11 11p11 11 + z z=z == ,, where p p= , where where p == n1nn+ n +2nn22 11 + 1 11 1 11 p qppaqa q a + ++ b bb BB B n1nn11 n2nn22 ng xy - -1-gx21gy2 nngxy gxy 1g x21g y2 1 gx21gy2 CC HAPTER 888 CHAPTER HAPTER z-Test zTwo-Sample Two-Sample forfor the the Difference Difference Between Between Means Means z-Test Two-Sample Test for the Difference Between Means (Independent (Independent samples; samples; oror normally normally n1nnand n2nnÚ 3030 (Independent samples; and or normally Ú 30 11 and 22 Ú distributed distributed populations): populations): distributed populations): r r=r == 2 22 2 22 22 2 22 2n gxgx 2ng x- -1-gx2 11gx2 2ng 2ng y2yy-1-gy2 11gy2 2n gx2 2ng gy2 forfor the the Correlation Correlation Coefficient: Coefficient: t- Test t-t-Test Test for the Correlation Coefficient: t = tt == r rr -r2 r22 1 1-1 - r BB n nB n -2- 22 (d.f. (d.f. = =n (d.f. = nn -2-)22)) Equation Equation ofof a Regression aa Regression Line: Line: yn yn= + +b, Equation of Regression Line: yn =mx = mx mx + b, b, ng ng xy- -1-gx21gy2 1g x21g y2 nxy gxy 1 gx21gy2 where where and and mm = == where and m 2 22 2 22 n gx ng x- -1-gx2 11gx2 n gx gx2 gg ygyy gg xgxx b b=b =y = == - -m = yy -mx - mx mx -m m n nn n nn Two-Sample Two-Sample forfor the the Difference Difference Between Between Means Means t-Test t-t-Test Two-Sample Test for the Difference Between Means (Independent (Independent samples samples from from normally normally distributed distributed (Independent samples from normally distributed populations, populations, ): n1nnor n2nn6 3030 populations, or 6 30):): 11 or 22 6 Coefficient Coefficient ofof Determination: Determination: Coefficient of Determination: 2 22 n1y g y2 g1y - y2 y2 Explained Explained variation variation Explained variation g1y i nnii 22 r2 rr= == = == 2 Total Total variation variation Total variation gg 1y -ii -y2 g1y - y2 y222 i1y 1x1x -11 -x 1m m -2x2x22-22 - 1m 1m - 2m2m2222 11x 1 11 t =tt == sxs1s-xxx11-2-xx22 If If population variances variances are are equal, equal, d.f. = =n n 2- 22 d.f. If population population variances are equal, d.f. =1nn+ +2nn11 + 22 n i2ynyn2i2i222 gg 1y -ii -y g1y i1y and and and Standard Standard Error Error ofof Estimate: Estimate: se ss= = Standard Error of Estimate: = ee CC C n n-n -2- 22 2 22 2 22 1n1n 12s 1n 12s - 12s 12s + 1n - 12s 12s 1 + 11 + 2 22 11n11 21n22 1 1 1 1 1 1 # # # + ++ . .. sxs1s-xxx11-2-xx=22 == Interval Interval forfor c- Prediction cy: y: 6 66 y y6 yn yn+ n1nn11 n2nn22 Prediction Interval for c-Prediction y:yn ynyn -E -E E y 66 yn +E, + E, E, CC n1nn+ n 2- 22 BB C +2nnB 11 + 22 where where where If If population variances variances are are not not equal, equal, d.f.d.f. is is the If population population variances are not equal, d.f. is the the 2 22 n1x n1x x2 n1x - x2 x2 0 00 1 11 s21 ss2121 s22 ss2222 E = t 1d.f. E = t s s 1 1 + + + + 1d.f. = =n E = t s 1 + + 1d.f. = nn -22 - 22 22 c e c e c e 2 2 2 22 smaller smaller ofof n1nn1- or 11 or n2nn1- and 11 and sxs1s-xxx11-2-xx=22 == + ++ . .. 2 smaller of or and 11 22 CC C n nn ng ng x xx- -1-gx2 11gx2 ng gx2 n CC C1nn1 n2nn2 2 CC HAPTER 10 10 CHAPTER HAPTER 10 Test Test Statistic Statistic forfor the the Kruskal-Wallis Kruskal-Wallis Test: Test: Test Statistic for the Kruskal-Wallis Test: Given Given three three oror more more independent independent samples, samples, the the test test Given three or more independent samples, the test statistic statistic forfor the the Kruskal-Wallis Kruskal-Wallis test test is isis statistic for the Kruskal-Wallis test 2 22 1O1O - -E2 1O - E2 E2 gg g Chi-Square: xx == EE E 22 Chi-Square: Chi-Square:x 2 = R21R R2121 R22R R2222 Á R2kR R2k2k 1212 12 HH = == a aa + ++ + +Á + ++ b bb H + Á nknnkk N1N + +12 N1N N1N + 12 12n1nn11 n2nn22 Goodness-of-Fit Goodness-of-Fit Test: Test: d.f. = =k d.f. Goodness-of-Fit Test: d.f. = kk -1- 11 Test Test ofof Independence: Independence: Test of Independence: - -31N + +12. = =k - 31N 31N + 12. 121d.f. . 1d.f. 1d.f. = kk -12 - 12 12 d.f. = =1no. ofof rows - -121no. ofof columns - -12 d.f. rows columns d.f. = 1no. 1no. of rows - 121no. 121no. of columns - 12 12 Spearman Spearman Rank Rank Correlation Correlation Coefficient: Coefficient: Spearman Rank Correlation Coefficient: 22 2 22 and and s21 ssÚ s2,ss2d.f. = n 1, d.f. = n 1- 11 d.f. and Ú d.f. =1nn- 1, 1, d.f. =2nn11 Ú N N 11 DD 22 2,, d.f. N= D= Test Test Statistic Statistic forfor the the Runs Runs Test: Test: Test Statistic for the Runs Test: Two-Sample Two-Sample Test forfor Variances: Variances: F-FFF = Two-Sample Test for Variances: F-Test F 2 22 6 gd 66gd gd rs rr=ss =1= 1-1 -- 2 22 n1n n1n n1n- -12 - 12 12 s21 ss2121 ,, where ==2 , 2where where s2 ss222 gn gn x BxxBB gn x-ii -iAx iiAiAx MS MS SSSS MS SS B BB B BB FF = == , where ,,where MS MS =B == = == where F MS B B MS MS k kk kk - -1 11 MS k -1- 11 WW W One-Way One-Way Analysis Analysis ofof Variance Variance Test: Test: One-Way Analysis of Variance Test: 2 22 CC HAPTER 999 CHAPTER HAPTER Correlation Correlation Coefficient: Coefficient: Correlation Coefficient: 22 x 2x = 1 Copyright © 2012 Pearson Education, Inc. and and d.f. = =n d.f. and d.f. = nn -1- 11 and and q q= and q =1= 1-1 -p. - p. p. 22 Chi-Square Chi-Square Test Test forfor a Variance aaVariance or Standard Standard Deviation Deviation s2ssor s:s: Chi-Square Test for Variance or Standard Deviation s: Interval Interval forfor Population Population Variance Variance c- Confidence cs2s:s22:: Confidence Interval for Population Variance c-Confidence 2 22 d dgg 1d - -d2d2 d -m - dmmdd g1d 1d d2 gg dgdd tt == t = , where ,, where d d= where d == , s,,dssd= = d= n nn sd>ssd1n AA 1n A n nn- --1 11 d>>1n Two-Sample Two-Sample forfor the the Difference Difference Between Between z-Test zTwo-Sample Test for the Difference Between z-Test Proportions Proportions ( n(1(npn1,1p, must bebe atat least least 5):5): n2nqn22q Proportions and must be at least 5): pn,1nqn1,1qqn,,2npn2,2p, pand , and q must n n6 3030 population population is is normal oror nearly nearly normal, normal, and and .30.. n 66 population is normal normal or nearly normal, and 1d.f. = =n 1d.f. 1d.f. = nn -12 - 12 12 s21ss2121 s22ss2222 where where sxs1s-xxx11-2-xx= + ++ = where = 22 n1nn11 n2nn22 CC C Interval Interval forfor Population Population Standard Standard cc- Confidence Confidence Interval for Population Standard c-Confidence Deviation Deviation s:s: Deviation s: Transforming Transforming a zaa zScore toto anan Value: x-xx x= + +zs Transforming Score to an Value: z-Score x-Value: x =m = mm + zs zs x xx -m - mm forfor a Mean aa Mean unknown, t- Test t-t-Test m :mmt:: = ,, for ss tt == , for Test for Mean for unknown, s unknown, s> s> 1n s>1n 1n 1x1x x m -2x2x222-2 -1m - 1m 1m - 2m2m2222 11x11 1 11 z z=z == , ,, sxs1s-xxx11-2-xx22 zc zzc2 22 np n qnp naqnqnaab cbb Minimum Minimum Sample Sample Size Size toto Estimate Estimate :: n= p :ppn Minimum Sample Size to Estimate n =p = EE E forfor the the Difference Difference Between Between Means Means (Dependent (Dependent t-t-Test t- Test Test for the Difference Between Means (Dependent samples): samples): samples): x xx -m - mm z- zm :mmz:: z=z == ,, for ss Test forfor a Mean aa Mean known with with a aa , for z-Test s known Test for Mean for known with 1n s>s> 1n s> 1n normal normal population, population, oror forfor n nÚ 3030 normal population, or for n ÚÚ 30 n qnp np nqnqn p EE = =z=c zzcc E n n BB B n 2 22 2 22 1n1n - -12s 1n1n - -12s 1n - 12s 12s 1n - 12s 12s 2 22 6 6 s s 6 6 6 s 6 xR2xxR2R2 xL2xxL2L2 Standard Standard Score, Score, oror Score: z- zz-Score: Standard Score, or Score: CC HAPTER 777 CHAPTER HAPTER 2 22 gg 1n -ii -12s g1n - 12s 12s SSSS SS i ii i1n WW W and and = == MS = == MS and MS WW W NN - -k NN - -k N - kk N - kk ( d.f. ((d.f. d.f. = k d.f. = N - -k d.f. = kk -1, - 1, 1, d.f. =N N -)kk)) N N DD N= D= CC HAPTER 11 11 CHAPTER HAPTER 11 Test Test Statistic Statistic forfor Sign Sign Test: Test: Test Statistic for Sign Test: When When the the test test statistic statistic is is the smaller smaller number number n n… 25,25, When the test statistic is the the smaller number n …… 25, ofof + +or - -signs. of or signs. + or - signs. 1x1x + +0.52 - -0.5n 1x + 0.52 0.52 - 0.5n 0.5n , where When When isx is the ,, where n n7 25,25, z z=z == xx When where is the the n 77 25, 2n 2n 2n 2 22 smaller smaller number number ofof and and isn is the total total + +or - -signs nn smaller number of or signs and is the the total + or - signs number number ofof + +and - -signs. number of and signs. + and - signs. Test Test Statistic Statistic forfor Wilcoxon Wilcoxon Rank Rank Sum Sum Test: Test: Test Statistic for Wilcoxon Rank Sum Test: RR - -m R - RmmRR sum sum ofof the the ranks ranks forfor the the z z=z == , where ,, where RR = == where sum of the ranks for the R sRssRR n1n1n n 12 n111n +2nn+ + 12 12 11n+ 11 + 22 + smaller smaller sample, sample, mRmmR=R == , ,, smaller sample, 2 22 n1nn121n1n n 12 n221n +2nn+ + 12 12 11n+ 11 + 22 + sRssR=R == ,, and n1nn… n2nn22 , and and … 11 … 12 BB 12 B 12 When When and and the the test test statistic statistic is is n1nn… 2020 n2nn… 20,20, G, When and the test statistic is G, … 20 … 20, G, 11 … 22 … the the number number ofof runs. runs. the number of runs. When When oror the the test test statistic statistic is isis n1nn7 2020 n2nn7 20,20, When or the test statistic 7 20 7 20, 11 7 22 7 GG - -m G - GmmGG ofof runs, runs, z z=z == , where ,, where GG = =number where number of runs, G = number sGssGG n121nn22 2n2n 12n and mGmmG= + +1, and = + 1, 1, and G= n1nn+ n +2nn22 11 + 2n2n n121n12n n n n2212n 12n -1nn-2n2n2222 12n 1n121nn22 11 . .. sGssG= = G= 2 2 2 BB B 1n1n n 22 1n n 12 +2n2n221n +2nn- 12 12 11n+ 11 + 11n+ 11 + 22 - 0321693620_INS_p1-4a.qxp 0321693620_INS_p1-4a.qxp 12/1/10 11:09 AMAM Page 1 11 0321693620_INS_p1-4a.qxp12/1/10 12/1/1011:09 11:09 AMPage Page Key Key Formulas KeyFormulas Formulas Key Key Formulas KeyFormulas Formulas Key Key Formulas KeyFormulas Formulas Key Key Formulas KeyFormulas Formulas From From Larson/Farber Larson/Farber Elementary Elementary Statistics: Statistics: Picturing Picturing thethe World, World, Fifth Fifth Edition Edition From Larson/Farber Elementary Statistics: Picturing the World, Fifth Edition ©© 2012 Prentice Prentice Hall Hall © 2012 2012 Prentice Hall From From Larson/Farber Larson/Farber Elementary Elementary Statistics: Statistics: Picturing Picturing thethe World, World, Fifth Fifth Edition Edition From Larson/Farber Elementary Statistics: Picturing the World, Fifth Edition ©© 2012 Prentice Prentice Hall Hall © 2012 2012 Prentice Hall From From Larson/Farber Larson/Farber Elementary Elementary Statistics: Statistics: Picturing Picturing thethe World, World, Fifth Fifth Edition Edition From Larson/Farber Elementary Statistics: Picturing the World, Fifth Edition ©© 2012 Prentice Prentice Hall Hall © 2012 2012 Prentice Hall From From Larson/Farber Larson/Farber Elementary Elementary Statistics: Statistics: Picturing Picturing thethe World, World, Fifth Fifth Edition Edition From Larson/Farber Elementary Statistics: Picturing the World, Fifth Edition ©© 2012 Prentice Prentice Hall Hall © 2012 2012 Prentice Hall CC HAPTER 222 CHAPTER HAPTER Class Class Width Width Class Width= == Sample Sample Standard Standard Deviation Deviation ofof a Frequency aa Frequency Distribution: Distribution: Sample Standard Deviation of Frequency Distribution: Range Range ofof data data Range of data Number Number ofof classes classes Number of classes s s=s == CC C n n-n -1- 11 1 round 11round 2 22 upup toto next next convenient convenient number number round up to next convenient number 1Lower 1Lower class class limit2 limit2 + +1Upper class class limit2 limit2 1Lower class limit2 + 1Upper 1Upper class limit2 Midpoint Midpoint = == Midpoint 2 22 Class Class frequency frequency Class frequency f ff Relative Relative Frequency Frequency = == = == Relative Frequency Sample Sample size size Sample size n nn gg xgxx m mm = == Population Population Mean: Mean: Population Mean: NN N x -m - mm Value Value - -Mean Value - Mean Mean x xStandard Standard Score: Score: z z=z == = == Standard Score: s ss Standard Standard deviation deviation Standard deviation CC HAPTER 333 CHAPTER HAPTER Classical Classical (or(or Theoretical) Theoretical) Probability: Probability: Classical (or Theoretical) Probability: Number Number ofof outcomes outcomes inin event event EE Number of outcomes in event E P1E2 P1E2 = == P1E2 Total Total number number ofof outcomes outcomes Total number of outcomes inin sample sample space space in sample space gg xgxx x x= Sample Sample Mean: Mean: x == Sample Mean: n nn # w2# #w2 g g1x 1x g 1x w2 x x= Weighted Weighted Mean: Mean: Weighted Mean: x == gg wgww Empirical Empirical (or(or Statistical) Statistical) Probability: Probability: Empirical (or Statistical) Probability: Frequency Frequency ofof event event EE Frequency of event E f ff P1E2 P1E2 = == = == P1E2 n nn Total Total frequency frequency Total frequency # f2# #f2 g1x g1x g1x f2 Mean Mean ofof a Frequency aa Frequency Distribution: Distribution: x x= x == Mean of Frequency Distribution: n nn = =1Maximum entry2 entry2 - -1Minimum entry2 entry2 Range Range = 1Maximum 1Maximum entry2 - 1Minimum 1Minimum entry2 Range 2 22 gg 1x - -m2 g1x 1x - m2 m2 Population Population Variance: Variance: Population Variance: s == NN N 2 22 gg 1x - -m2 g1x 1x - m2 m2 2 22 gg 1x - -x2 g1x 1x - x2 x2 Sample Sample Variance: Variance: Sample Variance: s == n nn -1- 11 2 22 s s=s == 2s 2s = == Sample Sample Standard Standard Deviation: Deviation: 2s Sample Standard Deviation: g1x g1x - - x2 g1x 2 22 -x2x2 CC C n n-n -1- 11 Empirical Empirical Rule Rule 68-95-99.7 68-95-99.7 Rule) Rule) For For data data with with a aa Empirical Rule(or(or (or 68-95-99.7 Rule) For data with (symmetric) (symmetric) bell-shaped bell-shaped distribution: distribution: (symmetric) bell-shaped distribution: 1. 1. About About 68% 68% ofof thethe data data lies lies between between and m mm - -s m mm + +s. 1. About 68% of the data lies between and -s s and + s. s. m mm - -2s 2. 2. About About 95% 95% ofof the the data data lies lies between between and and 2. About 95% of the data lies between and - 2s 2s m mm + +2s. + 2s. 2s. m mm - -3s 3. 3. About About 99.7% 99.7% ofof the the data data lies lies between between and and 3. About 99.7% of the data lies between and - 3s 3s m mm + +3s. + 3s. 3s. Chebychev’s Chebychev’s Theorem Theorem The portion portion ofof any any data data setset lying lying Chebychev’s TheoremThe The portion of any data set lying within within standard deviations deviations ofof the the mean mean is is at kk 1k1k 7 77 1212 within standard deviations of the mean is at at k standard 1k 12 1 11 least least 1 1-1 -- 2 . 22.. least k kk 22 2 22 s2ss= == gg 1x - -m2 P1x2 P1x2 g1x 1x - m2 m2 P1x2 Standard Standard Deviation Deviation ofof a Discrete aa Discrete Random Random Variable: Variable: Standard Deviation of Discrete Random Variable: 2 22 2 22 s ss = == 2s 2s g1x P1x2 P1x2 = == 22 g1x - -m2 2s 2g 1x - m2 m2 P1x2 Expected Expected Value: Value: E1x2 E1x2 = =m = == gg xP1x2 Expected Value: E1x2 = mm gxP1x2 xP1x2 Binomial Binomial Probability Probability ofof successes inin trials: xx nn Binomial Probability of successes in trials: x successes n trials: n!n! n! xx P1x2 pxxxpqpxnxq-qnxn--= == pxpqpxnxq-qnxn--xx P1x2 = =n=CnnxCC P1x2 1n1n - -x2!x! 1n - x2!x! x2!x! Population Population Parameters Parameters ofof a Binomial aa Binomial Distribution: Distribution: Population Parameters of Binomial Distribution: 22 Variance: Variance: s2ss= =npq Variance: = npq npq Standard Standard Deviation: Deviation: 1npq s ss = == 1npq Standard Deviation: 1npq P1A P1A oror B2B2 = =P1A2 + +P1B2 - -P1A and and B2B2 P1A or B2 = P1A2 P1A2 + P1B2 P1B2 - P1A P1A and B2 22 s2 s= Variance Variance ofof a Discrete aa Discrete Random Random Variable: Variable: Variance of Discrete Random Variable: AA B:B: Probability Probability ofof occurrence occurrence ofof both both events events and and A B: Probability of occurrence of both events and Probability Probability ofof occurrence occurrence ofof either either oror or both: both: AA BB Probability of occurrence of either or or both: A B or CC C NN N Mean Mean ofof a Discrete aa Discrete Random Random Variable: Variable: m mm = == g gxP1x2 xP1x2 gxP1x2 Mean of Discrete Random Variable: Mean: Mean: m mm = =np Mean: = np np # P1B2 # #P1B2 P1A P1A and and B2B2 = =P1A2 A BB if if and and are P1B2 if A and are P1A and B2 = P1A2 P1A2 A B are independent independent independent Population Population Standard Standard Deviation: Deviation: Population Standard Deviation: CC HAPTER 444 CHAPTER HAPTER P1E¿2 P1E¿2 = =1= 1-1 -P1E2 Probability Probability ofof a Complement: aa Complement: P1E¿2 - P1E2 P1E2 Probability of Complement: # P1B # #P1B P1A P1A and and B2B2 = =P1A2 P1B P1A and B2 = P1A2 P1A2 ƒ Aƒ2A ƒ A22 22 s2s= 2 22 s ss = == 2s 2s = == 2s g1x g1x - - x2 f g1x 2 22 -x2x2 ff if if and and are P1A P1A oror B2B2 = =P1A2 + +P1B2 A BB if A and are P1A or B2 = P1A2 P1A2 + P1B2 P1B2 A B are mutually mutually exclusive exclusive mutually exclusive Permutations Permutations ofof objects taken taken a time: aa time: nn r at rr at Permutations of objects taken at time: n objects P=rr nPnnrP n!n! n! == , where ,,where r r…r …… n nn where 1n1n 1n r2! - r2! r2! Distinguishable Distinguishable Permutations: Permutations: , ,, n1nnalike, n 2nnalike, ÁÁ Distinguishable Permutations: alike, alike, Á 11 alike, 22 alike, alike: nknnalike: alike: kk n!n! n! , , Á Á !nkk!! , n1n!n1#1!n!#2#n!n2#2!n!#2#n!n2Á 2!! nkn Á Á where where n1nn+ n n + +n where +2nn+ +3nn+ +Á +knnk=k =n = nn 11 + 22 + 33 + Combination Combination ofof objects taken taken a time: aa time: nn r at rr at Combination of objects taken at time: n objects n!n! n! C=rr == nCnn rC 1n1n - -r2!r! 1n - r2!r! r2!r! Geometric Geometric Distribution: Distribution: The The probability probability that that the the first first Geometric Distribution: The probability that the first x -x1x--11 success success will will occur occur onon trial trial number number isx is xx , ,, P1x2 = =p1q2 success will occur on trial number is P1x2 P1x2 = p1q2 p1q2 where where q q= where q =1= 1-1 -p. - p. p. Poisson Poisson Distribution: Distribution: The The probability probability ofof exactly exactly x xx Poisson Distribution: The probability of exactly xe-m mxmemx-m e-m occurrences occurrences inin anan interval interval is is , where ,, where P1x2 = == occurrences in an interval is P1x2 where P1x2 x!x! x! and and is the mean mean number number ofof occurences occurences e eLe L2.71828 mm and is the the mean number of occurences L 2.71828 2.71828 m is per per interval interval unit. unit. per interval unit. CC HAPTER 555 CHAPTER HAPTER s ss sxss= = xx = 1n 1n 1n x -m - xmmxx x xx -m - mm Value - -Mean Value Value - Mean Mean x xz-Score = == z-Score = == = == z-Score sxssxx Standard Error Standard Error Standard Error 1n s>s> 1n s> 1n CC HAPTER 666 CHAPTER HAPTER Interval Interval forfor :: xc- Confidence cm :mmx 6 66 m mm 6 66 x x+ Confidence Interval for c-Confidence x -E -E E x +E, + E, E, s ss EE = =z=c zzcc if if s where where is known and and the the population population is isis E s is where if s is known known and the population 1n 1n 1n s ss n nÚ 30,30, EE = =t=c ttcc if if normally normally distributed distributed oror oror the n ÚÚ 30, E normally distributed or or if the the 1n 1n 1n population population is is normally oror approximately approximately normally normally population is normally normally or approximately normally ss n n6 3030 distributed, distributed, is unknown, and and s is n 66 30 distributed, is unknown, unknown, and zcs zzccss2 22 a aa b bb m :mmn Minimum Minimum Sample Sample Size Size toto Estimate Estimate :: n= n == Minimum Sample Size to Estimate EE E p,p, Point Point Estimate Estimate forfor the the population population proportion proportion ofof p, Point Estimate for the population proportion of x xx n= np n == p successes: successes: p successes: n nn cc- Confidence pp Interval Interval forfor Population Population Proportion Proportion (when c-Confidence p (when Confidence Interval for Population Proportion (when np nnp n+ n -E n +E, npnp Ú ÚÚ 5 and 55 and nqnq Ú ÚÚ 52:52: p 6 66 p p6 p where where np nq 52: p -E E p 66 p + E, E, and where x -m - mm Value Value - -Mean Value - Mean Mean x x= == z z=z == s ss Standard Standard deviation deviation Standard deviation Central Limit Theorem population 3030 Central Limit Theorem (n((nÚ oror population is isis n ÚÚ 30 Central Limit Theorem or population normally distributed): normally distributed): normally distributed): mxmm= m = mm xx = s 22 s s Variance the Sampling Distribution: Variance ofof the Sampling Distribution: = Variance of the Sampling Distribution: s2xss= xx = n nn x xx -m - mm forfor a Mean aa Mean unknown, t- Test t-t-Test m :mmt:: = ,, for ss tt == , for Test for Mean for unknown, s unknown, s> s> 1n s>1n 1n Test forfor a Proportion aa Proportion (when :52:: z- zpp npnp Ú ÚÚ 5 and 55 and nqnq Ú ÚÚ 5252 Test for Proportion (when and z-Test p (when np nq np np n -p mpm nmpnpn p np np n -p pp z z= = == z == sps 1pq>n 1pq>n nspnpn 1pq>n 2 22 2 22 1n1n - -12s 1n1n - -12s 1n - 12s 12s 1n - 12s 12s 6 6 s s 6 6 6 s 6 CC CC C xR2xxR2R2 C xL2xxL2L2 and and d.f. = =n d.f. and d.f. = nn -1- 11 n1p nnn 2p n2p n222-2 -1p 1p p p - 1p -2p2p2222 x x1xx+ +2xx22 11p 11 11p11 11 + z z=z == ,, where p p= , where where p == n1nn+ n +2nn22 11 + 1 11 1 11 p qppaqa q a + ++ b bb BB B n1nn11 n2nn22 and and q q= and q =1= 1-1 -p. - p. p. 22 Chi-Square Chi-Square Test Test forfor a Variance aaVariance or Standard Standard Deviation Deviation s2ssor s:s: Chi-Square Test for Variance or Standard Deviation s: 2 22 1n1n - -12s 1n - 12s 12s 1d.f. 1d.f. = =n x == 1d.f. = nn -12 - 12 12 2 22 s ss ng xy - -1-gx21gy2 nngxy gxy 1g x21g y2 1 gx21gy2 CC HAPTER 888 CHAPTER HAPTER z-Test zTwo-Sample Two-Sample forfor the the Difference Difference Between Between Means Means z-Test Two-Sample Test for the Difference Between Means (Independent (Independent samples; samples; oror normally normally n1nnand n2nnÚ 3030 (Independent samples; and or normally Ú 30 11 and 22 Ú distributed distributed populations): populations): distributed populations): r r=r == 2 22 2 22 22 2 22 2n gxgx 2ng x- -1-gx2 11gx2 2ng 2ng y2yy-1-gy2 11gy2 2n gx2 2ng gy2 forfor the the Correlation Correlation Coefficient: Coefficient: t- Test t-t-Test Test for the Correlation Coefficient: t = tt == r rr -r2 r22 1 1-1 - r BB n nB n -2- 22 (d.f. (d.f. = =n (d.f. = nn -2-)22)) Equation Equation ofof a Regression aa Regression Line: Line: yn yn= + +b, Equation of Regression Line: yn =mx = mx mx + b, b, ng ng xy- -1-gx21gy2 1g x21g y2 nxy gxy 1 gx21gy2 where where and and mm = == where and m 2 22 2 22 n gx ng x- -1-gx2 11gx2 n gx gx2 gg ygyy gg xgxx b b=b =y = == - -m = yy -mx - mx mx -m m n nn n nn Two-Sample Two-Sample forfor the the Difference Difference Between Between Means Means t-Test t-t-Test Two-Sample Test for the Difference Between Means (Independent (Independent samples samples from from normally normally distributed distributed (Independent samples from normally distributed populations, populations, ): n1nnor n2nn6 3030 populations, or 6 30):): 11 or 22 6 Coefficient Coefficient ofof Determination: Determination: Coefficient of Determination: 2 22 n1y g y2 g1y - y2 y2 Explained Explained variation variation Explained variation g1y i nnii 22 r2 rr= == = == 2 Total Total variation variation Total variation gg 1y -ii -y2 g1y - y2 y222 i1y 1x1x -11 -x 1m m -2x2x22-22 - 1m 1m - 2m2m2222 11x 1 11 t =tt == sxs1s-xxx11-2-xx22 If If population variances variances are are equal, equal, d.f. = =n n 2- 22 d.f. If population population variances are equal, d.f. =1nn+ +2nn11 + 22 n i2ynyn2i2i222 gg 1y -ii -y g1y i1y and and and Standard Standard Error Error ofof Estimate: Estimate: se ss= = Standard Error of Estimate: = ee CC C n n-n -2- 22 2 22 2 22 1n1n 12s 1n 12s - 12s 12s + 1n - 12s 12s 1 + 11 + 2 22 11n11 21n22 1 1 1 1 1 1 # # # + ++ . .. sxs1s-xxx11-2-xx=22 == Interval Interval forfor c- Prediction cy: y: 6 66 y y6 yn yn+ n1nn11 n2nn22 Prediction Interval for c-Prediction y:yn ynyn -E -E E y 66 yn +E, + E, E, CC n1nn+ n 2- 22 BB C +2nnB 11 + 22 where where where If If population variances variances are are not not equal, equal, d.f.d.f. is is the If population population variances are not equal, d.f. is the the 2 22 n1x n1x x2 n1x - x2 x2 0 00 1 11 s21 ss2121 s22 ss2222 E = t 1d.f. E = t s s 1 1 + + + + 1d.f. = =n E = t s 1 + + 1d.f. = nn -22 - 22 22 c e c e c e 2 2 2 22 smaller smaller ofof n1nn1- or 11 or n2nn1- and 11 and sxs1s-xxx11-2-xx=22 == + ++ . .. 2 smaller of or and 11 22 CC C n nn ng ng x xx- -1-gx2 11gx2 ng gx2 n CC C1nn1 n2nn2 1 2 Copyright © 2012 Pearson Education, Inc. CC HAPTER 10 10 CHAPTER HAPTER 10 Test Test Statistic Statistic forfor the the Kruskal-Wallis Kruskal-Wallis Test: Test: Test Statistic for the Kruskal-Wallis Test: Given Given three three oror more more independent independent samples, samples, the the test test Given three or more independent samples, the test statistic statistic forfor the the Kruskal-Wallis Kruskal-Wallis test test is isis statistic for the Kruskal-Wallis test 2 22 1O1O - -E2 1O - E2 E2 gg g Chi-Square: xx == EE E 22 Chi-Square: Chi-Square:x 2 = R21R R2121 R22R R2222 Á R2kR R2k2k 1212 12 HH = == a aa + ++ + +Á + ++ b bb H + Á nknnkk N1N + +12 N1N N1N + 12 12n1nn11 n2nn22 Goodness-of-Fit Goodness-of-Fit Test: Test: d.f. = =k d.f. Goodness-of-Fit Test: d.f. = kk -1- 11 Test Test ofof Independence: Independence: Test of Independence: - -31N + +12. = =k - 31N 31N + 12. 121d.f. . 1d.f. 1d.f. = kk -12 - 12 12 d.f. = =1no. ofof rows - -121no. ofof columns - -12 d.f. rows columns d.f. = 1no. 1no. of rows - 121no. 121no. of columns - 12 12 Spearman Spearman Rank Rank Correlation Correlation Coefficient: Coefficient: Spearman Rank Correlation Coefficient: 22 2 22 and and s21 ssÚ s2,ss2d.f. = n 1, d.f. = n 1- 11 d.f. and Ú d.f. =1nn- 1, 1, d.f. =2nn11 Ú N N 11 DD 22 2,, d.f. N= D= Test Test Statistic Statistic forfor the the Runs Runs Test: Test: Test Statistic for the Runs Test: Two-Sample Two-Sample Test forfor Variances: Variances: F-FFF = Two-Sample Test for Variances: F-Test F 2 22 6 gd 66gd gd rs rr=ss =1= 1-1 -- 2 22 n1n n1n n1n- -12 - 12 12 s21 ss2121 ,, where ==2 , 2where where s2 ss222 gn gn x BxxBB gn x-ii -iAx iiAiAx MS MS SSSS MS SS B BB B BB FF = == , where ,,where MS MS =B == = == where F MS B B MS MS k kk kk - -1 11 MS k -1- 11 WW W One-Way One-Way Analysis Analysis ofof Variance Variance Test: Test: One-Way Analysis of Variance Test: 2 22 CC HAPTER 999 CHAPTER HAPTER Correlation Correlation Coefficient: Coefficient: Correlation Coefficient: 22 x 2x = Interval Interval forfor Population Population Variance Variance c- Confidence cs2s:s22:: Confidence Interval for Population Variance c-Confidence 2 22 d dgg 1d - -d2d2 d -m - dmmdd g1d 1d d2 gg dgdd tt == t = , where ,, where d d= where d == , s,,dssd= = d= n nn sd>ssd1n AA 1n A n nn- --1 11 d>>1n Two-Sample Two-Sample forfor the the Difference Difference Between Between z-Test zTwo-Sample Test for the Difference Between z-Test Proportions Proportions ( n(1(npn1,1p, must bebe atat least least 5):5): n2nqn22q Proportions and must be at least 5): pn,1nqn1,1qqn,,2npn2,2p, pand , and q must n n6 3030 population population is is normal oror nearly nearly normal, normal, and and .30.. n 66 population is normal normal or nearly normal, and 1d.f. = =n 1d.f. 1d.f. = nn -12 - 12 12 s21ss2121 s22ss2222 where where sxs1s-xxx11-2-xx= + ++ = where = 22 n1nn11 n2nn22 CC C zc zzc2 22 np n qnp naqnqnaab cbb Minimum Minimum Sample Sample Size Size toto Estimate Estimate :: n= p :ppn Minimum Sample Size to Estimate n =p = EE E forfor the the Difference Difference Between Between Means Means (Dependent (Dependent t-t-Test t- Test Test for the Difference Between Means (Dependent samples): samples): samples): x xx -m - mm z- zm :mmz:: z=z == ,, for ss Test forfor a Mean aa Mean known with with a aa , for z-Test s known Test for Mean for known with 1n s>s> 1n s> 1n normal normal population, population, oror forfor n nÚ 3030 normal population, or for n ÚÚ 30 1x1x x m -2x2x222-2 -1m - 1m 1m - 2m2m2222 11x11 1 11 z z=z == , ,, sxs1s-xxx11-2-xx22 Interval Interval forfor Population Population Standard Standard cc- Confidence Confidence Interval for Population Standard c-Confidence Deviation Deviation s:s: Deviation s: Transforming Transforming a zaa zScore toto anan Value: x-xx x= + +zs Transforming Score to an Value: z-Score x-Value: x =m = mm + zs zs CC HAPTER 777 CHAPTER HAPTER n qnp np nqnqn p EE = =z=c zzcc E n n BB B n 2 22 2 22 1n1n - -12s 1n1n - -12s 1n - 12s 12s 1n - 12s 12s 2 22 6 6 s s 6 6 6 s 6 xR2xxR2R2 xL2xxL2L2 Standard Standard Score, Score, oror Score: z- zz-Score: Standard Score, or Score: Mean the Sampling Distribution: Mean ofof the Sampling Distribution: Mean of the Sampling Distribution: Standard Deviation the Sampling Standard Deviation ofof the Sampling Standard Deviation of the Sampling Distribution (Standard Error): Distribution (Standard Error): Distribution (Standard Error): 2 22 gg 1n -ii -12s g1n - 12s 12s SSSS SS i ii i1n WW W and and = == MS = == MS and MS WW W NN - -k NN - -k N - kk N - kk ( d.f. ((d.f. d.f. = k d.f. = N - -k d.f. = kk -1, - 1, 1, d.f. =N N -)kk)) N N DD N= D= CC HAPTER 11 11 CHAPTER HAPTER 11 Test Test Statistic Statistic forfor Sign Sign Test: Test: Test Statistic for Sign Test: When When the the test test statistic statistic is is the smaller smaller number number n n… 25,25, When the test statistic is the the smaller number n …… 25, ofof + +or - -signs. of or signs. + or - signs. 1x1x + +0.52 - -0.5n 1x + 0.52 0.52 - 0.5n 0.5n , where When When isx is the ,, where n n7 25,25, z z=z == xx When where is the the n 77 25, 2n 2n 2n 2 22 smaller smaller number number ofof and and isn is the total total + +or - -signs nn smaller number of or signs and is the the total + or - signs number number ofof + +and - -signs. number of and signs. + and - signs. Test Test Statistic Statistic forfor Wilcoxon Wilcoxon Rank Rank Sum Sum Test: Test: Test Statistic for Wilcoxon Rank Sum Test: RR - -m R - RmmRR sum sum ofof the the ranks ranks forfor the the z z=z == , where ,, where RR = == where sum of the ranks for the R sRssRR n1n1n n 12 n111n +2nn+ + 12 12 11n+ 11 + 22 + smaller smaller sample, sample, mRmmR=R == , ,, smaller sample, 2 22 n1nn121n1n n 12 n221n +2nn+ + 12 12 11n+ 11 + 22 + sRssR=R == ,, and n1nn… n2nn22 , and and … 11 … 12 BB 12 B 12 When When and and the the test test statistic statistic is is n1nn… 2020 n2nn… 20,20, G, When and the test statistic is G, … 20 … 20, G, 11 … 22 … the the number number ofof runs. runs. the number of runs. When When oror the the test test statistic statistic is isis n1nn7 2020 n2nn7 20,20, When or the test statistic 7 20 7 20, 11 7 22 7 GG - -m G - GmmGG ofof runs, runs, z z=z == , where ,, where GG = =number where number of runs, G = number sGssGG n121nn22 2n2n 12n and mGmmG= + +1, and = + 1, 1, and G= n1nn+ n +2nn22 11 + 2n2n n121n12n n n n2212n 12n -1nn-2n2n2222 12n 1n121nn22 11 . .. sGssG= = G= 2 2 2 BB B 1n1n n 22 1n n 12 +2n2n221n +2nn- 12 12 11n+ 11 + 11n+ 11 + 22 - 0321693620_INS_p1-4a.qxp 0321693620_INS_p1-4a.qxp 12/1/10 11:09 AMAM Page 1 11 0321693620_INS_p1-4a.qxp12/1/10 12/1/1011:09 11:09 AMPage Page Key Key Formulas KeyFormulas Formulas Key Key Formulas KeyFormulas Formulas Key Key Formulas KeyFormulas Formulas Key Key Formulas KeyFormulas Formulas From From Larson/Farber Larson/Farber Elementary Elementary Statistics: Statistics: Picturing Picturing thethe World, World, Fifth Fifth Edition Edition From Larson/Farber Elementary Statistics: Picturing the World, Fifth Edition ©© 2012 Prentice Prentice Hall Hall © 2012 2012 Prentice Hall From From Larson/Farber Larson/Farber Elementary Elementary Statistics: Statistics: Picturing Picturing thethe World, World, Fifth Fifth Edition Edition From Larson/Farber Elementary Statistics: Picturing the World, Fifth Edition ©© 2012 Prentice Prentice Hall Hall © 2012 2012 Prentice Hall From From Larson/Farber Larson/Farber Elementary Elementary Statistics: Statistics: Picturing Picturing thethe World, World, Fifth Fifth Edition Edition From Larson/Farber Elementary Statistics: Picturing the World, Fifth Edition ©© 2012 Prentice Prentice Hall Hall © 2012 2012 Prentice Hall From From Larson/Farber Larson/Farber Elementary Elementary Statistics: Statistics: Picturing Picturing thethe World, World, Fifth Fifth Edition Edition From Larson/Farber Elementary Statistics: Picturing the World, Fifth Edition ©© 2012 Prentice Prentice Hall Hall © 2012 2012 Prentice Hall CC HAPTER 222 CHAPTER HAPTER Class Class Width Width Class Width= == Sample Sample Standard Standard Deviation Deviation ofof a Frequency aa Frequency Distribution: Distribution: Sample Standard Deviation of Frequency Distribution: Range Range ofof data data Range of data Number Number ofof classes classes Number of classes s s=s == CC C n n-n -1- 11 1 round 11round 2 22 upup toto next next convenient convenient number number round up to next convenient number 1Lower 1Lower class class limit2 limit2 + +1Upper class class limit2 limit2 1Lower class limit2 + 1Upper 1Upper class limit2 Midpoint Midpoint = == Midpoint 2 22 Class Class frequency frequency Class frequency f ff Relative Relative Frequency Frequency = == = == Relative Frequency Sample Sample size size Sample size n nn gg xgxx m mm = == Population Population Mean: Mean: Population Mean: NN N x -m - mm Value Value - -Mean Value - Mean Mean x xStandard Standard Score: Score: z z=z == = == Standard Score: s ss Standard Standard deviation deviation Standard deviation CC HAPTER 333 CHAPTER HAPTER Classical Classical (or(or Theoretical) Theoretical) Probability: Probability: Classical (or Theoretical) Probability: Number Number ofof outcomes outcomes inin event event EE Number of outcomes in event E P1E2 P1E2 = == P1E2 Total Total number number ofof outcomes outcomes Total number of outcomes inin sample sample space space in sample space gg xgxx x x= Sample Sample Mean: Mean: x == Sample Mean: n nn # w2# #w2 g g1x 1x g 1x w2 x x= Weighted Weighted Mean: Mean: Weighted Mean: x == gg wgww Empirical Empirical (or(or Statistical) Statistical) Probability: Probability: Empirical (or Statistical) Probability: Frequency Frequency ofof event event EE Frequency of event E f ff P1E2 P1E2 = == = == P1E2 n nn Total Total frequency frequency Total frequency # f2# #f2 g1x g1x g1x f2 Mean Mean ofof a Frequency aa Frequency Distribution: Distribution: x x= x == Mean of Frequency Distribution: n nn = =1Maximum entry2 entry2 - -1Minimum entry2 entry2 Range Range = 1Maximum 1Maximum entry2 - 1Minimum 1Minimum entry2 Range 2 22 gg 1x - -m2 g1x 1x - m2 m2 Population Population Variance: Variance: Population Variance: s == NN N 2 22 gg 1x - -m2 g1x 1x - m2 m2 2 22 gg 1x - -x2 g1x 1x - x2 x2 Sample Sample Variance: Variance: Sample Variance: s == n nn -1- 11 2 22 s s=s == 2s 2s = == Sample Sample Standard Standard Deviation: Deviation: 2s Sample Standard Deviation: g1x g1x - - x2 g1x 2 22 -x2x2 CC C n n-n -1- 11 Empirical Empirical Rule Rule 68-95-99.7 68-95-99.7 Rule) Rule) For For data data with with a aa Empirical Rule(or(or (or 68-95-99.7 Rule) For data with (symmetric) (symmetric) bell-shaped bell-shaped distribution: distribution: (symmetric) bell-shaped distribution: 1. 1. About About 68% 68% ofof thethe data data lies lies between between and m mm - -s m mm + +s. 1. About 68% of the data lies between and -s s and + s. s. m mm - -2s 2. 2. About About 95% 95% ofof the the data data lies lies between between and and 2. About 95% of the data lies between and - 2s 2s m mm + +2s. + 2s. 2s. m mm - -3s 3. 3. About About 99.7% 99.7% ofof the the data data lies lies between between and and 3. About 99.7% of the data lies between and - 3s 3s m mm + +3s. + 3s. 3s. Chebychev’s Chebychev’s Theorem Theorem The portion portion ofof any any data data setset lying lying Chebychev’s TheoremThe The portion of any data set lying within within standard deviations deviations ofof the the mean mean is is at kk 1k1k 7 77 1212 within standard deviations of the mean is at at k standard 1k 12 1 11 least least 1 1-1 -- 2 . 22.. least k kk 22 2 22 s2ss= == gg 1x - -m2 P1x2 P1x2 g1x 1x - m2 m2 P1x2 Standard Standard Deviation Deviation ofof a Discrete aa Discrete Random Random Variable: Variable: Standard Deviation of Discrete Random Variable: 2 22 2 22 s ss = == 2s 2s g1x P1x2 P1x2 = == 22 g1x - -m2 2s 2g 1x - m2 m2 P1x2 Expected Expected Value: Value: E1x2 E1x2 = =m = == gg xP1x2 Expected Value: E1x2 = mm gxP1x2 xP1x2 Binomial Binomial Probability Probability ofof successes inin trials: xx nn Binomial Probability of successes in trials: x successes n trials: n!n! n! xx P1x2 pxxxpqpxnxq-qnxn--= == pxpqpxnxq-qnxn--xx P1x2 = =n=CnnxCC P1x2 1n1n - -x2!x! 1n - x2!x! x2!x! Population Population Parameters Parameters ofof a Binomial aa Binomial Distribution: Distribution: Population Parameters of Binomial Distribution: 22 Variance: Variance: s2ss= =npq Variance: = npq npq Standard Standard Deviation: Deviation: 1npq s ss = == 1npq Standard Deviation: 1npq P1A P1A oror B2B2 = =P1A2 + +P1B2 - -P1A and and B2B2 P1A or B2 = P1A2 P1A2 + P1B2 P1B2 - P1A P1A and B2 22 s2 s= Variance Variance ofof a Discrete aa Discrete Random Random Variable: Variable: Variance of Discrete Random Variable: AA B:B: Probability Probability ofof occurrence occurrence ofof both both events events and and A B: Probability of occurrence of both events and Probability Probability ofof occurrence occurrence ofof either either oror or both: both: AA BB Probability of occurrence of either or or both: A B or CC C NN N Mean Mean ofof a Discrete aa Discrete Random Random Variable: Variable: m mm = == g gxP1x2 xP1x2 gxP1x2 Mean of Discrete Random Variable: Mean: Mean: m mm = =np Mean: = np np # P1B2 # #P1B2 P1A P1A and and B2B2 = =P1A2 A BB if if and and are P1B2 if A and are P1A and B2 = P1A2 P1A2 A B are independent independent independent Population Population Standard Standard Deviation: Deviation: Population Standard Deviation: CC HAPTER 444 CHAPTER HAPTER P1E¿2 P1E¿2 = =1= 1-1 -P1E2 Probability Probability ofof a Complement: aa Complement: P1E¿2 - P1E2 P1E2 Probability of Complement: # P1B # #P1B P1A P1A and and B2B2 = =P1A2 P1B P1A and B2 = P1A2 P1A2 ƒ Aƒ2A ƒ A22 22 s2s= 2 22 s ss = == 2s 2s = == 2s g1x g1x - - x2 f g1x 2 22 -x2x2 ff if if and and are P1A P1A oror B2B2 = =P1A2 + +P1B2 A BB if A and are P1A or B2 = P1A2 P1A2 + P1B2 P1B2 A B are mutually mutually exclusive exclusive mutually exclusive Permutations Permutations ofof objects taken taken a time: aa time: nn r at rr at Permutations of objects taken at time: n objects P=rr nPnnrP n!n! n! == , where ,,where r r…r …… n nn where 1n1n 1n r2! - r2! r2! Distinguishable Distinguishable Permutations: Permutations: , ,, n1nnalike, n 2nnalike, ÁÁ Distinguishable Permutations: alike, alike, Á 11 alike, 22 alike, alike: nknnalike: alike: kk n!n! n! , , Á Á !nkk!! , n1n!n1#1!n!#2#n!n2#2!n!#2#n!n2Á 2!! nkn Á Á where where n1nn+ n n + +n where +2nn+ +3nn+ +Á +knnk=k =n = nn 11 + 22 + 33 + Combination Combination ofof objects taken taken a time: aa time: nn r at rr at Combination of objects taken at time: n objects n!n! n! C=rr == nCnn rC 1n1n - -r2!r! 1n - r2!r! r2!r! Geometric Geometric Distribution: Distribution: The The probability probability that that the the first first Geometric Distribution: The probability that the first x -x1x--11 success success will will occur occur onon trial trial number number isx is xx , ,, P1x2 = =p1q2 success will occur on trial number is P1x2 P1x2 = p1q2 p1q2 where where q q= where q =1= 1-1 -p. - p. p. Poisson Poisson Distribution: Distribution: The The probability probability ofof exactly exactly x xx Poisson Distribution: The probability of exactly xe-m mxmemx-m e-m occurrences occurrences inin anan interval interval is is , where ,, where P1x2 = == occurrences in an interval is P1x2 where P1x2 x!x! x! and and is the mean mean number number ofof occurences occurences e eLe L2.71828 mm and is the the mean number of occurences L 2.71828 2.71828 m is per per interval interval unit. unit. per interval unit. CC HAPTER 555 CHAPTER HAPTER s ss sxss= = xx = 1n 1n 1n x -m - xmmxx x xx -m - mm Value - -Mean Value Value - Mean Mean x xz-Score = == z-Score = == = == z-Score sxssxx Standard Error Standard Error Standard Error 1n s>s> 1n s> 1n CC HAPTER 666 CHAPTER HAPTER Interval Interval forfor :: xc- Confidence cm :mmx 6 66 m mm 6 66 x x+ Confidence Interval for c-Confidence x -E -E E x +E, + E, E, s ss EE = =z=c zzcc if if s where where is known and and the the population population is isis E s is where if s is known known and the population 1n 1n 1n s ss n nÚ 30,30, EE = =t=c ttcc if if normally normally distributed distributed oror oror the n ÚÚ 30, E normally distributed or or if the the 1n 1n 1n population population is is normally oror approximately approximately normally normally population is normally normally or approximately normally ss n n6 3030 distributed, distributed, is unknown, and and s is n 66 30 distributed, is unknown, unknown, and zcs zzccss2 22 a aa b bb m :mmn Minimum Minimum Sample Sample Size Size toto Estimate Estimate :: n= n == Minimum Sample Size to Estimate EE E p,p, Point Point Estimate Estimate forfor the the population population proportion proportion ofof p, Point Estimate for the population proportion of x xx n= np n == p successes: successes: p successes: n nn cc- Confidence pp Interval Interval forfor Population Population Proportion Proportion (when c-Confidence p (when Confidence Interval for Population Proportion (when np nnp n+ n -E n +E, npnp Ú ÚÚ 5 and 55 and nqnq Ú ÚÚ 52:52: p 6 66 p p6 p where where np nq 52: p -E E p 66 p + E, E, and where x -m - mm Value Value - -Mean Value - Mean Mean x x= == z z=z == s ss Standard Standard deviation deviation Standard deviation Central Limit Theorem population 3030 Central Limit Theorem (n((nÚ oror population is isis n ÚÚ 30 Central Limit Theorem or population normally distributed): normally distributed): normally distributed): mxmm= m = mm xx = s 22 s s Variance the Sampling Distribution: Variance ofof the Sampling Distribution: = Variance of the Sampling Distribution: s2xss= xx = n nn x xx -m - mm forfor a Mean aa Mean unknown, t- Test t-t-Test m :mmt:: = ,, for ss tt == , for Test for Mean for unknown, s unknown, s> s> 1n s>1n 1n Test forfor a Proportion aa Proportion (when :52:: z- zpp npnp Ú ÚÚ 5 and 55 and nqnq Ú ÚÚ 5252 Test for Proportion (when and z-Test p (when np nq np np n -p mpm nmpnpn p np np n -p pp z z= = == z == sps 1pq>n 1pq>n nspnpn 1pq>n 2 22 2 22 1n1n - -12s 1n1n - -12s 1n - 12s 12s 1n - 12s 12s 6 6 s s 6 6 6 s 6 CC CC C xR2xxR2R2 C xL2xxL2L2 and and d.f. = =n d.f. and d.f. = nn -1- 11 n1p nnn 2p n2p n222-2 -1p 1p p p - 1p -2p2p2222 x x1xx+ +2xx22 11p 11 11p11 11 + z z=z == ,, where p p= , where where p == n1nn+ n +2nn22 11 + 1 11 1 11 p qppaqa q a + ++ b bb BB B n1nn11 n2nn22 and and q q= and q =1= 1-1 -p. - p. p. 22 Chi-Square Chi-Square Test Test forfor a Variance aaVariance or Standard Standard Deviation Deviation s2ssor s:s: Chi-Square Test for Variance or Standard Deviation s: 2 22 1n1n - -12s 1n - 12s 12s 1d.f. 1d.f. = =n x == 1d.f. = nn -12 - 12 12 2 22 s ss ng xy - -1-gx21gy2 nngxy gxy 1g x21g y2 1 gx21gy2 CC HAPTER 888 CHAPTER HAPTER z-Test zTwo-Sample Two-Sample forfor the the Difference Difference Between Between Means Means z-Test Two-Sample Test for the Difference Between Means (Independent (Independent samples; samples; oror normally normally n1nnand n2nnÚ 3030 (Independent samples; and or normally Ú 30 11 and 22 Ú distributed distributed populations): populations): distributed populations): r r=r == 2 22 2 22 22 2 22 2n gxgx 2ng x- -1-gx2 11gx2 2ng 2ng y2yy-1-gy2 11gy2 2n gx2 2ng gy2 forfor the the Correlation Correlation Coefficient: Coefficient: t- Test t-t-Test Test for the Correlation Coefficient: t = tt == r rr -r2 r22 1 1-1 - r BB n nB n -2- 22 (d.f. (d.f. = =n (d.f. = nn -2-)22)) Equation Equation ofof a Regression aa Regression Line: Line: yn yn= + +b, Equation of Regression Line: yn =mx = mx mx + b, b, ng ng xy- -1-gx21gy2 1g x21g y2 nxy gxy 1 gx21gy2 where where and and mm = == where and m 2 22 2 22 n gx ng x- -1-gx2 11gx2 n gx gx2 gg ygyy gg xgxx b b=b =y = == - -m = yy -mx - mx mx -m m n nn n nn Two-Sample Two-Sample forfor the the Difference Difference Between Between Means Means t-Test t-t-Test Two-Sample Test for the Difference Between Means (Independent (Independent samples samples from from normally normally distributed distributed (Independent samples from normally distributed populations, populations, ): n1nnor n2nn6 3030 populations, or 6 30):): 11 or 22 6 Coefficient Coefficient ofof Determination: Determination: Coefficient of Determination: 2 22 n1y g y2 g1y - y2 y2 Explained Explained variation variation Explained variation g1y i nnii 22 r2 rr= == = == 2 Total Total variation variation Total variation gg 1y -ii -y2 g1y - y2 y222 i1y 1x1x -11 -x 1m m -2x2x22-22 - 1m 1m - 2m2m2222 11x 1 11 t =tt == sxs1s-xxx11-2-xx22 If If population variances variances are are equal, equal, d.f. = =n n 2- 22 d.f. If population population variances are equal, d.f. =1nn+ +2nn11 + 22 n i2ynyn2i2i222 gg 1y -ii -y g1y i1y and and and Standard Standard Error Error ofof Estimate: Estimate: se ss= = Standard Error of Estimate: = ee CC C n n-n -2- 22 2 22 2 22 1n1n 12s 1n 12s - 12s 12s + 1n - 12s 12s 1 + 11 + 2 22 11n11 21n22 1 1 1 1 1 1 # # # + ++ . .. sxs1s-xxx11-2-xx=22 == Interval Interval forfor c- Prediction cy: y: 6 66 y y6 yn yn+ n1nn11 n2nn22 Prediction Interval for c-Prediction y:yn ynyn -E -E E y 66 yn +E, + E, E, CC n1nn+ n 2- 22 BB C +2nnB 11 + 22 where where where If If population variances variances are are not not equal, equal, d.f.d.f. is is the If population population variances are not equal, d.f. is the the 2 22 n1x n1x x2 n1x - x2 x2 0 00 1 11 s21 ss2121 s22 ss2222 E = t 1d.f. E = t s s 1 1 + + + + 1d.f. = =n E = t s 1 + + 1d.f. = nn -22 - 22 22 c e c e c e 2 2 2 22 smaller smaller ofof n1nn1- or 11 or n2nn1- and 11 and sxs1s-xxx11-2-xx=22 == + ++ . .. 2 smaller of or and 11 22 CC C n nn ng ng x xx- -1-gx2 11gx2 ng gx2 n CC C1nn1 n2nn2 1 2 CC HAPTER 10 10 CHAPTER HAPTER 10 Test Test Statistic Statistic forfor the the Kruskal-Wallis Kruskal-Wallis Test: Test: Test Statistic for the Kruskal-Wallis Test: Given Given three three oror more more independent independent samples, samples, the the test test Given three or more independent samples, the test statistic statistic forfor the the Kruskal-Wallis Kruskal-Wallis test test is isis statistic for the Kruskal-Wallis test 2 22 1O1O - -E2 1O - E2 E2 gg g Chi-Square: xx == EE E 22 Chi-Square: Chi-Square:x 2 = R21R R2121 R22R R2222 Á R2kR R2k2k 1212 12 HH = == a aa + ++ + +Á + ++ b bb H + Á nknnkk N1N + +12 N1N N1N + 12 12n1nn11 n2nn22 Goodness-of-Fit Goodness-of-Fit Test: Test: d.f. = =k d.f. Goodness-of-Fit Test: d.f. = kk -1- 11 Test Test ofof Independence: Independence: Test of Independence: - -31N + +12. = =k - 31N 31N + 12. 121d.f. . 1d.f. 1d.f. = kk -12 - 12 12 d.f. = =1no. ofof rows - -121no. ofof columns - -12 d.f. rows columns d.f. = 1no. 1no. of rows - 121no. 121no. of columns - 12 12 Spearman Spearman Rank Rank Correlation Correlation Coefficient: Coefficient: Spearman Rank Correlation Coefficient: 22 2 22 and and s21 ssÚ s2,ss2d.f. = n 1, d.f. = n 1- 11 d.f. and Ú d.f. =1nn- 1, 1, d.f. =2nn11 Ú N N 11 DD 22 2,, d.f. N= D= Test Test Statistic Statistic forfor the the Runs Runs Test: Test: Test Statistic for the Runs Test: Two-Sample Two-Sample Test forfor Variances: Variances: F-FFF = Two-Sample Test for Variances: F-Test F 2 22 6 gd 66gd gd rs rr=ss =1= 1-1 -- 2 22 n1n n1n n1n- -12 - 12 12 s21 ss2121 ,, where ==2 , 2where where s2 ss222 gn gn x BxxBB gn x-ii -iAx iiAiAx MS MS SSSS MS SS B BB B BB FF = == , where ,,where MS MS =B == = == where F MS B B MS MS k kk kk - -1 11 MS k -1- 11 WW W One-Way One-Way Analysis Analysis ofof Variance Variance Test: Test: One-Way Analysis of Variance Test: 2 22 CC HAPTER 999 CHAPTER HAPTER Correlation Correlation Coefficient: Coefficient: Correlation Coefficient: 22 x 2x = Interval Interval forfor Population Population Variance Variance c- Confidence cs2s:s22:: Confidence Interval for Population Variance c-Confidence 2 22 d dgg 1d - -d2d2 d -m - dmmdd g1d 1d d2 gg dgdd tt == t = , where ,, where d d= where d == , s,,dssd= = d= n nn sd>ssd1n AA 1n A n nn- --1 11 d>>1n Two-Sample Two-Sample forfor the the Difference Difference Between Between z-Test zTwo-Sample Test for the Difference Between z-Test Proportions Proportions ( n(1(npn1,1p, must bebe atat least least 5):5): n2nqn22q Proportions and must be at least 5): pn,1nqn1,1qqn,,2npn2,2p, pand , and q must n n6 3030 population population is is normal oror nearly nearly normal, normal, and and .30.. n 66 population is normal normal or nearly normal, and 1d.f. = =n 1d.f. 1d.f. = nn -12 - 12 12 s21ss2121 s22ss2222 where where sxs1s-xxx11-2-xx= + ++ = where = 22 n1nn11 n2nn22 CC C zc zzc2 22 np n qnp naqnqnaab cbb Minimum Minimum Sample Sample Size Size toto Estimate Estimate :: n= p :ppn Minimum Sample Size to Estimate n =p = EE E forfor the the Difference Difference Between Between Means Means (Dependent (Dependent t-t-Test t- Test Test for the Difference Between Means (Dependent samples): samples): samples): x xx -m - mm z- zm :mmz:: z=z == ,, for ss Test forfor a Mean aa Mean known with with a aa , for z-Test s known Test for Mean for known with 1n s>s> 1n s> 1n normal normal population, population, oror forfor n nÚ 3030 normal population, or for n ÚÚ 30 1x1x x m -2x2x222-2 -1m - 1m 1m - 2m2m2222 11x11 1 11 z z=z == , ,, sxs1s-xxx11-2-xx22 Interval Interval forfor Population Population Standard Standard cc- Confidence Confidence Interval for Population Standard c-Confidence Deviation Deviation s:s: Deviation s: Transforming Transforming a zaa zScore toto anan Value: x-xx x= + +zs Transforming Score to an Value: z-Score x-Value: x =m = mm + zs zs CC HAPTER 777 CHAPTER HAPTER n qnp np nqnqn p EE = =z=c zzcc E n n BB B n 2 22 2 22 1n1n - -12s 1n1n - -12s 1n - 12s 12s 1n - 12s 12s 2 22 6 6 s s 6 6 6 s 6 xR2xxR2R2 xL2xxL2L2 Standard Standard Score, Score, oror Score: z- zz-Score: Standard Score, or Score: Mean the Sampling Distribution: Mean ofof the Sampling Distribution: Mean of the Sampling Distribution: Standard Deviation the Sampling Standard Deviation ofof the Sampling Standard Deviation of the Sampling Distribution (Standard Error): Distribution (Standard Error): Distribution (Standard Error): 2 22 gg 1n -ii -12s g1n - 12s 12s SSSS SS i ii i1n WW W and and = == MS = == MS and MS WW W NN - -k NN - -k N - kk N - kk When When and and the the test test statistic statistic is is n1nn… 2020 n2nn… 20,20, G, When and the test statistic is G, … 20 … 20, G, 11 … 22 … the the number number ofof runs. runs. the number of runs. When When oror the the test test statistic statistic is isis n1nn7 2020 n2nn7 20,20, When or the test statistic 7 20 7 20, 11 7 22 7 GG - -m G - GmmGG ofof runs, runs, z z=z == , where ,, where GG = =number where number of runs, G = number sGssGG n121nn22 2n2n 12n and mGmmG= + +1, and = + 1, 1, and G= n1nn+ n +2nn22 11 + ( d.f. ((d.f. d.f. = k d.f. = N - -k d.f. = kk -1, - 1, 1, d.f. =N N -)kk)) N N DD N= D= 2n2n n121n12n n n n2212n 12n -1nn-2n2n2222 12n 1n121nn22 11 . .. sGssG= = G= 2 2 2 BB B 1n1n n 22 1n n 12 +2n2n221n +2nn- 12 12 11n+ 11 + 11n+ 11 + 22 - CC HAPTER 11 11 CHAPTER HAPTER 11 Test Test Statistic Statistic forfor Sign Sign Test: Test: Test Statistic for Sign Test: When When the the test test statistic statistic is is the smaller smaller number number n n… 25,25, When the test statistic is the the smaller number n …… 25, ofof + +or - -signs. of or signs. + or - signs. 1x1x + +0.52 - -0.5n 1x + 0.52 0.52 - 0.5n 0.5n , where When When isx is the ,, where n n7 25,25, z z=z == xx When where is the the n 77 25, 2n 2n 2n 2 22 smaller smaller number number ofof and and isn is the total total + +or - -signs nn smaller number of or signs and is the the total + or - signs number number ofof + +and - -signs. number of and signs. + and - signs. Test Test Statistic Statistic forfor Wilcoxon Wilcoxon Rank Rank Sum Sum Test: Test: Test Statistic for Wilcoxon Rank Sum Test: RR - -m R - RmmRR sum sum ofof the the ranks ranks forfor the the z z=z == , where ,, where RR = == where sum of the ranks for the R sRssRR n1n1n n 12 n111n +2nn+ + 12 12 11n+ 11 + 22 + smaller smaller sample, sample, mRmmR=R == , ,, smaller sample, 2 22 n1nn121n1n n 12 n221n +2nn+ + 12 12 11n+ 11 + 22 + sRssR=R == ,, and n1nn… n2nn22 , and and … 11 … 12 BB 12 B 12 Copyright © 2012 Pearson Education, Inc. 41831S4_INS p5-8 AM 11 1 41831S4_INS p5-8 11/8/07 10:03 AMAMPage Page 41831S4_INS p5-811/8/07 11/8/0710:03 10:03 Page Table Table 44— 4— Standard Standard Normal Normal Distribution Distribution (continued) (continued) Table — Standard Normal Distribution (continued) Table Table 44— — Standard Standard Normal Normal Distribution Distribution Table 4— Standard Normal Distribution Table Table Table 6— 6— 6— Chi-Square Chi-Square Chi-Square Distribution Distribution Distribution Area Area Area Area Area Area 11 1 αα α 22 2 zz z zz z .09 .09.09 .0002 .0002 .0002 .0003 .0003 .0003 .0005 .0005 .0005 .0007 .0007 .0007 .0010 .0010 .0010 .0014 .0014 .0014 .0019 .0019 .0019 .0026 .0026 .0026 .0036 .0036 .0036 .0048 .0048 .0048 .0064 .0064 .0064 .0084 .0084 .0084 .0110 .0110 .0110 .0143 .0143 .0143 .0183 .0183 .0183 .0233 .0233 .0233 .0294 .0294 .0294 .0367 .0367 .0367 .0455 .0455 .0455 .0559 .0559 .0559 .0681 .0681 .0681 .0823 .0823 .0823 .0985 .0985 .0985 .1170 .1170 .1170 .1379 .1379 .1379 .1611 .1611 .1611 .1867 .1867 .1867 .2148 .2148 .2148 .2451 .2451 .2451 .2776 .2776 .2776 .3121 .3121 .3121 .3483 .3483 .3483 .3859 .3859 .3859 .4247 .4247 .4247 .4641 .4641 .4641 −t −t −t 00 0 zz z zz z 00 0 zz z 3.4 � � 3.4 �3.4 3.3 � � 3.3 �3.3 3.2 � � 3.2 �3.2 3.1 � � 3.1 �3.1 3.0 � � 3.0 �3.0 2.9 � � 2.9 �2.9 2.8 � � 2.8 �2.8 2.7 � � 2.7 �2.7 2.6 � � 2.6 �2.6 2.5 � � 2.5 �2.5 2.4 � � 2.4 �2.4 2.3 � � 2.3 �2.3 2.2 � � 2.2 �2.2 2.1 � � 2.1 �2.1 2.0 � � 2.0 �2.0 1.9 � � 1.9 �1.9 1.8 � � 1.8 �1.8 1.7 � � 1.7 �1.7 1.6 � � 1.6 �1.6 1.5 � � 1.5 �1.5 1.4 � � 1.4 �1.4 1.3 � � 1.3 �1.3 1.2 � � 1.2 �1.2 1.1 � � 1.1 �1.1 1.0 � � 1.0 �1.0 0.9 � � 0.9 �0.9 0.8 � � 0.8 �0.8 0.7 � � 0.7 �0.7 0.6 � � 0.6 �0.6 0.5 � � 0.5 �0.5 0.4 � � 0.4 �0.4 0.3 � � 0.3 �0.3 0.2 � � 0.2 �0.2 0.1 � � 0.1 �0.1 �0.0 0.0 � � 0.0 Table Table Table 5— 5— 5— t-Distribution t-Distribution t-Distribution .08 .08.08 .0003 .0003 .0003 .0004 .0004 .0004 .0005 .0005 .0005 .0007 .0007 .0007 .0010 .0010 .0010 .0014 .0014 .0014 .0020 .0020 .0020 .0027 .0027 .0027 .0037 .0037 .0037 .0049 .0049 .0049 .0066 .0066 .0066 .0087 .0087 .0087 .0113 .0113 .0113 .0146 .0146 .0146 .0188 .0188 .0188 .0239 .0239 .0239 .0301 .0301 .0301 .0375 .0375 .0375 .0465 .0465 .0465 .0571 .0571 .0571 .0694 .0694 .0694 .0838 .0838 .0838 .1003 .1003 .1003 .1190 .1190 .1190 .1401 .1401 .1401 .1635 .1635 .1635 .1894 .1894 .1894 .2177 .2177 .2177 .2483 .2483 .2483 .2810 .2810 .2810 .3156 .3156 .3156 .3520 .3520 .3520 .3897 .3897 .3897 .4286 .4286 .4286 .4681 .4681 .4681 .07 .07.07 .0003 .0003 .0003 .0004 .0004 .0004 .0005 .0005 .0005 .0008 .0008 .0008 .0011 .0011 .0011 .0015 .0015 .0015 .0021 .0021 .0021 .0028 .0028 .0028 .0038 .0038 .0038 .0051 .0051 .0051 .0068 .0068 .0068 .0089 .0089 .0089 .0116 .0116 .0116 .0150 .0150 .0150 .0192 .0192 .0192 .0244 .0244 .0244 .0307 .0307 .0307 .0384 .0384 .0384 .0475 .0475 .0475 .0582 .0582 .0582 .0708 .0708 .0708 .0853 .0853 .0853 .1020 .1020 .1020 .1210 .1210 .1210 .1423 .1423 .1423 .1660 .1660 .1660 .1922 .1922 .1922 .2206 .2206 .2206 .2514 .2514 .2514 .2843 .2843 .2843 .3192 .3192 .3192 .3557 .3557 .3557 .3936 .3936 .3936 .4325 .4325 .4325 .4721 .4721 .4721 .06 .06.06 .0003 .0003 .0003 .0004 .0004 .0004 .0006 .0006 .0006 .0008 .0008 .0008 .0011 .0011 .0011 .0015 .0015 .0015 .0021 .0021 .0021 .0029 .0029 .0029 .0039 .0039 .0039 .0052 .0052 .0052 .0069 .0069 .0069 .0091 .0091 .0091 .0119 .0119 .0119 .0154 .0154 .0154 .0197 .0197 .0197 .0250 .0250 .0250 .0314 .0314 .0314 .0392 .0392 .0392 .0485 .0485 .0485 .0594 .0594 .0594 .0721 .0721 .0721 .0869 .0869 .0869 .1038 .1038 .1038 .1230 .1230 .1230 .1446 .1446 .1446 .1685 .1685 .1685 .1949 .1949 .1949 .2236 .2236 .2236 .2546 .2546 .2546 .2877 .2877 .2877 .3228 .3228 .3228 .3594 .3594 .3594 .3974 .3974 .3974 .4364 .4364 .4364 .4761 .4761 .4761 .05 .05.05 .0003 .0003 .0003 .0004 .0004 .0004 .0006 .0006 .0006 .0008 .0008 .0008 .0011 .0011 .0011 .0016 .0016 .0016 .0022 .0022 .0022 .0030 .0030 .0030 .0040 .0040 .0040 .0054 .0054 .0054 .0071 .0071 .0071 .0094 .0094 .0094 .0122 .0122 .0122 .0158 .0158 .0158 .0202 .0202 .0202 .0256 .0256 .0256 .0322 .0322 .0322 .0401 .0401 .0401 .0495 .0495 .0495 .0606 .0606 .0606 .0735 .0735 .0735 .0885 .0885 .0885 .1056 .1056 .1056 .1251 .1251 .1251 .1469 .1469 .1469 .1711 .1711 .1711 .1977 .1977 .1977 .2266 .2266 .2266 .2578 .2578 .2578 .2912 .2912 .2912 .3264 .3264 .3264 .3632 .3632 .3632 .4013 .4013 .4013 .4404 .4404 .4404 .4801 .4801 .4801 .04 .04.04 .03 .03.03 .0003 .0003 .0003 .0003 .0003 .0003 .0004 .0004 .0004 .0004 .0004 .0004 .0006 .0006 .0006 .0006 .0006 .0006 .0008 .0008 .0009 .0008 .0009 .0009 .0012 .0012 .0012 .0012 .0012 .0012 .0016 .0016 .0017 .0016 .0017 .0017 .0023 .0023 .0023 .0023 .0023 .0023 .0031 .0031 .0032 .0031 .0032 .0032 .0041 .0043 .0041 .0041 .0043 .0043 .0055 .0055 .0057 .0055 .0057 .0057 .0073 .0075 .0073 .0073 .0075 .0075 .0096 .0096 .0099 .0096 .0099 .0099 .0125 .0129 .0125 .0125 .0129 .0129 .0162 .0162 .0166 .0162 .0166 .0166 .0207 .0212 .0207 .0207 .0212 .0212 .0262 .0262 .0268 .0262 .0268 .0268 .0329 .0336 .0329 .0329 .0336 .0336 .0409 .0409 .0418 .0409 .0418 .0418 .0505 .0516 .0505 .0505 .0516 .0516 .0618 .0618 .0630 .0618 .0630 .0630 .0749 .0764 .0749 .0749 .0764 .0764 .0901 .0901 .0918 .0901 .0918 .0918 .1075 .1093 .1075 .1075 .1093 .1093 .1271 .1271 .1292 .1271 .1292 .1292 .1492 .1515 .1492 .1492 .1515 .1515 .1736 .1736 .1762 .1736 .1762 .1762 .2005 .2033 .2005 .2005 .2033 .2033 .2296 .2296 .2327 .2296 .2327 .2327 .2611 .2643 .2611 .2611 .2643 .2643 .2946 .2946 .2981 .2946 .2981 .2981 .3300 .3336 .3300 .3300 .3336 .3336 .3669 .3669 .3707 .3669 .3707 .3707 .4052 .4090 .4052 .4052 .4090 .4090 .4443 .4443 .4483 .4443 .4483 .4483 .4840 .4840 .4880 .4840 .4880 .4880 Critical Critical Values Values Critical Values cc c Level Level of ofConfidence Confidence Level of Confidence 0.80 0.80 0.80 0.90 0.90 0.90 0.95 0.95 0.95 0.99 0.99 0.99 zzcc zc 1.28 1.28 1.28 1.645 1.645 1.645 1.96 1.96 1.96 2.575 2.575 2.575 cc c −z −zcc−z c zz==0z0= 0 zzcc z c zz z Copyright © 2012 Pearson Education, Inc. .02 .02.02 .0003 .0003 .0003 .0005 .0005 .0005 .0006 .0006 .0006 .0009 .0009 .0009 .0013 .0013 .0013 .0018 .0018 .0018 .0024 .0024 .0024 .0033 .0033 .0033 .0044 .0044 .0044 .0059 .0059 .0059 .0078 .0078 .0078 .0102 .0102 .0102 .0132 .0132 .0132 .0170 .0170 .0170 .0217 .0217 .0217 .0274 .0274 .0274 .0344 .0344 .0344 .0427 .0427 .0427 .0526 .0526 .0526 .0643 .0643 .0643 .0778 .0778 .0778 .0934 .0934 .0934 .1112 .1112 .1112 .1314 .1314 .1314 .1539 .1539 .1539 .1788 .1788 .1788 .2061 .2061 .2061 .2358 .2358 .2358 .2676 .2676 .2676 .3015 .3015 .3015 .3372 .3372 .3372 .3745 .3745 .3745 .4129 .4129 .4129 .4522 .4522 .4522 .4920 .4920 .4920 .01 .01.01 .0003 .0003 .0003 .0005 .0005 .0005 .0007 .0007 .0007 .0009 .0009 .0009 .0013 .0013 .0013 .0018 .0018 .0018 .0025 .0025 .0025 .0034 .0034 .0034 .0045 .0045 .0045 .0060 .0060 .0060 .0080 .0080 .0080 .0104 .0104 .0104 .0136 .0136 .0136 .0174 .0174 .0174 .0222 .0222 .0222 .0281 .0281 .0281 .0351 .0351 .0351 .0436 .0436 .0436 .0537 .0537 .0537 .0655 .0655 .0655 .0793 .0793 .0793 .0951 .0951 .0951 .1131 .1131 .1131 .1335 .1335 .1335 .1562 .1562 .1562 .1814 .1814 .1814 .2090 .2090 .2090 .2389 .2389 .2389 .2709 .2709 .2709 .3050 .3050 .3050 .3409 .3409 .3409 .3783 .3783 .3783 .4168 .4168 .4168 .4562 .4562 .4562 .4960 .4960 .4960 .00 .00.00 .0003 .0003 .0003 .0005 .0005 .0005 .0007 .0007 .0007 .0010 .0010 .0010 .0013 .0013 .0013 .0019 .0019 .0019 .0026 .0026 .0026 .0035 .0035 .0035 .0047 .0047 .0047 .0062 .0062 .0062 .0082 .0082 .0082 .0107 .0107 .0107 .0139 .0139 .0139 .0179 .0179 .0179 .0228 .0228 .0228 .0287 .0287 .0287 .0359 .0359 .0359 .0446 .0446 .0446 .0548 .0548 .0548 .0668 .0668 .0668 .0808 .0808 .0808 .0968 .0968 .0968 .1151 .1151 .1151 .1357 .1357 .1357 .1587 .1587 .1587 .1841 .1841 .1841 .2119 .2119 .2119 .2420 .2420 .2420 .2743 .2743 .2743 .3085 .3085 .3085 .3446 .3446 .3446 .3821 .3821 .3821 .4207 .4207 .4207 .4602 .4602 .4602 .5000 .5000 .5000 zz z 0.0 0.00.0 0.1 0.10.1 0.20.2 0.2 0.3 0.30.3 0.40.4 0.4 0.5 0.50.5 0.60.6 0.6 0.7 0.70.7 0.80.8 0.8 0.9 0.90.9 1.01.0 1.0 1.1 1.11.1 1.21.2 1.2 1.3 1.31.3 1.41.4 1.4 1.5 1.51.5 1.61.6 1.6 1.7 1.71.7 1.81.8 1.8 1.9 1.91.9 2.02.0 2.0 2.1 2.12.1 2.22.2 2.2 2.3 2.32.3 2.42.4 2.4 2.5 2.52.5 2.62.6 2.6 2.7 2.72.7 2.82.8 2.8 2.9 2.92.9 3.03.0 3.0 3.1 3.13.1 3.23.2 3.2 3.3 3.33.3 3.4 3.43.4 .00 .00.00 .5000 .5000 .5000 .5398 .5398 .5398 .5793 .5793 .5793 .6179 .6179 .6179 .6554 .6554 .6554 .6915 .6915 .6915 .7257 .7257 .7257 .7580 .7580 .7580 .7881 .7881 .7881 .8159 .8159 .8159 .8413 .8413 .8413 .8643 .8643 .8643 .8849 .8849 .8849 .9032 .9032 .9032 .9192 .9192 .9192 .9332 .9332 .9332 .9452 .9452 .9452 .9554 .9554 .9554 .9641 .9641 .9641 .9713 .9713 .9713 .9772 .9772 .9772 .9821 .9821 .9821 .9861 .9861 .9861 .9893 .9893 .9893 .9918 .9918 .9918 .9938 .9938 .9938 .9953 .9953 .9953 .9965 .9965 .9965 .9974 .9974 .9974 .9981 .9981 .9981 .9987 .9987 .9987 .9990 .9990 .9990 .9993 .9993 .9993 .9995 .9995 .9995 .9997 .9997 .9997 .01 .01.01 .5040 .5040 .5040 .5438 .5438 .5438 .5832 .5832 .5832 .6217 .6217 .6217 .6591 .6591 .6591 .6950 .6950 .6950 .7291 .7291 .7291 .7611 .7611 .7611 .7910 .7910 .7910 .8186 .8186 .8186 .8438 .8438 .8438 .8665 .8665 .8665 .8869 .8869 .8869 .9049 .9049 .9049 .9207 .9207 .9207 .9345 .9345 .9345 .9463 .9463 .9463 .9564 .9564 .9564 .9649 .9649 .9649 .9719 .9719 .9719 .9778 .9778 .9778 .9826 .9826 .9826 .9864 .9864 .9864 .9896 .9896 .9896 .9920 .9920 .9920 .9940 .9940 .9940 .9955 .9955 .9955 .9966 .9966 .9966 .9975 .9975 .9975 .9982 .9982 .9982 .9987 .9987 .9987 .9991 .9991 .9991 .9993 .9993 .9993 .9995 .9995 .9995 .9997 .9997 .9997 tt t c-confidence c-confidence interval interval c-confidence interval .02 .02.02 .5080 .5080 .5080 .5478 .5478 .5478 .5871 .5871 .5871 .6255 .6255 .6255 .6628 .6628 .6628 .6985 .6985 .6985 .7324 .7324 .7324 .7642 .7642 .7642 .7939 .7939 .7939 .8212 .8212 .8212 .8461 .8461 .8461 .8686 .8686 .8686 .8888 .8888 .8888 .9066 .9066 .9066 .9222 .9222 .9222 .9357 .9357 .9357 .9474 .9474 .9474 .9573 .9573 .9573 .9656 .9656 .9656 .9726 .9726 .9726 .9783 .9783 .9783 .9830 .9830 .9830 .9868 .9868 .9868 .9898 .9898 .9898 .9922 .9922 .9922 .9941 .9941 .9941 .9956 .9956 .9956 .9967 .9967 .9967 .9976 .9976 .9976 .9982 .9982 .9982 .9987 .9987 .9987 .9991 .9991 .9991 .9994 .9994 .9994 .9995 .9995 .9995 .9997 .9997 .9997 .03 .03.03 .5120 .5120 .5120 .5517 .5517 .5517 .5910 .5910 .5910 .6293 .6293 .6293 .6664 .6664 .6664 .7019 .7019 .7019 .7357 .7357 .7357 .7673 .7673 .7673 .7967 .7967 .7967 .8238 .8238 .8238 .8485 .8485 .8485 .8708 .8708 .8708 .8907 .8907 .8907 .9082 .9082 .9082 .9236 .9236 .9236 .9370 .9370 .9370 .9484 .9484 .9484 .9582 .9582 .9582 .9664 .9664 .9664 .9732 .9732 .9732 .9788 .9788 .9788 .9834 .9834 .9834 .9871 .9871 .9871 .9901 .9901 .9901 .9925 .9925 .9925 .9943 .9943 .9943 .9957 .9957 .9957 .9968 .9968 .9968 .9977 .9977 .9977 .9983 .9983 .9983 .9988 .9988 .9988 .9991 .9991 .9991 .9994 .9994 .9994 .9996 .9996 .9996 .9997 .9997 .9997 .04 .04.04 .5160 .5160 .5160 .5557 .5557 .5557 .5948 .5948 .5948 .6331 .6331 .6331 .6700 .6700 .6700 .7054 .7054 .7054 .7389 .7389 .7389 .7704 .7704 .7704 .7995 .7995 .7995 .8264 .8264 .8264 .8508 .8508 .8508 .8729 .8729 .8729 .8925 .8925 .8925 .9099 .9099 .9099 .9251 .9251 .9251 .9382 .9382 .9382 .9495 .9495 .9495 .9591 .9591 .9591 .9671 .9671 .9671 .9738 .9738 .9738 .9793 .9793 .9793 .9838 .9838 .9838 .9875 .9875 .9875 .9904 .9904 .9904 .9927 .9927 .9927 .9945 .9945 .9945 .9959 .9959 .9959 .9969 .9969 .9969 .9977 .9977 .9977 .9984 .9984 .9984 .9988 .9988 .9988 .9992 .9992 .9992 .9994 .9994 .9994 .9996 .9996 .9996 .9997 .9997 .9997 .05 .05.05 .5199 .5199 .5199 .5596 .5596 .5596 .5987 .5987 .5987 .6368 .6368 .6368 .6736 .6736 .6736 .7088 .7088 .7088 .7422 .7422 .7422 .7734 .7734 .7734 .8023 .8023 .8023 .8289 .8289 .8289 .8531 .8531 .8531 .8749 .8749 .8749 .8944 .8944 .8944 .9115 .9115 .9115 .9265 .9265 .9265 .9394 .9394 .9394 .9505 .9505 .9505 .9599 .9599 .9599 .9678 .9678 .9678 .9744 .9744 .9744 .9798 .9798 .9798 .9842 .9842 .9842 .9878 .9878 .9878 .9906 .9906 .9906 .9929 .9929 .9929 .9946 .9946 .9946 .9960 .9960 .9960 .9970 .9970 .9970 .9978 .9978 .9978 .9984 .9984 .9984 .9989 .9989 .9989 .9992 .9992 .9992 .9994 .9994 .9994 .9996 .9996 .9996 .9997 .9997 .9997 .06 .06.06 .5239 .5239 .5239 .5636 .5636 .5636 .6026 .6026 .6026 .6406 .6406 .6406 .6772 .6772 .6772 .7123 .7123 .7123 .7454 .7454 .7454 .7764 .7764 .7764 .8051 .8051 .8051 .8315 .8315 .8315 .8554 .8554 .8554 .8770 .8770 .8770 .8962 .8962 .8962 .9131 .9131 .9131 .9279 .9279 .9279 .9406 .9406 .9406 .9515 .9515 .9515 .9608 .9608 .9608 .9686 .9686 .9686 .9750 .9750 .9750 .9803 .9803 .9803 .9846 .9846 .9846 .9881 .9881 .9881 .9909 .9909 .9909 .9931 .9931 .9931 .9948 .9948 .9948 .9961 .9961 .9961 .9971 .9971 .9971 .9979 .9979 .9979 .9985 .9985 .9985 .9989 .9989 .9989 .9992 .9992 .9992 .9994 .9994 .9994 .9996 .9996 .9996 .9997 .9997 .9997 .07 .07.07 .5279 .5279 .5279 .5675 .5675 .5675 .6064 .6064 .6064 .6443 .6443 .6443 .6808 .6808 .6808 .7157 .7157 .7157 .7486 .7486 .7486 .7794 .7794 .7794 .8078 .8078 .8078 .8340 .8340 .8340 .8577 .8577 .8577 .8790 .8790 .8790 .8980 .8980 .8980 .9147 .9147 .9147 .9292 .9292 .9292 .9418 .9418 .9418 .9525 .9525 .9525 .9616 .9616 .9616 .9693 .9693 .9693 .9756 .9756 .9756 .9808 .9808 .9808 .9850 .9850 .9850 .9884 .9884 .9884 .9911 .9911 .9911 .9932 .9932 .9932 .9949 .9949 .9949 .9962 .9962 .9962 .9972 .9972 .9972 .9979 .9979 .9979 .9985 .9985 .9985 .9989 .9989 .9989 .9992 .9992 .9992 .9995 .9995 .9995 .9996 .9996 .9996 .9997 .9997 .9997 .08 .08.08 .5319 .5319 .5319 .5714 .5714 .5714 .6103 .6103 .6103 .6480 .6480 .6480 .6844 .6844 .6844 .7190 .7190 .7190 .7517 .7517 .7517 .7823 .7823 .7823 .8106 .8106 .8106 .8365 .8365 .8365 .8599 .8599 .8599 .8810 .8810 .8810 .8997 .8997 .8997 .9162 .9162 .9162 .9306 .9306 .9306 .9429 .9429 .9429 .9535 .9535 .9535 .9625 .9625 .9625 .9699 .9699 .9699 .9761 .9761 .9761 .9812 .9812 .9812 .9854 .9854 .9854 .9887 .9887 .9887 .9913 .9913 .9913 .9934 .9934 .9934 .9951 .9951 .9951 .9963 .9963 .9963 .9973 .9973 .9973 .9980 .9980 .9980 .9986 .9986 .9986 .9990 .9990 .9990 .9993 .9993 .9993 .9995 .9995 .9995 .9996 .9996 .9996 .9997 .9997 .9997 .09 .09.09 .5359 .5359 .5359 .5753 .5753 .5753 .6141 .6141 .6141 .6517 .6517 .6517 .6879 .6879 .6879 .7224 .7224 .7224 .7549 .7549 .7549 .7852 .7852 .7852 .8133 .8133 .8133 .8389 .8389 .8389 .8621 .8621 .8621 .8830 .8830 .8830 .9015 .9015 .9015 .9177 .9177 .9177 .9319 .9319 .9319 .9441 .9441 .9441 .9545 .9545 .9545 .9633 .9633 .9633 .9706 .9706 .9706 .9767 .9767 .9767 .9817 .9817 .9817 .9857 .9857 .9857 .9890 .9890 .9890 .9916 .9916 .9916 .9936 .9936 .9936 .9952 .9952 .9952 .9964 .9964 .9964 .9974 .9974 .9974 .9981 .9981 .9981 .9986 .9986 .9986 .9990 .9990 .9990 .9993 .9993 .9993 .9995 .9995 .9995 .9997 .9997 .9997 .9998 .9998 .9998 d.f. d.f.d.f. 11 1 22 2 33 3 44 4 55 5 66 6 77 7 88 8 99 9 10 1010 11 1111 12 1212 13 1313 14 1414 15 1515 16 1616 17 1717 18 1818 19 1919 20 2020 21 2121 22 2222 23 2323 24 2424 25 2525 26 2626 27 2727 28 2828 29 2929 q qq tt t αα α tt −t −t −t Left-tailed Left-tailed test test Left-tailed test Level Level Level of of of confidence, confidence, 0.50 cc c 0.50 confidence, 0.50 One One tail, tail, 0.25 0.25 AA A One tail, 0.25 Two Two tails, tails, 0.50 AA A 0.50 Two tails, 0.50 1.000 1.000 1.000 .816 .816 .816 .765 .765 .765 .741 .741 .741 .727 .727 .727 .718 .718 .718 .711 .711 .711 .706 .706 .706 .703 .703 .703 .700 .700 .700 .697 .697 .697 .695 .695 .695 .694 .694 .694 .692 .692 .692 .691 .691 .691 .690 .690 .690 .689 .689 .689 .688 .688 .688 .688 .688 .688 .687 .687 .687 .686 .686 .686 .686 .686 .686 .685 .685 .685 .685 .685 .685 .684 .684 .684 .684 .684 .684 .684 .684 .684 .683 .683 .683 .683 .683 .683 .674 .674 .674 0.80 0.80 0.80 0.10 0.10 0.10 0.20 0.20 0.20 3.078 3.078 3.078 1.886 1.886 1.886 1.638 1.638 1.638 1.533 1.533 1.533 1.476 1.476 1.476 1.440 1.440 1.440 1.415 1.415 1.415 1.397 1.397 1.397 1.383 1.383 1.383 1.372 1.372 1.372 1.363 1.363 1.363 1.356 1.356 1.356 1.350 1.350 1.350 1.345 1.345 1.345 1.341 1.341 1.341 1.337 1.337 1.337 1.333 1.333 1.333 1.330 1.330 1.330 1.328 1.328 1.328 1.325 1.325 1.325 1.323 1.323 1.323 1.321 1.321 1.321 1.319 1.319 1.319 1.318 1.318 1.318 1.316 1.316 1.316 1.315 1.315 1.315 1.314 1.314 1.314 1.313 1.313 1.313 1.311 1.311 1.311 1.282 1.282 1.282 t αα α tt t Right-tailed Right-tailed test test Right-tailed test 0.90 0.90 0.95 0.98 0.99 0.90 0.95 0.95 0.98 0.98 0.99 0.99 0.05 0.05 0.025 0.01 0.005 0.05 0.025 0.025 0.01 0.01 0.005 0.005 0.10 0.10 0.05 0.02 0.01 0.10 0.05 0.05 0.02 0.02 0.01 0.01 6.31412.706 12.70631.821 31.82163.657 63.657 6.314 6.314 12.706 31.821 63.657 2.920 2.920 4.303 6.965 9.925 2.920 4.303 4.303 6.965 6.965 9.925 9.925 2.353 2.353 3.182 4.541 5.841 2.353 3.182 3.182 4.541 4.541 5.841 5.841 2.132 2.132 2.776 3.747 4.604 2.132 2.776 2.776 3.747 3.747 4.604 4.604 2.015 2.015 2.571 3.365 4.032 2.015 2.571 2.571 3.365 3.365 4.032 4.032 1.943 1.943 2.447 3.143 3.707 1.943 2.447 2.447 3.143 3.143 3.707 3.707 1.895 1.895 2.365 2.998 3.499 1.895 2.365 2.365 2.998 2.998 3.499 3.499 1.860 1.860 2.306 2.896 3.355 1.860 2.306 2.306 2.896 2.896 3.355 3.355 1.833 1.833 2.262 2.821 3.250 1.833 2.262 2.262 2.821 2.821 3.250 3.250 1.812 1.812 2.228 2.764 3.169 1.812 2.228 2.228 2.764 2.764 3.169 3.169 1.796 1.796 2.201 2.718 3.106 1.796 2.201 2.201 2.718 2.718 3.106 3.106 1.782 1.782 2.179 2.681 3.055 1.782 2.179 2.179 2.681 2.681 3.055 3.055 1.771 1.771 2.160 2.650 3.012 1.771 2.160 2.160 2.650 2.650 3.012 3.012 1.761 1.761 2.145 2.624 2.977 1.761 2.145 2.145 2.624 2.624 2.977 2.977 1.753 1.753 2.131 2.602 2.947 1.753 2.131 2.131 2.602 2.602 2.947 2.947 1.746 1.746 2.120 2.583 2.921 1.746 2.120 2.120 2.583 2.583 2.921 2.921 1.740 1.740 2.110 2.567 2.898 1.740 2.110 2.110 2.567 2.567 2.898 2.898 1.734 1.734 2.101 2.552 2.878 1.734 2.101 2.101 2.552 2.552 2.878 2.878 1.729 1.729 2.093 2.539 2.861 1.729 2.093 2.093 2.539 2.539 2.861 2.861 1.725 1.725 2.086 2.528 2.845 1.725 2.086 2.086 2.528 2.528 2.845 2.845 1.721 1.721 2.080 2.518 2.831 1.721 2.080 2.080 2.518 2.518 2.831 2.831 1.717 1.717 2.074 2.508 2.819 1.717 2.074 2.074 2.508 2.508 2.819 2.819 1.714 1.714 2.069 2.500 2.807 1.714 2.069 2.069 2.500 2.500 2.807 2.807 1.711 1.711 2.064 2.492 2.797 1.711 2.064 2.064 2.492 2.492 2.797 2.797 1.708 1.708 2.060 2.485 2.787 1.708 2.060 2.060 2.485 2.485 2.787 2.787 1.706 1.706 2.056 2.479 2.779 1.706 2.056 2.056 2.479 2.479 2.779 2.779 1.703 1.703 2.052 2.473 2.771 1.703 2.052 2.052 2.473 2.473 2.771 2.771 1.701 1.701 2.048 2.467 2.763 1.701 2.048 2.048 2.467 2.467 2.763 2.763 1.699 1.699 2.045 2.462 2.756 1.699 2.045 2.045 2.462 2.462 2.756 2.756 1.645 1.645 1.960 2.326 2.576 1.645 1.960 1.960 2.326 2.326 2.576 2.576 tt t −t −t −t 11 1 αα α 22 2 tt t Two-tailed Two-tailed test test Two-tailed test tt 11 1 αα α 22 2 αα α t χχ22 χ 2 χχ22 χ 2 Degrees Degrees of of of Degrees freedom freedom freedom 11 1 22 2 33 3 44 4 55 5 66 6 77 7 88 8 99 9 1010 10 1111 11 1212 12 1313 13 1414 14 1515 15 1616 16 1717 17 1818 18 1919 19 2020 20 2121 21 2222 22 2323 23 2424 24 2525 25 2626 26 2727 27 2828 28 2929 29 3030 30 4040 40 5050 50 6060 60 7070 70 8080 80 9090 90 100 100 100 χχ22 χ 2 LL Right tail Right tailtail Right 11 1 αα α 22 2 L χχ22 χ 2 RR χχ22 χ 2 R Two tails Two tails Two tails AA A 0.995 0.99 0.975 0.95 0.90 0.10 0.05 0.025 0.01 0.005 0.995 0.99 0.99 0.975 0.975 0.95 0.95 0.90 0.90 0.10 0.10 0.05 0.05 0.025 0.025 0.01 0.01 0.005 0.005 0.995 —— —— 0.001 0.001 0.004 0.016 2.706 3.841 5.024 6.635 7.879 0.001 0.004 0.004 0.016 0.016 2.706 2.706 3.841 3.841 5.024 5.024 6.635 6.635 7.879 7.879 — — 0.010 0.020 0.051 0.103 0.211 4.605 5.991 7.378 9.210 10.597 0.010 0.020 0.020 0.051 0.051 0.103 0.103 0.211 0.211 4.605 4.605 5.991 5.991 7.378 7.378 9.210 9.21010.597 10.597 0.010 0.072 0.115 0.216 0.352 0.584 6.251 7.815 9.348 11.345 12.838 0.072 0.115 0.115 0.216 0.216 0.352 0.352 0.584 0.584 6.251 6.251 7.815 7.815 9.348 9.34811.345 11.34512.838 12.838 0.072 0.207 0.297 0.484 0.711 1.064 7.779 9.488 11.143 13.277 14.860 0.207 0.297 0.297 0.484 0.484 0.711 0.711 1.064 1.064 7.779 7.779 9.488 9.48811.143 11.14313.277 13.27714.860 14.860 0.207 0.412 0.554 0.831 1.145 1.610 9.236 11.071 12.833 15.086 16.750 0.412 0.554 0.554 0.831 0.831 1.145 1.145 1.610 1.610 9.236 9.23611.071 11.07112.833 12.83315.086 15.08616.750 16.750 0.412 0.676 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.812 18.548 0.676 0.872 0.872 1.237 1.237 1.635 1.635 2.204 2.20410.645 10.64512.592 12.59214.449 14.44916.812 16.81218.548 18.548 0.676 0.989 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.475 20.278 0.989 1.239 1.239 1.690 1.690 2.167 2.167 2.833 2.83312.017 12.01714.067 14.06716.013 16.01318.475 18.47520.278 20.278 0.989 1.344 1.646 2.180 2.733 3.490 13.362 15.507 17.535 20.090 21.955 1.344 1.646 1.646 2.180 2.180 2.733 2.733 3.490 3.49013.362 13.36215.507 15.50717.535 17.53520.090 20.09021.955 21.955 1.344 1.735 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666 23.589 1.735 2.088 2.088 2.700 2.700 3.325 3.325 4.168 4.16814.684 14.68416.919 16.91919.023 19.02321.666 21.66623.589 23.589 1.735 2.156 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 25.188 2.156 2.558 2.558 3.247 3.247 3.940 3.940 4.865 4.86515.987 15.98718.307 18.30720.483 20.48323.209 23.20925.188 25.188 2.156 2.603 3.053 3.816 4.575 5.578 17.275 19.675 21.920 24.725 26.757 2.603 3.053 3.053 3.816 3.816 4.575 4.575 5.578 5.57817.275 17.27519.675 19.67521.920 21.92024.725 24.72526.757 26.757 2.603 3.074 3.571 4.404 5.226 6.304 18.549 21.026 23.337 26.217 28.299 3.074 3.571 3.571 4.404 4.404 5.226 5.226 6.304 6.30418.549 18.54921.026 21.02623.337 23.33726.217 26.21728.299 28.299 3.074 3.565 4.107 5.009 5.892 7.042 19.812 22.362 24.736 27.688 29.819 3.565 4.107 4.107 5.009 5.009 5.892 5.892 7.042 7.04219.812 19.81222.362 22.36224.736 24.73627.688 27.68829.819 29.819 3.565 4.075 4.660 5.629 6.571 7.790 21.064 23.685 26.119 29.141 31.319 4.075 4.660 4.660 5.629 5.629 6.571 6.571 7.790 7.79021.064 21.06423.685 23.68526.119 26.11929.141 29.14131.319 31.319 4.075 4.601 5.229 6.262 7.261 8.547 22.307 24.996 27.488 30.578 32.801 4.601 5.229 5.229 6.262 6.262 7.261 7.261 8.547 8.54722.307 22.30724.996 24.99627.488 27.48830.578 30.57832.801 32.801 4.601 5.142 5.812 6.908 7.962 9.312 23.542 26.296 28.845 32.000 34.267 5.142 5.812 5.812 6.908 6.908 7.962 7.962 9.312 9.31223.542 23.54226.296 26.29628.845 28.84532.000 32.00034.267 34.267 5.142 5.697 6.408 7.564 8.672 10.085 24.769 27.587 30.191 33.409 35.718 5.697 6.408 6.408 7.564 7.564 8.672 8.67210.085 10.08524.769 24.76927.587 27.58730.191 30.19133.409 33.40935.718 35.718 5.697 6.265 7.015 8.231 9.390 10.865 25.989 28.869 31.526 34.805 37.156 6.265 7.015 7.015 8.231 8.231 9.390 9.39010.865 10.86525.989 25.98928.869 28.86931.526 31.52634.805 34.80537.156 37.156 6.265 6.844 7.633 8.907 10.117 11.651 27.204 30.144 32.852 36.191 38.582 6.844 7.633 7.633 8.907 8.90710.117 10.11711.651 11.65127.204 27.20430.144 30.14432.852 32.85236.191 36.19138.582 38.582 6.844 7.434 8.260 9.591 10.851 12.443 28.412 31.410 34.170 37.566 39.997 7.434 8.260 8.260 9.591 9.59110.851 10.85112.443 12.44328.412 28.41231.410 31.41034.170 34.17037.566 37.56639.997 39.997 7.434 8.034 8.897 10.283 11.591 13.240 29.615 32.671 35.479 38.932 41.401 8.034 8.897 8.89710.283 10.28311.591 11.59113.240 13.24029.615 29.61532.671 32.67135.479 35.47938.932 38.93241.401 41.401 8.034 8.643 9.542 10.982 12.338 14.042 30.813 33.924 36.781 40.289 42.796 8.643 9.542 9.54210.982 10.98212.338 12.33814.042 14.04230.813 30.81333.924 33.92436.781 36.78140.289 40.28942.796 42.796 8.643 9.260 9.26010.196 10.196 10.19611.689 11.689 11.68913.091 13.091 13.09114.848 14.848 14.84832.007 32.007 32.00735.172 35.172 35.17238.076 38.076 38.07641.638 41.638 41.63844.181 44.181 44.181 9.260 9.886 10.856 12.401 13.848 15.659 33.196 36.415 39.364 42.980 45.559 9.88610.856 10.85612.401 12.40113.848 13.84815.659 15.65933.196 33.19636.415 36.41539.364 39.36442.980 42.98045.559 45.559 9.886 10.520 11.524 13.120 14.611 16.473 34.382 37.652 40.646 44.314 46.928 10.52011.524 11.52413.120 13.12014.611 14.61116.473 16.47334.382 34.38237.652 37.65240.646 40.64644.314 44.31446.928 46.928 10.520 11.160 12.198 13.844 15.379 17.292 35.563 38.885 41.923 45.642 48.290 11.16012.198 12.19813.844 13.84415.379 15.37917.292 17.29235.563 35.56338.885 38.88541.923 41.92345.642 45.64248.290 48.290 11.160 11.808 12.879 14.573 16.151 18.114 36.741 40.113 43.194 46.963 49.645 11.80812.879 12.87914.573 14.57316.151 16.15118.114 18.11436.741 36.74140.113 40.11343.194 43.19446.963 46.96349.645 49.645 11.808 12.461 13.565 15.308 16.928 18.939 37.916 41.337 44.461 48.278 50.993 12.46113.565 13.56515.308 15.30816.928 16.92818.939 18.93937.916 37.91641.337 41.33744.461 44.46148.278 48.27850.993 50.993 12.461 13.121 13.12114.257 14.257 14.25716.047 16.047 16.04717.708 17.708 17.70819.768 19.768 19.76839.087 39.087 39.08742.557 42.557 42.55745.722 45.722 45.72249.588 49.588 49.58852.336 52.336 52.336 13.121 13.787 14.954 16.791 18.493 20.599 40.256 43.773 46.979 50.892 53.672 13.78714.954 14.95416.791 16.79118.493 18.49320.599 20.59940.256 40.25643.773 43.77346.979 46.97950.892 50.89253.672 53.672 13.787 20.707 22.164 24.433 26.509 29.051 51.805 55.758 59.342 63.691 66.766 20.70722.164 22.16424.433 24.43326.509 26.50929.051 29.05151.805 51.80555.758 55.75859.342 59.34263.691 63.69166.766 66.766 20.707 27.991 29.707 32.357 34.764 37.689 63.167 67.505 71.420 76.154 79.490 27.99129.707 29.70732.357 32.35734.764 34.76437.689 37.68963.167 63.16767.505 67.50571.420 71.42076.154 76.15479.490 79.490 27.991 35.534 37.485 40.482 43.188 46.459 74.397 79.082 83.298 88.379 91.952 35.53437.485 37.48540.482 40.48243.188 43.18846.459 46.45974.397 74.39779.082 79.08283.298 83.29888.379 88.37991.952 91.952 35.534 43.275 45.442 48.758 51.739 55.329 85.527 90.531 95.023 100.425 104.215 43.27545.442 45.44248.758 48.75851.739 51.73955.329 55.32985.527 85.52790.531 90.53195.023 95.023 100.425 104.215 43.275 100.425 104.215 51.172 51.17253.540 53.540 53.54057.153 57.153 57.15360.391 60.391 60.39164.278 64.278 64.27896.578 96.578 96.578 101.879 101.879 106.629 106.629 112.329 112.329 116.321 116.321 51.172 101.879 106.629 112.329 116.321 59.196 59.19661.754 61.754 61.75465.647 65.647 65.64769.126 69.126 69.12673.291 73.291 73.291 107.565 107.565 113.145 113.145 118.136 118.136 124.116 124.116 128.299 128.299 59.196 107.565 113.145 118.136 124.116 128.299 67.328 70.065 74.222 77.929 82.358 118.498 124.342 129.561 135.807 140.169 67.32870.065 70.06574.222 74.22277.929 77.92982.358 82.358 118.498 124.342 129.561 135.807 140.169 67.328 118.498 124.342 129.561 135.807 140.169 41831S4_INS p5-8 AM 11 1 41831S4_INS p5-8 11/8/07 10:03 AMAMPage Page 41831S4_INS p5-811/8/07 11/8/0710:03 10:03 Page Table Table 44— 4— Standard Standard Normal Normal Distribution Distribution (continued) (continued) Table — Standard Normal Distribution (continued) Table Table 44— — Standard Standard Normal Normal Distribution Distribution Table 4— Standard Normal Distribution Table Table Table 6— 6— 6— Chi-Square Chi-Square Chi-Square Distribution Distribution Distribution Area Area Area Area Area Area 11 1 αα α 22 2 zz z zz z .09 .09.09 .0002 .0002 .0002 .0003 .0003 .0003 .0005 .0005 .0005 .0007 .0007 .0007 .0010 .0010 .0010 .0014 .0014 .0014 .0019 .0019 .0019 .0026 .0026 .0026 .0036 .0036 .0036 .0048 .0048 .0048 .0064 .0064 .0064 .0084 .0084 .0084 .0110 .0110 .0110 .0143 .0143 .0143 .0183 .0183 .0183 .0233 .0233 .0233 .0294 .0294 .0294 .0367 .0367 .0367 .0455 .0455 .0455 .0559 .0559 .0559 .0681 .0681 .0681 .0823 .0823 .0823 .0985 .0985 .0985 .1170 .1170 .1170 .1379 .1379 .1379 .1611 .1611 .1611 .1867 .1867 .1867 .2148 .2148 .2148 .2451 .2451 .2451 .2776 .2776 .2776 .3121 .3121 .3121 .3483 .3483 .3483 .3859 .3859 .3859 .4247 .4247 .4247 .4641 .4641 .4641 −t −t −t 00 0 zz z zz z 00 0 zz z 3.4 � � 3.4 �3.4 3.3 � � 3.3 �3.3 3.2 � � 3.2 �3.2 3.1 � � 3.1 �3.1 3.0 � � 3.0 �3.0 2.9 � � 2.9 �2.9 2.8 � � 2.8 �2.8 2.7 � � 2.7 �2.7 2.6 � � 2.6 �2.6 2.5 � � 2.5 �2.5 2.4 � � 2.4 �2.4 2.3 � � 2.3 �2.3 2.2 � � 2.2 �2.2 2.1 � � 2.1 �2.1 2.0 � � 2.0 �2.0 1.9 � � 1.9 �1.9 1.8 � � 1.8 �1.8 1.7 � � 1.7 �1.7 1.6 � � 1.6 �1.6 1.5 � � 1.5 �1.5 1.4 � � 1.4 �1.4 1.3 � � 1.3 �1.3 1.2 � � 1.2 �1.2 1.1 � � 1.1 �1.1 1.0 � � 1.0 �1.0 0.9 � � 0.9 �0.9 0.8 � � 0.8 �0.8 0.7 � � 0.7 �0.7 0.6 � � 0.6 �0.6 0.5 � � 0.5 �0.5 0.4 � � 0.4 �0.4 0.3 � � 0.3 �0.3 0.2 � � 0.2 �0.2 0.1 � � 0.1 �0.1 �0.0 0.0 � � 0.0 Table Table Table 5— 5— 5— t-Distribution t-Distribution t-Distribution .08 .08.08 .0003 .0003 .0003 .0004 .0004 .0004 .0005 .0005 .0005 .0007 .0007 .0007 .0010 .0010 .0010 .0014 .0014 .0014 .0020 .0020 .0020 .0027 .0027 .0027 .0037 .0037 .0037 .0049 .0049 .0049 .0066 .0066 .0066 .0087 .0087 .0087 .0113 .0113 .0113 .0146 .0146 .0146 .0188 .0188 .0188 .0239 .0239 .0239 .0301 .0301 .0301 .0375 .0375 .0375 .0465 .0465 .0465 .0571 .0571 .0571 .0694 .0694 .0694 .0838 .0838 .0838 .1003 .1003 .1003 .1190 .1190 .1190 .1401 .1401 .1401 .1635 .1635 .1635 .1894 .1894 .1894 .2177 .2177 .2177 .2483 .2483 .2483 .2810 .2810 .2810 .3156 .3156 .3156 .3520 .3520 .3520 .3897 .3897 .3897 .4286 .4286 .4286 .4681 .4681 .4681 .07 .07.07 .0003 .0003 .0003 .0004 .0004 .0004 .0005 .0005 .0005 .0008 .0008 .0008 .0011 .0011 .0011 .0015 .0015 .0015 .0021 .0021 .0021 .0028 .0028 .0028 .0038 .0038 .0038 .0051 .0051 .0051 .0068 .0068 .0068 .0089 .0089 .0089 .0116 .0116 .0116 .0150 .0150 .0150 .0192 .0192 .0192 .0244 .0244 .0244 .0307 .0307 .0307 .0384 .0384 .0384 .0475 .0475 .0475 .0582 .0582 .0582 .0708 .0708 .0708 .0853 .0853 .0853 .1020 .1020 .1020 .1210 .1210 .1210 .1423 .1423 .1423 .1660 .1660 .1660 .1922 .1922 .1922 .2206 .2206 .2206 .2514 .2514 .2514 .2843 .2843 .2843 .3192 .3192 .3192 .3557 .3557 .3557 .3936 .3936 .3936 .4325 .4325 .4325 .4721 .4721 .4721 .06 .06.06 .0003 .0003 .0003 .0004 .0004 .0004 .0006 .0006 .0006 .0008 .0008 .0008 .0011 .0011 .0011 .0015 .0015 .0015 .0021 .0021 .0021 .0029 .0029 .0029 .0039 .0039 .0039 .0052 .0052 .0052 .0069 .0069 .0069 .0091 .0091 .0091 .0119 .0119 .0119 .0154 .0154 .0154 .0197 .0197 .0197 .0250 .0250 .0250 .0314 .0314 .0314 .0392 .0392 .0392 .0485 .0485 .0485 .0594 .0594 .0594 .0721 .0721 .0721 .0869 .0869 .0869 .1038 .1038 .1038 .1230 .1230 .1230 .1446 .1446 .1446 .1685 .1685 .1685 .1949 .1949 .1949 .2236 .2236 .2236 .2546 .2546 .2546 .2877 .2877 .2877 .3228 .3228 .3228 .3594 .3594 .3594 .3974 .3974 .3974 .4364 .4364 .4364 .4761 .4761 .4761 .05 .05.05 .0003 .0003 .0003 .0004 .0004 .0004 .0006 .0006 .0006 .0008 .0008 .0008 .0011 .0011 .0011 .0016 .0016 .0016 .0022 .0022 .0022 .0030 .0030 .0030 .0040 .0040 .0040 .0054 .0054 .0054 .0071 .0071 .0071 .0094 .0094 .0094 .0122 .0122 .0122 .0158 .0158 .0158 .0202 .0202 .0202 .0256 .0256 .0256 .0322 .0322 .0322 .0401 .0401 .0401 .0495 .0495 .0495 .0606 .0606 .0606 .0735 .0735 .0735 .0885 .0885 .0885 .1056 .1056 .1056 .1251 .1251 .1251 .1469 .1469 .1469 .1711 .1711 .1711 .1977 .1977 .1977 .2266 .2266 .2266 .2578 .2578 .2578 .2912 .2912 .2912 .3264 .3264 .3264 .3632 .3632 .3632 .4013 .4013 .4013 .4404 .4404 .4404 .4801 .4801 .4801 .04 .04.04 .03 .03.03 .0003 .0003 .0003 .0003 .0003 .0003 .0004 .0004 .0004 .0004 .0004 .0004 .0006 .0006 .0006 .0006 .0006 .0006 .0008 .0008 .0009 .0008 .0009 .0009 .0012 .0012 .0012 .0012 .0012 .0012 .0016 .0016 .0017 .0016 .0017 .0017 .0023 .0023 .0023 .0023 .0023 .0023 .0031 .0031 .0032 .0031 .0032 .0032 .0041 .0043 .0041 .0041 .0043 .0043 .0055 .0055 .0057 .0055 .0057 .0057 .0073 .0075 .0073 .0073 .0075 .0075 .0096 .0096 .0099 .0096 .0099 .0099 .0125 .0129 .0125 .0125 .0129 .0129 .0162 .0162 .0166 .0162 .0166 .0166 .0207 .0212 .0207 .0207 .0212 .0212 .0262 .0262 .0268 .0262 .0268 .0268 .0329 .0336 .0329 .0329 .0336 .0336 .0409 .0409 .0418 .0409 .0418 .0418 .0505 .0516 .0505 .0505 .0516 .0516 .0618 .0618 .0630 .0618 .0630 .0630 .0749 .0764 .0749 .0749 .0764 .0764 .0901 .0901 .0918 .0901 .0918 .0918 .1075 .1093 .1075 .1075 .1093 .1093 .1271 .1271 .1292 .1271 .1292 .1292 .1492 .1515 .1492 .1492 .1515 .1515 .1736 .1736 .1762 .1736 .1762 .1762 .2005 .2033 .2005 .2005 .2033 .2033 .2296 .2296 .2327 .2296 .2327 .2327 .2611 .2643 .2611 .2611 .2643 .2643 .2946 .2946 .2981 .2946 .2981 .2981 .3300 .3336 .3300 .3300 .3336 .3336 .3669 .3669 .3707 .3669 .3707 .3707 .4052 .4090 .4052 .4052 .4090 .4090 .4443 .4443 .4483 .4443 .4483 .4483 .4840 .4840 .4880 .4840 .4880 .4880 .02 .02.02 .0003 .0003 .0003 .0005 .0005 .0005 .0006 .0006 .0006 .0009 .0009 .0009 .0013 .0013 .0013 .0018 .0018 .0018 .0024 .0024 .0024 .0033 .0033 .0033 .0044 .0044 .0044 .0059 .0059 .0059 .0078 .0078 .0078 .0102 .0102 .0102 .0132 .0132 .0132 .0170 .0170 .0170 .0217 .0217 .0217 .0274 .0274 .0274 .0344 .0344 .0344 .0427 .0427 .0427 .0526 .0526 .0526 .0643 .0643 .0643 .0778 .0778 .0778 .0934 .0934 .0934 .1112 .1112 .1112 .1314 .1314 .1314 .1539 .1539 .1539 .1788 .1788 .1788 .2061 .2061 .2061 .2358 .2358 .2358 .2676 .2676 .2676 .3015 .3015 .3015 .3372 .3372 .3372 .3745 .3745 .3745 .4129 .4129 .4129 .4522 .4522 .4522 .4920 .4920 .4920 .01 .01.01 .0003 .0003 .0003 .0005 .0005 .0005 .0007 .0007 .0007 .0009 .0009 .0009 .0013 .0013 .0013 .0018 .0018 .0018 .0025 .0025 .0025 .0034 .0034 .0034 .0045 .0045 .0045 .0060 .0060 .0060 .0080 .0080 .0080 .0104 .0104 .0104 .0136 .0136 .0136 .0174 .0174 .0174 .0222 .0222 .0222 .0281 .0281 .0281 .0351 .0351 .0351 .0436 .0436 .0436 .0537 .0537 .0537 .0655 .0655 .0655 .0793 .0793 .0793 .0951 .0951 .0951 .1131 .1131 .1131 .1335 .1335 .1335 .1562 .1562 .1562 .1814 .1814 .1814 .2090 .2090 .2090 .2389 .2389 .2389 .2709 .2709 .2709 .3050 .3050 .3050 .3409 .3409 .3409 .3783 .3783 .3783 .4168 .4168 .4168 .4562 .4562 .4562 .4960 .4960 .4960 .00 .00.00 .0003 .0003 .0003 .0005 .0005 .0005 .0007 .0007 .0007 .0010 .0010 .0010 .0013 .0013 .0013 .0019 .0019 .0019 .0026 .0026 .0026 .0035 .0035 .0035 .0047 .0047 .0047 .0062 .0062 .0062 .0082 .0082 .0082 .0107 .0107 .0107 .0139 .0139 .0139 .0179 .0179 .0179 .0228 .0228 .0228 .0287 .0287 .0287 .0359 .0359 .0359 .0446 .0446 .0446 .0548 .0548 .0548 .0668 .0668 .0668 .0808 .0808 .0808 .0968 .0968 .0968 .1151 .1151 .1151 .1357 .1357 .1357 .1587 .1587 .1587 .1841 .1841 .1841 .2119 .2119 .2119 .2420 .2420 .2420 .2743 .2743 .2743 .3085 .3085 .3085 .3446 .3446 .3446 .3821 .3821 .3821 .4207 .4207 .4207 .4602 .4602 .4602 .5000 .5000 .5000 zz z 0.0 0.00.0 0.1 0.10.1 0.20.2 0.2 0.3 0.30.3 0.40.4 0.4 0.5 0.50.5 0.60.6 0.6 0.7 0.70.7 0.80.8 0.8 0.9 0.90.9 1.01.0 1.0 1.1 1.11.1 1.21.2 1.2 1.3 1.31.3 1.41.4 1.4 1.5 1.51.5 1.61.6 1.6 1.7 1.71.7 1.81.8 1.8 1.9 1.91.9 2.02.0 2.0 2.1 2.12.1 2.22.2 2.2 2.3 2.32.3 2.42.4 2.4 2.5 2.52.5 2.62.6 2.6 2.7 2.72.7 2.82.8 2.8 2.9 2.92.9 3.03.0 3.0 3.1 3.13.1 3.23.2 3.2 3.3 3.33.3 3.4 3.43.4 .00 .00.00 .5000 .5000 .5000 .5398 .5398 .5398 .5793 .5793 .5793 .6179 .6179 .6179 .6554 .6554 .6554 .6915 .6915 .6915 .7257 .7257 .7257 .7580 .7580 .7580 .7881 .7881 .7881 .8159 .8159 .8159 .8413 .8413 .8413 .8643 .8643 .8643 .8849 .8849 .8849 .9032 .9032 .9032 .9192 .9192 .9192 .9332 .9332 .9332 .9452 .9452 .9452 .9554 .9554 .9554 .9641 .9641 .9641 .9713 .9713 .9713 .9772 .9772 .9772 .9821 .9821 .9821 .9861 .9861 .9861 .9893 .9893 .9893 .9918 .9918 .9918 .9938 .9938 .9938 .9953 .9953 .9953 .9965 .9965 .9965 .9974 .9974 .9974 .9981 .9981 .9981 .9987 .9987 .9987 .9990 .9990 .9990 .9993 .9993 .9993 .9995 .9995 .9995 .9997 .9997 .9997 .01 .01.01 .5040 .5040 .5040 .5438 .5438 .5438 .5832 .5832 .5832 .6217 .6217 .6217 .6591 .6591 .6591 .6950 .6950 .6950 .7291 .7291 .7291 .7611 .7611 .7611 .7910 .7910 .7910 .8186 .8186 .8186 .8438 .8438 .8438 .8665 .8665 .8665 .8869 .8869 .8869 .9049 .9049 .9049 .9207 .9207 .9207 .9345 .9345 .9345 .9463 .9463 .9463 .9564 .9564 .9564 .9649 .9649 .9649 .9719 .9719 .9719 .9778 .9778 .9778 .9826 .9826 .9826 .9864 .9864 .9864 .9896 .9896 .9896 .9920 .9920 .9920 .9940 .9940 .9940 .9955 .9955 .9955 .9966 .9966 .9966 .9975 .9975 .9975 .9982 .9982 .9982 .9987 .9987 .9987 .9991 .9991 .9991 .9993 .9993 .9993 .9995 .9995 .9995 .9997 .9997 .9997 zzcc zc 1.28 1.28 1.28 1.645 1.645 1.645 1.96 1.96 1.96 2.575 2.575 2.575 .02 .02.02 .5080 .5080 .5080 .5478 .5478 .5478 .5871 .5871 .5871 .6255 .6255 .6255 .6628 .6628 .6628 .6985 .6985 .6985 .7324 .7324 .7324 .7642 .7642 .7642 .7939 .7939 .7939 .8212 .8212 .8212 .8461 .8461 .8461 .8686 .8686 .8686 .8888 .8888 .8888 .9066 .9066 .9066 .9222 .9222 .9222 .9357 .9357 .9357 .9474 .9474 .9474 .9573 .9573 .9573 .9656 .9656 .9656 .9726 .9726 .9726 .9783 .9783 .9783 .9830 .9830 .9830 .9868 .9868 .9868 .9898 .9898 .9898 .9922 .9922 .9922 .9941 .9941 .9941 .9956 .9956 .9956 .9967 .9967 .9967 .9976 .9976 .9976 .9982 .9982 .9982 .9987 .9987 .9987 .9991 .9991 .9991 .9994 .9994 .9994 .9995 .9995 .9995 .9997 .9997 .9997 .03 .03.03 .5120 .5120 .5120 .5517 .5517 .5517 .5910 .5910 .5910 .6293 .6293 .6293 .6664 .6664 .6664 .7019 .7019 .7019 .7357 .7357 .7357 .7673 .7673 .7673 .7967 .7967 .7967 .8238 .8238 .8238 .8485 .8485 .8485 .8708 .8708 .8708 .8907 .8907 .8907 .9082 .9082 .9082 .9236 .9236 .9236 .9370 .9370 .9370 .9484 .9484 .9484 .9582 .9582 .9582 .9664 .9664 .9664 .9732 .9732 .9732 .9788 .9788 .9788 .9834 .9834 .9834 .9871 .9871 .9871 .9901 .9901 .9901 .9925 .9925 .9925 .9943 .9943 .9943 .9957 .9957 .9957 .9968 .9968 .9968 .9977 .9977 .9977 .9983 .9983 .9983 .9988 .9988 .9988 .9991 .9991 .9991 .9994 .9994 .9994 .9996 .9996 .9996 .9997 .9997 .9997 .04 .04.04 .5160 .5160 .5160 .5557 .5557 .5557 .5948 .5948 .5948 .6331 .6331 .6331 .6700 .6700 .6700 .7054 .7054 .7054 .7389 .7389 .7389 .7704 .7704 .7704 .7995 .7995 .7995 .8264 .8264 .8264 .8508 .8508 .8508 .8729 .8729 .8729 .8925 .8925 .8925 .9099 .9099 .9099 .9251 .9251 .9251 .9382 .9382 .9382 .9495 .9495 .9495 .9591 .9591 .9591 .9671 .9671 .9671 .9738 .9738 .9738 .9793 .9793 .9793 .9838 .9838 .9838 .9875 .9875 .9875 .9904 .9904 .9904 .9927 .9927 .9927 .9945 .9945 .9945 .9959 .9959 .9959 .9969 .9969 .9969 .9977 .9977 .9977 .9984 .9984 .9984 .9988 .9988 .9988 .9992 .9992 .9992 .9994 .9994 .9994 .9996 .9996 .9996 .9997 .9997 .9997 .05 .05.05 .5199 .5199 .5199 .5596 .5596 .5596 .5987 .5987 .5987 .6368 .6368 .6368 .6736 .6736 .6736 .7088 .7088 .7088 .7422 .7422 .7422 .7734 .7734 .7734 .8023 .8023 .8023 .8289 .8289 .8289 .8531 .8531 .8531 .8749 .8749 .8749 .8944 .8944 .8944 .9115 .9115 .9115 .9265 .9265 .9265 .9394 .9394 .9394 .9505 .9505 .9505 .9599 .9599 .9599 .9678 .9678 .9678 .9744 .9744 .9744 .9798 .9798 .9798 .9842 .9842 .9842 .9878 .9878 .9878 .9906 .9906 .9906 .9929 .9929 .9929 .9946 .9946 .9946 .9960 .9960 .9960 .9970 .9970 .9970 .9978 .9978 .9978 .9984 .9984 .9984 .9989 .9989 .9989 .9992 .9992 .9992 .9994 .9994 .9994 .9996 .9996 .9996 .9997 .9997 .9997 .06 .06.06 .5239 .5239 .5239 .5636 .5636 .5636 .6026 .6026 .6026 .6406 .6406 .6406 .6772 .6772 .6772 .7123 .7123 .7123 .7454 .7454 .7454 .7764 .7764 .7764 .8051 .8051 .8051 .8315 .8315 .8315 .8554 .8554 .8554 .8770 .8770 .8770 .8962 .8962 .8962 .9131 .9131 .9131 .9279 .9279 .9279 .9406 .9406 .9406 .9515 .9515 .9515 .9608 .9608 .9608 .9686 .9686 .9686 .9750 .9750 .9750 .9803 .9803 .9803 .9846 .9846 .9846 .9881 .9881 .9881 .9909 .9909 .9909 .9931 .9931 .9931 .9948 .9948 .9948 .9961 .9961 .9961 .9971 .9971 .9971 .9979 .9979 .9979 .9985 .9985 .9985 .9989 .9989 .9989 .9992 .9992 .9992 .9994 .9994 .9994 .9996 .9996 .9996 .9997 .9997 .9997 cc c −z −zcc−z c zz==0z0= 0 zzcc z c t c-confidence c-confidence interval interval c-confidence interval Critical Critical Values Values Critical Values cc c Level Level of ofConfidence Confidence Level of Confidence 0.80 0.80 0.80 0.90 0.90 0.90 0.95 0.95 0.95 0.99 0.99 0.99 tt zz z Copyright © 2012 Pearson Education, Inc. .07 .07.07 .5279 .5279 .5279 .5675 .5675 .5675 .6064 .6064 .6064 .6443 .6443 .6443 .6808 .6808 .6808 .7157 .7157 .7157 .7486 .7486 .7486 .7794 .7794 .7794 .8078 .8078 .8078 .8340 .8340 .8340 .8577 .8577 .8577 .8790 .8790 .8790 .8980 .8980 .8980 .9147 .9147 .9147 .9292 .9292 .9292 .9418 .9418 .9418 .9525 .9525 .9525 .9616 .9616 .9616 .9693 .9693 .9693 .9756 .9756 .9756 .9808 .9808 .9808 .9850 .9850 .9850 .9884 .9884 .9884 .9911 .9911 .9911 .9932 .9932 .9932 .9949 .9949 .9949 .9962 .9962 .9962 .9972 .9972 .9972 .9979 .9979 .9979 .9985 .9985 .9985 .9989 .9989 .9989 .9992 .9992 .9992 .9995 .9995 .9995 .9996 .9996 .9996 .9997 .9997 .9997 .08 .08.08 .5319 .5319 .5319 .5714 .5714 .5714 .6103 .6103 .6103 .6480 .6480 .6480 .6844 .6844 .6844 .7190 .7190 .7190 .7517 .7517 .7517 .7823 .7823 .7823 .8106 .8106 .8106 .8365 .8365 .8365 .8599 .8599 .8599 .8810 .8810 .8810 .8997 .8997 .8997 .9162 .9162 .9162 .9306 .9306 .9306 .9429 .9429 .9429 .9535 .9535 .9535 .9625 .9625 .9625 .9699 .9699 .9699 .9761 .9761 .9761 .9812 .9812 .9812 .9854 .9854 .9854 .9887 .9887 .9887 .9913 .9913 .9913 .9934 .9934 .9934 .9951 .9951 .9951 .9963 .9963 .9963 .9973 .9973 .9973 .9980 .9980 .9980 .9986 .9986 .9986 .9990 .9990 .9990 .9993 .9993 .9993 .9995 .9995 .9995 .9996 .9996 .9996 .9997 .9997 .9997 .09 .09.09 .5359 .5359 .5359 .5753 .5753 .5753 .6141 .6141 .6141 .6517 .6517 .6517 .6879 .6879 .6879 .7224 .7224 .7224 .7549 .7549 .7549 .7852 .7852 .7852 .8133 .8133 .8133 .8389 .8389 .8389 .8621 .8621 .8621 .8830 .8830 .8830 .9015 .9015 .9015 .9177 .9177 .9177 .9319 .9319 .9319 .9441 .9441 .9441 .9545 .9545 .9545 .9633 .9633 .9633 .9706 .9706 .9706 .9767 .9767 .9767 .9817 .9817 .9817 .9857 .9857 .9857 .9890 .9890 .9890 .9916 .9916 .9916 .9936 .9936 .9936 .9952 .9952 .9952 .9964 .9964 .9964 .9974 .9974 .9974 .9981 .9981 .9981 .9986 .9986 .9986 .9990 .9990 .9990 .9993 .9993 .9993 .9995 .9995 .9995 .9997 .9997 .9997 .9998 .9998 .9998 d.f. d.f.d.f. 11 1 22 2 33 3 44 4 55 5 66 6 77 7 88 8 99 9 10 1010 11 1111 12 1212 13 1313 14 1414 15 1515 16 1616 17 1717 18 1818 19 1919 20 2020 21 2121 22 2222 23 2323 24 2424 25 2525 26 2626 27 2727 28 2828 29 2929 q qq tt t αα α tt −t −t −t Left-tailed Left-tailed test test Left-tailed test Level Level Level of of of confidence, confidence, 0.50 cc c 0.50 confidence, 0.50 One One tail, tail, 0.25 0.25 AA A One tail, 0.25 Two Two tails, tails, 0.50 AA A 0.50 Two tails, 0.50 1.000 1.000 1.000 .816 .816 .816 .765 .765 .765 .741 .741 .741 .727 .727 .727 .718 .718 .718 .711 .711 .711 .706 .706 .706 .703 .703 .703 .700 .700 .700 .697 .697 .697 .695 .695 .695 .694 .694 .694 .692 .692 .692 .691 .691 .691 .690 .690 .690 .689 .689 .689 .688 .688 .688 .688 .688 .688 .687 .687 .687 .686 .686 .686 .686 .686 .686 .685 .685 .685 .685 .685 .685 .684 .684 .684 .684 .684 .684 .684 .684 .684 .683 .683 .683 .683 .683 .683 .674 .674 .674 0.80 0.80 0.80 0.10 0.10 0.10 0.20 0.20 0.20 3.078 3.078 3.078 1.886 1.886 1.886 1.638 1.638 1.638 1.533 1.533 1.533 1.476 1.476 1.476 1.440 1.440 1.440 1.415 1.415 1.415 1.397 1.397 1.397 1.383 1.383 1.383 1.372 1.372 1.372 1.363 1.363 1.363 1.356 1.356 1.356 1.350 1.350 1.350 1.345 1.345 1.345 1.341 1.341 1.341 1.337 1.337 1.337 1.333 1.333 1.333 1.330 1.330 1.330 1.328 1.328 1.328 1.325 1.325 1.325 1.323 1.323 1.323 1.321 1.321 1.321 1.319 1.319 1.319 1.318 1.318 1.318 1.316 1.316 1.316 1.315 1.315 1.315 1.314 1.314 1.314 1.313 1.313 1.313 1.311 1.311 1.311 1.282 1.282 1.282 t αα α tt t Right-tailed Right-tailed test test Right-tailed test 0.90 0.90 0.95 0.98 0.99 0.90 0.95 0.95 0.98 0.98 0.99 0.99 0.05 0.05 0.025 0.01 0.005 0.05 0.025 0.025 0.01 0.01 0.005 0.005 0.10 0.10 0.05 0.02 0.01 0.10 0.05 0.05 0.02 0.02 0.01 0.01 6.31412.706 12.70631.821 31.82163.657 63.657 6.314 6.314 12.706 31.821 63.657 2.920 2.920 4.303 6.965 9.925 2.920 4.303 4.303 6.965 6.965 9.925 9.925 2.353 2.353 3.182 4.541 5.841 2.353 3.182 3.182 4.541 4.541 5.841 5.841 2.132 2.132 2.776 3.747 4.604 2.132 2.776 2.776 3.747 3.747 4.604 4.604 2.015 2.015 2.571 3.365 4.032 2.015 2.571 2.571 3.365 3.365 4.032 4.032 1.943 1.943 2.447 3.143 3.707 1.943 2.447 2.447 3.143 3.143 3.707 3.707 1.895 1.895 2.365 2.998 3.499 1.895 2.365 2.365 2.998 2.998 3.499 3.499 1.860 1.860 2.306 2.896 3.355 1.860 2.306 2.306 2.896 2.896 3.355 3.355 1.833 1.833 2.262 2.821 3.250 1.833 2.262 2.262 2.821 2.821 3.250 3.250 1.812 1.812 2.228 2.764 3.169 1.812 2.228 2.228 2.764 2.764 3.169 3.169 1.796 1.796 2.201 2.718 3.106 1.796 2.201 2.201 2.718 2.718 3.106 3.106 1.782 1.782 2.179 2.681 3.055 1.782 2.179 2.179 2.681 2.681 3.055 3.055 1.771 1.771 2.160 2.650 3.012 1.771 2.160 2.160 2.650 2.650 3.012 3.012 1.761 1.761 2.145 2.624 2.977 1.761 2.145 2.145 2.624 2.624 2.977 2.977 1.753 1.753 2.131 2.602 2.947 1.753 2.131 2.131 2.602 2.602 2.947 2.947 1.746 1.746 2.120 2.583 2.921 1.746 2.120 2.120 2.583 2.583 2.921 2.921 1.740 1.740 2.110 2.567 2.898 1.740 2.110 2.110 2.567 2.567 2.898 2.898 1.734 1.734 2.101 2.552 2.878 1.734 2.101 2.101 2.552 2.552 2.878 2.878 1.729 1.729 2.093 2.539 2.861 1.729 2.093 2.093 2.539 2.539 2.861 2.861 1.725 1.725 2.086 2.528 2.845 1.725 2.086 2.086 2.528 2.528 2.845 2.845 1.721 1.721 2.080 2.518 2.831 1.721 2.080 2.080 2.518 2.518 2.831 2.831 1.717 1.717 2.074 2.508 2.819 1.717 2.074 2.074 2.508 2.508 2.819 2.819 1.714 1.714 2.069 2.500 2.807 1.714 2.069 2.069 2.500 2.500 2.807 2.807 1.711 1.711 2.064 2.492 2.797 1.711 2.064 2.064 2.492 2.492 2.797 2.797 1.708 1.708 2.060 2.485 2.787 1.708 2.060 2.060 2.485 2.485 2.787 2.787 1.706 1.706 2.056 2.479 2.779 1.706 2.056 2.056 2.479 2.479 2.779 2.779 1.703 1.703 2.052 2.473 2.771 1.703 2.052 2.052 2.473 2.473 2.771 2.771 1.701 1.701 2.048 2.467 2.763 1.701 2.048 2.048 2.467 2.467 2.763 2.763 1.699 1.699 2.045 2.462 2.756 1.699 2.045 2.045 2.462 2.462 2.756 2.756 1.645 1.645 1.960 2.326 2.576 1.645 1.960 1.960 2.326 2.326 2.576 2.576 tt t −t −t −t 11 1 αα α 22 2 tt t Two-tailed Two-tailed test test Two-tailed test tt 11 1 αα α 22 2 αα α t χχ22 χ 2 χχ22 χ 2 Degrees Degrees of of of Degrees freedom freedom freedom 11 1 22 2 33 3 44 4 55 5 66 6 77 7 88 8 99 9 1010 10 1111 11 1212 12 1313 13 1414 14 1515 15 1616 16 1717 17 1818 18 1919 19 2020 20 2121 21 2222 22 2323 23 2424 24 2525 25 2626 26 2727 27 2828 28 2929 29 3030 30 4040 40 5050 50 6060 60 7070 70 8080 80 9090 90 100 100 100 χχ22 χ 2 LL Right tail Right tailtail Right 11 1 αα α 22 2 L χχ22 χ 2 RR χχ22 χ 2 R Two tails Two tails Two tails AA A 0.995 0.99 0.975 0.95 0.90 0.10 0.05 0.025 0.01 0.005 0.995 0.99 0.99 0.975 0.975 0.95 0.95 0.90 0.90 0.10 0.10 0.05 0.05 0.025 0.025 0.01 0.01 0.005 0.005 0.995 —— —— 0.001 0.001 0.004 0.016 2.706 3.841 5.024 6.635 7.879 0.001 0.004 0.004 0.016 0.016 2.706 2.706 3.841 3.841 5.024 5.024 6.635 6.635 7.879 7.879 — — 0.010 0.020 0.051 0.103 0.211 4.605 5.991 7.378 9.210 10.597 0.010 0.020 0.020 0.051 0.051 0.103 0.103 0.211 0.211 4.605 4.605 5.991 5.991 7.378 7.378 9.210 9.21010.597 10.597 0.010 0.072 0.115 0.216 0.352 0.584 6.251 7.815 9.348 11.345 12.838 0.072 0.115 0.115 0.216 0.216 0.352 0.352 0.584 0.584 6.251 6.251 7.815 7.815 9.348 9.34811.345 11.34512.838 12.838 0.072 0.207 0.297 0.484 0.711 1.064 7.779 9.488 11.143 13.277 14.860 0.207 0.297 0.297 0.484 0.484 0.711 0.711 1.064 1.064 7.779 7.779 9.488 9.48811.143 11.14313.277 13.27714.860 14.860 0.207 0.412 0.554 0.831 1.145 1.610 9.236 11.071 12.833 15.086 16.750 0.412 0.554 0.554 0.831 0.831 1.145 1.145 1.610 1.610 9.236 9.23611.071 11.07112.833 12.83315.086 15.08616.750 16.750 0.412 0.676 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.812 18.548 0.676 0.872 0.872 1.237 1.237 1.635 1.635 2.204 2.20410.645 10.64512.592 12.59214.449 14.44916.812 16.81218.548 18.548 0.676 0.989 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.475 20.278 0.989 1.239 1.239 1.690 1.690 2.167 2.167 2.833 2.83312.017 12.01714.067 14.06716.013 16.01318.475 18.47520.278 20.278 0.989 1.344 1.646 2.180 2.733 3.490 13.362 15.507 17.535 20.090 21.955 1.344 1.646 1.646 2.180 2.180 2.733 2.733 3.490 3.49013.362 13.36215.507 15.50717.535 17.53520.090 20.09021.955 21.955 1.344 1.735 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666 23.589 1.735 2.088 2.088 2.700 2.700 3.325 3.325 4.168 4.16814.684 14.68416.919 16.91919.023 19.02321.666 21.66623.589 23.589 1.735 2.156 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 25.188 2.156 2.558 2.558 3.247 3.247 3.940 3.940 4.865 4.86515.987 15.98718.307 18.30720.483 20.48323.209 23.20925.188 25.188 2.156 2.603 3.053 3.816 4.575 5.578 17.275 19.675 21.920 24.725 26.757 2.603 3.053 3.053 3.816 3.816 4.575 4.575 5.578 5.57817.275 17.27519.675 19.67521.920 21.92024.725 24.72526.757 26.757 2.603 3.074 3.571 4.404 5.226 6.304 18.549 21.026 23.337 26.217 28.299 3.074 3.571 3.571 4.404 4.404 5.226 5.226 6.304 6.30418.549 18.54921.026 21.02623.337 23.33726.217 26.21728.299 28.299 3.074 3.565 4.107 5.009 5.892 7.042 19.812 22.362 24.736 27.688 29.819 3.565 4.107 4.107 5.009 5.009 5.892 5.892 7.042 7.04219.812 19.81222.362 22.36224.736 24.73627.688 27.68829.819 29.819 3.565 4.075 4.660 5.629 6.571 7.790 21.064 23.685 26.119 29.141 31.319 4.075 4.660 4.660 5.629 5.629 6.571 6.571 7.790 7.79021.064 21.06423.685 23.68526.119 26.11929.141 29.14131.319 31.319 4.075 4.601 5.229 6.262 7.261 8.547 22.307 24.996 27.488 30.578 32.801 4.601 5.229 5.229 6.262 6.262 7.261 7.261 8.547 8.54722.307 22.30724.996 24.99627.488 27.48830.578 30.57832.801 32.801 4.601 5.142 5.812 6.908 7.962 9.312 23.542 26.296 28.845 32.000 34.267 5.142 5.812 5.812 6.908 6.908 7.962 7.962 9.312 9.31223.542 23.54226.296 26.29628.845 28.84532.000 32.00034.267 34.267 5.142 5.697 6.408 7.564 8.672 10.085 24.769 27.587 30.191 33.409 35.718 5.697 6.408 6.408 7.564 7.564 8.672 8.67210.085 10.08524.769 24.76927.587 27.58730.191 30.19133.409 33.40935.718 35.718 5.697 6.265 7.015 8.231 9.390 10.865 25.989 28.869 31.526 34.805 37.156 6.265 7.015 7.015 8.231 8.231 9.390 9.39010.865 10.86525.989 25.98928.869 28.86931.526 31.52634.805 34.80537.156 37.156 6.265 6.844 7.633 8.907 10.117 11.651 27.204 30.144 32.852 36.191 38.582 6.844 7.633 7.633 8.907 8.90710.117 10.11711.651 11.65127.204 27.20430.144 30.14432.852 32.85236.191 36.19138.582 38.582 6.844 7.434 8.260 9.591 10.851 12.443 28.412 31.410 34.170 37.566 39.997 7.434 8.260 8.260 9.591 9.59110.851 10.85112.443 12.44328.412 28.41231.410 31.41034.170 34.17037.566 37.56639.997 39.997 7.434 8.034 8.897 10.283 11.591 13.240 29.615 32.671 35.479 38.932 41.401 8.034 8.897 8.89710.283 10.28311.591 11.59113.240 13.24029.615 29.61532.671 32.67135.479 35.47938.932 38.93241.401 41.401 8.034 8.643 9.542 10.982 12.338 14.042 30.813 33.924 36.781 40.289 42.796 8.643 9.542 9.54210.982 10.98212.338 12.33814.042 14.04230.813 30.81333.924 33.92436.781 36.78140.289 40.28942.796 42.796 8.643 9.260 9.26010.196 10.196 10.19611.689 11.689 11.68913.091 13.091 13.09114.848 14.848 14.84832.007 32.007 32.00735.172 35.172 35.17238.076 38.076 38.07641.638 41.638 41.63844.181 44.181 44.181 9.260 9.886 10.856 12.401 13.848 15.659 33.196 36.415 39.364 42.980 45.559 9.88610.856 10.85612.401 12.40113.848 13.84815.659 15.65933.196 33.19636.415 36.41539.364 39.36442.980 42.98045.559 45.559 9.886 10.520 11.524 13.120 14.611 16.473 34.382 37.652 40.646 44.314 46.928 10.52011.524 11.52413.120 13.12014.611 14.61116.473 16.47334.382 34.38237.652 37.65240.646 40.64644.314 44.31446.928 46.928 10.520 11.160 12.198 13.844 15.379 17.292 35.563 38.885 41.923 45.642 48.290 11.16012.198 12.19813.844 13.84415.379 15.37917.292 17.29235.563 35.56338.885 38.88541.923 41.92345.642 45.64248.290 48.290 11.160 11.808 12.879 14.573 16.151 18.114 36.741 40.113 43.194 46.963 49.645 11.80812.879 12.87914.573 14.57316.151 16.15118.114 18.11436.741 36.74140.113 40.11343.194 43.19446.963 46.96349.645 49.645 11.808 12.461 13.565 15.308 16.928 18.939 37.916 41.337 44.461 48.278 50.993 12.46113.565 13.56515.308 15.30816.928 16.92818.939 18.93937.916 37.91641.337 41.33744.461 44.46148.278 48.27850.993 50.993 12.461 13.121 13.12114.257 14.257 14.25716.047 16.047 16.04717.708 17.708 17.70819.768 19.768 19.76839.087 39.087 39.08742.557 42.557 42.55745.722 45.722 45.72249.588 49.588 49.58852.336 52.336 52.336 13.121 13.787 14.954 16.791 18.493 20.599 40.256 43.773 46.979 50.892 53.672 13.78714.954 14.95416.791 16.79118.493 18.49320.599 20.59940.256 40.25643.773 43.77346.979 46.97950.892 50.89253.672 53.672 13.787 20.707 22.164 24.433 26.509 29.051 51.805 55.758 59.342 63.691 66.766 20.70722.164 22.16424.433 24.43326.509 26.50929.051 29.05151.805 51.80555.758 55.75859.342 59.34263.691 63.69166.766 66.766 20.707 27.991 29.707 32.357 34.764 37.689 63.167 67.505 71.420 76.154 79.490 27.99129.707 29.70732.357 32.35734.764 34.76437.689 37.68963.167 63.16767.505 67.50571.420 71.42076.154 76.15479.490 79.490 27.991 35.534 37.485 40.482 43.188 46.459 74.397 79.082 83.298 88.379 91.952 35.53437.485 37.48540.482 40.48243.188 43.18846.459 46.45974.397 74.39779.082 79.08283.298 83.29888.379 88.37991.952 91.952 35.534 43.275 45.442 48.758 51.739 55.329 85.527 90.531 95.023 100.425 104.215 43.27545.442 45.44248.758 48.75851.739 51.73955.329 55.32985.527 85.52790.531 90.53195.023 95.023 100.425 104.215 43.275 100.425 104.215 51.172 51.17253.540 53.540 53.54057.153 57.153 57.15360.391 60.391 60.39164.278 64.278 64.27896.578 96.578 96.578 101.879 101.879 106.629 106.629 112.329 112.329 116.321 116.321 51.172 101.879 106.629 112.329 116.321 59.196 59.19661.754 61.754 61.75465.647 65.647 65.64769.126 69.126 69.12673.291 73.291 73.291 107.565 107.565 113.145 113.145 118.136 118.136 124.116 124.116 128.299 128.299 59.196 107.565 113.145 118.136 124.116 128.299 67.328 70.065 74.222 77.929 82.358 118.498 124.342 129.561 135.807 140.169 67.32870.065 70.06574.222 74.22277.929 77.92982.358 82.358 118.498 124.342 129.561 135.807 140.169 67.328 118.498 124.342 129.561 135.807 140.169 41831S4_INS p5-8 AM 11 1 41831S4_INS p5-8 11/8/07 10:03 AMAMPage Page 41831S4_INS p5-811/8/07 11/8/0710:03 10:03 Page Table Table 44— 4— Standard Standard Normal Normal Distribution Distribution (continued) (continued) Table — Standard Normal Distribution (continued) Table Table 44— — Standard Standard Normal Normal Distribution Distribution Table 4— Standard Normal Distribution Table Table Table 6— 6— 6— Chi-Square Chi-Square Chi-Square Distribution Distribution Distribution Area Area Area Area Area Area 11 1 αα α 22 2 zz z zz z .09 .09.09 .0002 .0002 .0002 .0003 .0003 .0003 .0005 .0005 .0005 .0007 .0007 .0007 .0010 .0010 .0010 .0014 .0014 .0014 .0019 .0019 .0019 .0026 .0026 .0026 .0036 .0036 .0036 .0048 .0048 .0048 .0064 .0064 .0064 .0084 .0084 .0084 .0110 .0110 .0110 .0143 .0143 .0143 .0183 .0183 .0183 .0233 .0233 .0233 .0294 .0294 .0294 .0367 .0367 .0367 .0455 .0455 .0455 .0559 .0559 .0559 .0681 .0681 .0681 .0823 .0823 .0823 .0985 .0985 .0985 .1170 .1170 .1170 .1379 .1379 .1379 .1611 .1611 .1611 .1867 .1867 .1867 .2148 .2148 .2148 .2451 .2451 .2451 .2776 .2776 .2776 .3121 .3121 .3121 .3483 .3483 .3483 .3859 .3859 .3859 .4247 .4247 .4247 .4641 .4641 .4641 −t −t −t 00 0 zz z zz z 00 0 zz z 3.4 � � 3.4 �3.4 3.3 � � 3.3 �3.3 3.2 � � 3.2 �3.2 3.1 � � 3.1 �3.1 3.0 � � 3.0 �3.0 2.9 � � 2.9 �2.9 2.8 � � 2.8 �2.8 2.7 � � 2.7 �2.7 2.6 � � 2.6 �2.6 2.5 � � 2.5 �2.5 2.4 � � 2.4 �2.4 2.3 � � 2.3 �2.3 2.2 � � 2.2 �2.2 2.1 � � 2.1 �2.1 2.0 � � 2.0 �2.0 1.9 � � 1.9 �1.9 1.8 � � 1.8 �1.8 1.7 � � 1.7 �1.7 1.6 � � 1.6 �1.6 1.5 � � 1.5 �1.5 1.4 � � 1.4 �1.4 1.3 � � 1.3 �1.3 1.2 � � 1.2 �1.2 1.1 � � 1.1 �1.1 1.0 � � 1.0 �1.0 0.9 � � 0.9 �0.9 0.8 � � 0.8 �0.8 0.7 � � 0.7 �0.7 0.6 � � 0.6 �0.6 0.5 � � 0.5 �0.5 0.4 � � 0.4 �0.4 0.3 � � 0.3 �0.3 0.2 � � 0.2 �0.2 0.1 � � 0.1 �0.1 �0.0 0.0 � � 0.0 Table Table Table 5— 5— 5— t-Distribution t-Distribution t-Distribution .08 .08.08 .0003 .0003 .0003 .0004 .0004 .0004 .0005 .0005 .0005 .0007 .0007 .0007 .0010 .0010 .0010 .0014 .0014 .0014 .0020 .0020 .0020 .0027 .0027 .0027 .0037 .0037 .0037 .0049 .0049 .0049 .0066 .0066 .0066 .0087 .0087 .0087 .0113 .0113 .0113 .0146 .0146 .0146 .0188 .0188 .0188 .0239 .0239 .0239 .0301 .0301 .0301 .0375 .0375 .0375 .0465 .0465 .0465 .0571 .0571 .0571 .0694 .0694 .0694 .0838 .0838 .0838 .1003 .1003 .1003 .1190 .1190 .1190 .1401 .1401 .1401 .1635 .1635 .1635 .1894 .1894 .1894 .2177 .2177 .2177 .2483 .2483 .2483 .2810 .2810 .2810 .3156 .3156 .3156 .3520 .3520 .3520 .3897 .3897 .3897 .4286 .4286 .4286 .4681 .4681 .4681 .07 .07.07 .0003 .0003 .0003 .0004 .0004 .0004 .0005 .0005 .0005 .0008 .0008 .0008 .0011 .0011 .0011 .0015 .0015 .0015 .0021 .0021 .0021 .0028 .0028 .0028 .0038 .0038 .0038 .0051 .0051 .0051 .0068 .0068 .0068 .0089 .0089 .0089 .0116 .0116 .0116 .0150 .0150 .0150 .0192 .0192 .0192 .0244 .0244 .0244 .0307 .0307 .0307 .0384 .0384 .0384 .0475 .0475 .0475 .0582 .0582 .0582 .0708 .0708 .0708 .0853 .0853 .0853 .1020 .1020 .1020 .1210 .1210 .1210 .1423 .1423 .1423 .1660 .1660 .1660 .1922 .1922 .1922 .2206 .2206 .2206 .2514 .2514 .2514 .2843 .2843 .2843 .3192 .3192 .3192 .3557 .3557 .3557 .3936 .3936 .3936 .4325 .4325 .4325 .4721 .4721 .4721 .06 .06.06 .0003 .0003 .0003 .0004 .0004 .0004 .0006 .0006 .0006 .0008 .0008 .0008 .0011 .0011 .0011 .0015 .0015 .0015 .0021 .0021 .0021 .0029 .0029 .0029 .0039 .0039 .0039 .0052 .0052 .0052 .0069 .0069 .0069 .0091 .0091 .0091 .0119 .0119 .0119 .0154 .0154 .0154 .0197 .0197 .0197 .0250 .0250 .0250 .0314 .0314 .0314 .0392 .0392 .0392 .0485 .0485 .0485 .0594 .0594 .0594 .0721 .0721 .0721 .0869 .0869 .0869 .1038 .1038 .1038 .1230 .1230 .1230 .1446 .1446 .1446 .1685 .1685 .1685 .1949 .1949 .1949 .2236 .2236 .2236 .2546 .2546 .2546 .2877 .2877 .2877 .3228 .3228 .3228 .3594 .3594 .3594 .3974 .3974 .3974 .4364 .4364 .4364 .4761 .4761 .4761 .05 .05.05 .0003 .0003 .0003 .0004 .0004 .0004 .0006 .0006 .0006 .0008 .0008 .0008 .0011 .0011 .0011 .0016 .0016 .0016 .0022 .0022 .0022 .0030 .0030 .0030 .0040 .0040 .0040 .0054 .0054 .0054 .0071 .0071 .0071 .0094 .0094 .0094 .0122 .0122 .0122 .0158 .0158 .0158 .0202 .0202 .0202 .0256 .0256 .0256 .0322 .0322 .0322 .0401 .0401 .0401 .0495 .0495 .0495 .0606 .0606 .0606 .0735 .0735 .0735 .0885 .0885 .0885 .1056 .1056 .1056 .1251 .1251 .1251 .1469 .1469 .1469 .1711 .1711 .1711 .1977 .1977 .1977 .2266 .2266 .2266 .2578 .2578 .2578 .2912 .2912 .2912 .3264 .3264 .3264 .3632 .3632 .3632 .4013 .4013 .4013 .4404 .4404 .4404 .4801 .4801 .4801 .04 .04.04 .03 .03.03 .0003 .0003 .0003 .0003 .0003 .0003 .0004 .0004 .0004 .0004 .0004 .0004 .0006 .0006 .0006 .0006 .0006 .0006 .0008 .0008 .0009 .0008 .0009 .0009 .0012 .0012 .0012 .0012 .0012 .0012 .0016 .0016 .0017 .0016 .0017 .0017 .0023 .0023 .0023 .0023 .0023 .0023 .0031 .0031 .0032 .0031 .0032 .0032 .0041 .0043 .0041 .0041 .0043 .0043 .0055 .0055 .0057 .0055 .0057 .0057 .0073 .0075 .0073 .0073 .0075 .0075 .0096 .0096 .0099 .0096 .0099 .0099 .0125 .0129 .0125 .0125 .0129 .0129 .0162 .0162 .0166 .0162 .0166 .0166 .0207 .0212 .0207 .0207 .0212 .0212 .0262 .0262 .0268 .0262 .0268 .0268 .0329 .0336 .0329 .0329 .0336 .0336 .0409 .0409 .0418 .0409 .0418 .0418 .0505 .0516 .0505 .0505 .0516 .0516 .0618 .0618 .0630 .0618 .0630 .0630 .0749 .0764 .0749 .0749 .0764 .0764 .0901 .0901 .0918 .0901 .0918 .0918 .1075 .1093 .1075 .1075 .1093 .1093 .1271 .1271 .1292 .1271 .1292 .1292 .1492 .1515 .1492 .1492 .1515 .1515 .1736 .1736 .1762 .1736 .1762 .1762 .2005 .2033 .2005 .2005 .2033 .2033 .2296 .2296 .2327 .2296 .2327 .2327 .2611 .2643 .2611 .2611 .2643 .2643 .2946 .2946 .2981 .2946 .2981 .2981 .3300 .3336 .3300 .3300 .3336 .3336 .3669 .3669 .3707 .3669 .3707 .3707 .4052 .4090 .4052 .4052 .4090 .4090 .4443 .4443 .4483 .4443 .4483 .4483 .4840 .4840 .4880 .4840 .4880 .4880 Critical Critical Values Values Critical Values cc c Level Level of ofConfidence Confidence Level of Confidence 0.80 0.80 0.80 0.90 0.90 0.90 0.95 0.95 0.95 0.99 0.99 0.99 zzcc zc 1.28 1.28 1.28 1.645 1.645 1.645 1.96 1.96 1.96 2.575 2.575 2.575 .02 .02.02 .0003 .0003 .0003 .0005 .0005 .0005 .0006 .0006 .0006 .0009 .0009 .0009 .0013 .0013 .0013 .0018 .0018 .0018 .0024 .0024 .0024 .0033 .0033 .0033 .0044 .0044 .0044 .0059 .0059 .0059 .0078 .0078 .0078 .0102 .0102 .0102 .0132 .0132 .0132 .0170 .0170 .0170 .0217 .0217 .0217 .0274 .0274 .0274 .0344 .0344 .0344 .0427 .0427 .0427 .0526 .0526 .0526 .0643 .0643 .0643 .0778 .0778 .0778 .0934 .0934 .0934 .1112 .1112 .1112 .1314 .1314 .1314 .1539 .1539 .1539 .1788 .1788 .1788 .2061 .2061 .2061 .2358 .2358 .2358 .2676 .2676 .2676 .3015 .3015 .3015 .3372 .3372 .3372 .3745 .3745 .3745 .4129 .4129 .4129 .4522 .4522 .4522 .4920 .4920 .4920 .01 .01.01 .0003 .0003 .0003 .0005 .0005 .0005 .0007 .0007 .0007 .0009 .0009 .0009 .0013 .0013 .0013 .0018 .0018 .0018 .0025 .0025 .0025 .0034 .0034 .0034 .0045 .0045 .0045 .0060 .0060 .0060 .0080 .0080 .0080 .0104 .0104 .0104 .0136 .0136 .0136 .0174 .0174 .0174 .0222 .0222 .0222 .0281 .0281 .0281 .0351 .0351 .0351 .0436 .0436 .0436 .0537 .0537 .0537 .0655 .0655 .0655 .0793 .0793 .0793 .0951 .0951 .0951 .1131 .1131 .1131 .1335 .1335 .1335 .1562 .1562 .1562 .1814 .1814 .1814 .2090 .2090 .2090 .2389 .2389 .2389 .2709 .2709 .2709 .3050 .3050 .3050 .3409 .3409 .3409 .3783 .3783 .3783 .4168 .4168 .4168 .4562 .4562 .4562 .4960 .4960 .4960 .00 .00.00 .0003 .0003 .0003 .0005 .0005 .0005 .0007 .0007 .0007 .0010 .0010 .0010 .0013 .0013 .0013 .0019 .0019 .0019 .0026 .0026 .0026 .0035 .0035 .0035 .0047 .0047 .0047 .0062 .0062 .0062 .0082 .0082 .0082 .0107 .0107 .0107 .0139 .0139 .0139 .0179 .0179 .0179 .0228 .0228 .0228 .0287 .0287 .0287 .0359 .0359 .0359 .0446 .0446 .0446 .0548 .0548 .0548 .0668 .0668 .0668 .0808 .0808 .0808 .0968 .0968 .0968 .1151 .1151 .1151 .1357 .1357 .1357 .1587 .1587 .1587 .1841 .1841 .1841 .2119 .2119 .2119 .2420 .2420 .2420 .2743 .2743 .2743 .3085 .3085 .3085 .3446 .3446 .3446 .3821 .3821 .3821 .4207 .4207 .4207 .4602 .4602 .4602 .5000 .5000 .5000 zz z 0.0 0.00.0 0.1 0.10.1 0.20.2 0.2 0.3 0.30.3 0.40.4 0.4 0.5 0.50.5 0.60.6 0.6 0.7 0.70.7 0.80.8 0.8 0.9 0.90.9 1.01.0 1.0 1.1 1.11.1 1.21.2 1.2 1.3 1.31.3 1.41.4 1.4 1.5 1.51.5 1.61.6 1.6 1.7 1.71.7 1.81.8 1.8 1.9 1.91.9 2.02.0 2.0 2.1 2.12.1 2.22.2 2.2 2.3 2.32.3 2.42.4 2.4 2.5 2.52.5 2.62.6 2.6 2.7 2.72.7 2.82.8 2.8 2.9 2.92.9 3.03.0 3.0 3.1 3.13.1 3.23.2 3.2 3.3 3.33.3 3.4 3.43.4 .00 .00.00 .5000 .5000 .5000 .5398 .5398 .5398 .5793 .5793 .5793 .6179 .6179 .6179 .6554 .6554 .6554 .6915 .6915 .6915 .7257 .7257 .7257 .7580 .7580 .7580 .7881 .7881 .7881 .8159 .8159 .8159 .8413 .8413 .8413 .8643 .8643 .8643 .8849 .8849 .8849 .9032 .9032 .9032 .9192 .9192 .9192 .9332 .9332 .9332 .9452 .9452 .9452 .9554 .9554 .9554 .9641 .9641 .9641 .9713 .9713 .9713 .9772 .9772 .9772 .9821 .9821 .9821 .9861 .9861 .9861 .9893 .9893 .9893 .9918 .9918 .9918 .9938 .9938 .9938 .9953 .9953 .9953 .9965 .9965 .9965 .9974 .9974 .9974 .9981 .9981 .9981 .9987 .9987 .9987 .9990 .9990 .9990 .9993 .9993 .9993 .9995 .9995 .9995 .9997 .9997 .9997 .01 .01.01 .5040 .5040 .5040 .5438 .5438 .5438 .5832 .5832 .5832 .6217 .6217 .6217 .6591 .6591 .6591 .6950 .6950 .6950 .7291 .7291 .7291 .7611 .7611 .7611 .7910 .7910 .7910 .8186 .8186 .8186 .8438 .8438 .8438 .8665 .8665 .8665 .8869 .8869 .8869 .9049 .9049 .9049 .9207 .9207 .9207 .9345 .9345 .9345 .9463 .9463 .9463 .9564 .9564 .9564 .9649 .9649 .9649 .9719 .9719 .9719 .9778 .9778 .9778 .9826 .9826 .9826 .9864 .9864 .9864 .9896 .9896 .9896 .9920 .9920 .9920 .9940 .9940 .9940 .9955 .9955 .9955 .9966 .9966 .9966 .9975 .9975 .9975 .9982 .9982 .9982 .9987 .9987 .9987 .9991 .9991 .9991 .9993 .9993 .9993 .9995 .9995 .9995 .9997 .9997 .9997 tt t c-confidence c-confidence interval interval c-confidence interval .02 .02.02 .5080 .5080 .5080 .5478 .5478 .5478 .5871 .5871 .5871 .6255 .6255 .6255 .6628 .6628 .6628 .6985 .6985 .6985 .7324 .7324 .7324 .7642 .7642 .7642 .7939 .7939 .7939 .8212 .8212 .8212 .8461 .8461 .8461 .8686 .8686 .8686 .8888 .8888 .8888 .9066 .9066 .9066 .9222 .9222 .9222 .9357 .9357 .9357 .9474 .9474 .9474 .9573 .9573 .9573 .9656 .9656 .9656 .9726 .9726 .9726 .9783 .9783 .9783 .9830 .9830 .9830 .9868 .9868 .9868 .9898 .9898 .9898 .9922 .9922 .9922 .9941 .9941 .9941 .9956 .9956 .9956 .9967 .9967 .9967 .9976 .9976 .9976 .9982 .9982 .9982 .9987 .9987 .9987 .9991 .9991 .9991 .9994 .9994 .9994 .9995 .9995 .9995 .9997 .9997 .9997 .03 .03.03 .5120 .5120 .5120 .5517 .5517 .5517 .5910 .5910 .5910 .6293 .6293 .6293 .6664 .6664 .6664 .7019 .7019 .7019 .7357 .7357 .7357 .7673 .7673 .7673 .7967 .7967 .7967 .8238 .8238 .8238 .8485 .8485 .8485 .8708 .8708 .8708 .8907 .8907 .8907 .9082 .9082 .9082 .9236 .9236 .9236 .9370 .9370 .9370 .9484 .9484 .9484 .9582 .9582 .9582 .9664 .9664 .9664 .9732 .9732 .9732 .9788 .9788 .9788 .9834 .9834 .9834 .9871 .9871 .9871 .9901 .9901 .9901 .9925 .9925 .9925 .9943 .9943 .9943 .9957 .9957 .9957 .9968 .9968 .9968 .9977 .9977 .9977 .9983 .9983 .9983 .9988 .9988 .9988 .9991 .9991 .9991 .9994 .9994 .9994 .9996 .9996 .9996 .9997 .9997 .9997 .04 .04.04 .5160 .5160 .5160 .5557 .5557 .5557 .5948 .5948 .5948 .6331 .6331 .6331 .6700 .6700 .6700 .7054 .7054 .7054 .7389 .7389 .7389 .7704 .7704 .7704 .7995 .7995 .7995 .8264 .8264 .8264 .8508 .8508 .8508 .8729 .8729 .8729 .8925 .8925 .8925 .9099 .9099 .9099 .9251 .9251 .9251 .9382 .9382 .9382 .9495 .9495 .9495 .9591 .9591 .9591 .9671 .9671 .9671 .9738 .9738 .9738 .9793 .9793 .9793 .9838 .9838 .9838 .9875 .9875 .9875 .9904 .9904 .9904 .9927 .9927 .9927 .9945 .9945 .9945 .9959 .9959 .9959 .9969 .9969 .9969 .9977 .9977 .9977 .9984 .9984 .9984 .9988 .9988 .9988 .9992 .9992 .9992 .9994 .9994 .9994 .9996 .9996 .9996 .9997 .9997 .9997 .05 .05.05 .5199 .5199 .5199 .5596 .5596 .5596 .5987 .5987 .5987 .6368 .6368 .6368 .6736 .6736 .6736 .7088 .7088 .7088 .7422 .7422 .7422 .7734 .7734 .7734 .8023 .8023 .8023 .8289 .8289 .8289 .8531 .8531 .8531 .8749 .8749 .8749 .8944 .8944 .8944 .9115 .9115 .9115 .9265 .9265 .9265 .9394 .9394 .9394 .9505 .9505 .9505 .9599 .9599 .9599 .9678 .9678 .9678 .9744 .9744 .9744 .9798 .9798 .9798 .9842 .9842 .9842 .9878 .9878 .9878 .9906 .9906 .9906 .9929 .9929 .9929 .9946 .9946 .9946 .9960 .9960 .9960 .9970 .9970 .9970 .9978 .9978 .9978 .9984 .9984 .9984 .9989 .9989 .9989 .9992 .9992 .9992 .9994 .9994 .9994 .9996 .9996 .9996 .9997 .9997 .9997 .06 .06.06 .5239 .5239 .5239 .5636 .5636 .5636 .6026 .6026 .6026 .6406 .6406 .6406 .6772 .6772 .6772 .7123 .7123 .7123 .7454 .7454 .7454 .7764 .7764 .7764 .8051 .8051 .8051 .8315 .8315 .8315 .8554 .8554 .8554 .8770 .8770 .8770 .8962 .8962 .8962 .9131 .9131 .9131 .9279 .9279 .9279 .9406 .9406 .9406 .9515 .9515 .9515 .9608 .9608 .9608 .9686 .9686 .9686 .9750 .9750 .9750 .9803 .9803 .9803 .9846 .9846 .9846 .9881 .9881 .9881 .9909 .9909 .9909 .9931 .9931 .9931 .9948 .9948 .9948 .9961 .9961 .9961 .9971 .9971 .9971 .9979 .9979 .9979 .9985 .9985 .9985 .9989 .9989 .9989 .9992 .9992 .9992 .9994 .9994 .9994 .9996 .9996 .9996 .9997 .9997 .9997 .07 .07.07 .5279 .5279 .5279 .5675 .5675 .5675 .6064 .6064 .6064 .6443 .6443 .6443 .6808 .6808 .6808 .7157 .7157 .7157 .7486 .7486 .7486 .7794 .7794 .7794 .8078 .8078 .8078 .8340 .8340 .8340 .8577 .8577 .8577 .8790 .8790 .8790 .8980 .8980 .8980 .9147 .9147 .9147 .9292 .9292 .9292 .9418 .9418 .9418 .9525 .9525 .9525 .9616 .9616 .9616 .9693 .9693 .9693 .9756 .9756 .9756 .9808 .9808 .9808 .9850 .9850 .9850 .9884 .9884 .9884 .9911 .9911 .9911 .9932 .9932 .9932 .9949 .9949 .9949 .9962 .9962 .9962 .9972 .9972 .9972 .9979 .9979 .9979 .9985 .9985 .9985 .9989 .9989 .9989 .9992 .9992 .9992 .9995 .9995 .9995 .9996 .9996 .9996 .9997 .9997 .9997 .08 .08.08 .5319 .5319 .5319 .5714 .5714 .5714 .6103 .6103 .6103 .6480 .6480 .6480 .6844 .6844 .6844 .7190 .7190 .7190 .7517 .7517 .7517 .7823 .7823 .7823 .8106 .8106 .8106 .8365 .8365 .8365 .8599 .8599 .8599 .8810 .8810 .8810 .8997 .8997 .8997 .9162 .9162 .9162 .9306 .9306 .9306 .9429 .9429 .9429 .9535 .9535 .9535 .9625 .9625 .9625 .9699 .9699 .9699 .9761 .9761 .9761 .9812 .9812 .9812 .9854 .9854 .9854 .9887 .9887 .9887 .9913 .9913 .9913 .9934 .9934 .9934 .9951 .9951 .9951 .9963 .9963 .9963 .9973 .9973 .9973 .9980 .9980 .9980 .9986 .9986 .9986 .9990 .9990 .9990 .9993 .9993 .9993 .9995 .9995 .9995 .9996 .9996 .9996 .9997 .9997 .9997 .09 .09.09 .5359 .5359 .5359 .5753 .5753 .5753 .6141 .6141 .6141 .6517 .6517 .6517 .6879 .6879 .6879 .7224 .7224 .7224 .7549 .7549 .7549 .7852 .7852 .7852 .8133 .8133 .8133 .8389 .8389 .8389 .8621 .8621 .8621 .8830 .8830 .8830 .9015 .9015 .9015 .9177 .9177 .9177 .9319 .9319 .9319 .9441 .9441 .9441 .9545 .9545 .9545 .9633 .9633 .9633 .9706 .9706 .9706 .9767 .9767 .9767 .9817 .9817 .9817 .9857 .9857 .9857 .9890 .9890 .9890 .9916 .9916 .9916 .9936 .9936 .9936 .9952 .9952 .9952 .9964 .9964 .9964 .9974 .9974 .9974 .9981 .9981 .9981 .9986 .9986 .9986 .9990 .9990 .9990 .9993 .9993 .9993 .9995 .9995 .9995 .9997 .9997 .9997 .9998 .9998 .9998 d.f. d.f.d.f. 11 1 22 2 33 3 44 4 55 5 66 6 77 7 88 8 99 9 10 1010 11 1111 12 1212 13 1313 14 1414 15 1515 16 1616 17 1717 18 1818 19 1919 20 2020 21 2121 22 2222 23 2323 24 2424 25 2525 26 2626 27 2727 28 2828 29 2929 q qq tt t αα α tt −t −t −t Left-tailed Left-tailed test test Left-tailed test Level Level Level of of of confidence, confidence, 0.50 cc c 0.50 confidence, 0.50 One One tail, tail, 0.25 0.25 AA A One tail, 0.25 Two Two tails, tails, 0.50 AA A 0.50 Two tails, 0.50 1.000 1.000 1.000 .816 .816 .816 .765 .765 .765 .741 .741 .741 .727 .727 .727 .718 .718 .718 .711 .711 .711 .706 .706 .706 .703 .703 .703 .700 .700 .700 .697 .697 .697 .695 .695 .695 .694 .694 .694 .692 .692 .692 .691 .691 .691 .690 .690 .690 .689 .689 .689 .688 .688 .688 .688 .688 .688 .687 .687 .687 .686 .686 .686 .686 .686 .686 .685 .685 .685 .685 .685 .685 .684 .684 .684 .684 .684 .684 .684 .684 .684 .683 .683 .683 .683 .683 .683 .674 .674 .674 0.80 0.80 0.80 0.10 0.10 0.10 0.20 0.20 0.20 3.078 3.078 3.078 1.886 1.886 1.886 1.638 1.638 1.638 1.533 1.533 1.533 1.476 1.476 1.476 1.440 1.440 1.440 1.415 1.415 1.415 1.397 1.397 1.397 1.383 1.383 1.383 1.372 1.372 1.372 1.363 1.363 1.363 1.356 1.356 1.356 1.350 1.350 1.350 1.345 1.345 1.345 1.341 1.341 1.341 1.337 1.337 1.337 1.333 1.333 1.333 1.330 1.330 1.330 1.328 1.328 1.328 1.325 1.325 1.325 1.323 1.323 1.323 1.321 1.321 1.321 1.319 1.319 1.319 1.318 1.318 1.318 1.316 1.316 1.316 1.315 1.315 1.315 1.314 1.314 1.314 1.313 1.313 1.313 1.311 1.311 1.311 1.282 1.282 1.282 t αα α tt t Right-tailed Right-tailed test test Right-tailed test 0.90 0.90 0.95 0.98 0.99 0.90 0.95 0.95 0.98 0.98 0.99 0.99 0.05 0.05 0.025 0.01 0.005 0.05 0.025 0.025 0.01 0.01 0.005 0.005 0.10 0.10 0.05 0.02 0.01 0.10 0.05 0.05 0.02 0.02 0.01 0.01 6.31412.706 12.70631.821 31.82163.657 63.657 6.314 6.314 12.706 31.821 63.657 2.920 2.920 4.303 6.965 9.925 2.920 4.303 4.303 6.965 6.965 9.925 9.925 2.353 2.353 3.182 4.541 5.841 2.353 3.182 3.182 4.541 4.541 5.841 5.841 2.132 2.132 2.776 3.747 4.604 2.132 2.776 2.776 3.747 3.747 4.604 4.604 2.015 2.015 2.571 3.365 4.032 2.015 2.571 2.571 3.365 3.365 4.032 4.032 1.943 1.943 2.447 3.143 3.707 1.943 2.447 2.447 3.143 3.143 3.707 3.707 1.895 1.895 2.365 2.998 3.499 1.895 2.365 2.365 2.998 2.998 3.499 3.499 1.860 1.860 2.306 2.896 3.355 1.860 2.306 2.306 2.896 2.896 3.355 3.355 1.833 1.833 2.262 2.821 3.250 1.833 2.262 2.262 2.821 2.821 3.250 3.250 1.812 1.812 2.228 2.764 3.169 1.812 2.228 2.228 2.764 2.764 3.169 3.169 1.796 1.796 2.201 2.718 3.106 1.796 2.201 2.201 2.718 2.718 3.106 3.106 1.782 1.782 2.179 2.681 3.055 1.782 2.179 2.179 2.681 2.681 3.055 3.055 1.771 1.771 2.160 2.650 3.012 1.771 2.160 2.160 2.650 2.650 3.012 3.012 1.761 1.761 2.145 2.624 2.977 1.761 2.145 2.145 2.624 2.624 2.977 2.977 1.753 1.753 2.131 2.602 2.947 1.753 2.131 2.131 2.602 2.602 2.947 2.947 1.746 1.746 2.120 2.583 2.921 1.746 2.120 2.120 2.583 2.583 2.921 2.921 1.740 1.740 2.110 2.567 2.898 1.740 2.110 2.110 2.567 2.567 2.898 2.898 1.734 1.734 2.101 2.552 2.878 1.734 2.101 2.101 2.552 2.552 2.878 2.878 1.729 1.729 2.093 2.539 2.861 1.729 2.093 2.093 2.539 2.539 2.861 2.861 1.725 1.725 2.086 2.528 2.845 1.725 2.086 2.086 2.528 2.528 2.845 2.845 1.721 1.721 2.080 2.518 2.831 1.721 2.080 2.080 2.518 2.518 2.831 2.831 1.717 1.717 2.074 2.508 2.819 1.717 2.074 2.074 2.508 2.508 2.819 2.819 1.714 1.714 2.069 2.500 2.807 1.714 2.069 2.069 2.500 2.500 2.807 2.807 1.711 1.711 2.064 2.492 2.797 1.711 2.064 2.064 2.492 2.492 2.797 2.797 1.708 1.708 2.060 2.485 2.787 1.708 2.060 2.060 2.485 2.485 2.787 2.787 1.706 1.706 2.056 2.479 2.779 1.706 2.056 2.056 2.479 2.479 2.779 2.779 1.703 1.703 2.052 2.473 2.771 1.703 2.052 2.052 2.473 2.473 2.771 2.771 1.701 1.701 2.048 2.467 2.763 1.701 2.048 2.048 2.467 2.467 2.763 2.763 1.699 1.699 2.045 2.462 2.756 1.699 2.045 2.045 2.462 2.462 2.756 2.756 1.645 1.645 1.960 2.326 2.576 1.645 1.960 1.960 2.326 2.326 2.576 2.576 cc c −z −zcc−z c zz==0z0= 0 zzcc z c zz z Copyright © 2012 Pearson Education, Inc. tt t −t −t −t 11 1 αα α 22 2 tt t Two-tailed Two-tailed test test Two-tailed test tt 11 1 αα α 22 2 αα α t χχ22 χ 2 χχ22 χ 2 Degrees Degrees of of of Degrees freedom freedom freedom 11 1 22 2 33 3 44 4 55 5 66 6 77 7 88 8 99 9 1010 10 1111 11 1212 12 1313 13 1414 14 1515 15 1616 16 1717 17 1818 18 1919 19 2020 20 2121 21 2222 22 2323 23 2424 24 2525 25 2626 26 2727 27 2828 28 2929 29 3030 30 4040 40 5050 50 6060 60 7070 70 8080 80 9090 90 100 100 100 χχ22 χ 2 LL Right tail Right tailtail Right 11 1 αα α 22 2 L χχ22 χ 2 RR χχ22 χ 2 R Two tails Two tails Two tails AA A 0.995 0.99 0.975 0.95 0.90 0.10 0.05 0.025 0.01 0.005 0.995 0.99 0.99 0.975 0.975 0.95 0.95 0.90 0.90 0.10 0.10 0.05 0.05 0.025 0.025 0.01 0.01 0.005 0.005 0.995 —— —— 0.001 0.001 0.004 0.016 2.706 3.841 5.024 6.635 7.879 0.001 0.004 0.004 0.016 0.016 2.706 2.706 3.841 3.841 5.024 5.024 6.635 6.635 7.879 7.879 — — 0.010 0.020 0.051 0.103 0.211 4.605 5.991 7.378 9.210 10.597 0.010 0.020 0.020 0.051 0.051 0.103 0.103 0.211 0.211 4.605 4.605 5.991 5.991 7.378 7.378 9.210 9.21010.597 10.597 0.010 0.072 0.115 0.216 0.352 0.584 6.251 7.815 9.348 11.345 12.838 0.072 0.115 0.115 0.216 0.216 0.352 0.352 0.584 0.584 6.251 6.251 7.815 7.815 9.348 9.34811.345 11.34512.838 12.838 0.072 0.207 0.297 0.484 0.711 1.064 7.779 9.488 11.143 13.277 14.860 0.207 0.297 0.297 0.484 0.484 0.711 0.711 1.064 1.064 7.779 7.779 9.488 9.48811.143 11.14313.277 13.27714.860 14.860 0.207 0.412 0.554 0.831 1.145 1.610 9.236 11.071 12.833 15.086 16.750 0.412 0.554 0.554 0.831 0.831 1.145 1.145 1.610 1.610 9.236 9.23611.071 11.07112.833 12.83315.086 15.08616.750 16.750 0.412 0.676 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.812 18.548 0.676 0.872 0.872 1.237 1.237 1.635 1.635 2.204 2.20410.645 10.64512.592 12.59214.449 14.44916.812 16.81218.548 18.548 0.676 0.989 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.475 20.278 0.989 1.239 1.239 1.690 1.690 2.167 2.167 2.833 2.83312.017 12.01714.067 14.06716.013 16.01318.475 18.47520.278 20.278 0.989 1.344 1.646 2.180 2.733 3.490 13.362 15.507 17.535 20.090 21.955 1.344 1.646 1.646 2.180 2.180 2.733 2.733 3.490 3.49013.362 13.36215.507 15.50717.535 17.53520.090 20.09021.955 21.955 1.344 1.735 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666 23.589 1.735 2.088 2.088 2.700 2.700 3.325 3.325 4.168 4.16814.684 14.68416.919 16.91919.023 19.02321.666 21.66623.589 23.589 1.735 2.156 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 25.188 2.156 2.558 2.558 3.247 3.247 3.940 3.940 4.865 4.86515.987 15.98718.307 18.30720.483 20.48323.209 23.20925.188 25.188 2.156 2.603 3.053 3.816 4.575 5.578 17.275 19.675 21.920 24.725 26.757 2.603 3.053 3.053 3.816 3.816 4.575 4.575 5.578 5.57817.275 17.27519.675 19.67521.920 21.92024.725 24.72526.757 26.757 2.603 3.074 3.571 4.404 5.226 6.304 18.549 21.026 23.337 26.217 28.299 3.074 3.571 3.571 4.404 4.404 5.226 5.226 6.304 6.30418.549 18.54921.026 21.02623.337 23.33726.217 26.21728.299 28.299 3.074 3.565 4.107 5.009 5.892 7.042 19.812 22.362 24.736 27.688 29.819 3.565 4.107 4.107 5.009 5.009 5.892 5.892 7.042 7.04219.812 19.81222.362 22.36224.736 24.73627.688 27.68829.819 29.819 3.565 4.075 4.660 5.629 6.571 7.790 21.064 23.685 26.119 29.141 31.319 4.075 4.660 4.660 5.629 5.629 6.571 6.571 7.790 7.79021.064 21.06423.685 23.68526.119 26.11929.141 29.14131.319 31.319 4.075 4.601 5.229 6.262 7.261 8.547 22.307 24.996 27.488 30.578 32.801 4.601 5.229 5.229 6.262 6.262 7.261 7.261 8.547 8.54722.307 22.30724.996 24.99627.488 27.48830.578 30.57832.801 32.801 4.601 5.142 5.812 6.908 7.962 9.312 23.542 26.296 28.845 32.000 34.267 5.142 5.812 5.812 6.908 6.908 7.962 7.962 9.312 9.31223.542 23.54226.296 26.29628.845 28.84532.000 32.00034.267 34.267 5.142 5.697 6.408 7.564 8.672 10.085 24.769 27.587 30.191 33.409 35.718 5.697 6.408 6.408 7.564 7.564 8.672 8.67210.085 10.08524.769 24.76927.587 27.58730.191 30.19133.409 33.40935.718 35.718 5.697 6.265 7.015 8.231 9.390 10.865 25.989 28.869 31.526 34.805 37.156 6.265 7.015 7.015 8.231 8.231 9.390 9.39010.865 10.86525.989 25.98928.869 28.86931.526 31.52634.805 34.80537.156 37.156 6.265 6.844 7.633 8.907 10.117 11.651 27.204 30.144 32.852 36.191 38.582 6.844 7.633 7.633 8.907 8.90710.117 10.11711.651 11.65127.204 27.20430.144 30.14432.852 32.85236.191 36.19138.582 38.582 6.844 7.434 8.260 9.591 10.851 12.443 28.412 31.410 34.170 37.566 39.997 7.434 8.260 8.260 9.591 9.59110.851 10.85112.443 12.44328.412 28.41231.410 31.41034.170 34.17037.566 37.56639.997 39.997 7.434 8.034 8.897 10.283 11.591 13.240 29.615 32.671 35.479 38.932 41.401 8.034 8.897 8.89710.283 10.28311.591 11.59113.240 13.24029.615 29.61532.671 32.67135.479 35.47938.932 38.93241.401 41.401 8.034 8.643 9.542 10.982 12.338 14.042 30.813 33.924 36.781 40.289 42.796 8.643 9.542 9.54210.982 10.98212.338 12.33814.042 14.04230.813 30.81333.924 33.92436.781 36.78140.289 40.28942.796 42.796 8.643 9.260 9.26010.196 10.196 10.19611.689 11.689 11.68913.091 13.091 13.09114.848 14.848 14.84832.007 32.007 32.00735.172 35.172 35.17238.076 38.076 38.07641.638 41.638 41.63844.181 44.181 44.181 9.260 9.886 10.856 12.401 13.848 15.659 33.196 36.415 39.364 42.980 45.559 9.88610.856 10.85612.401 12.40113.848 13.84815.659 15.65933.196 33.19636.415 36.41539.364 39.36442.980 42.98045.559 45.559 9.886 10.520 11.524 13.120 14.611 16.473 34.382 37.652 40.646 44.314 46.928 10.52011.524 11.52413.120 13.12014.611 14.61116.473 16.47334.382 34.38237.652 37.65240.646 40.64644.314 44.31446.928 46.928 10.520 11.160 12.198 13.844 15.379 17.292 35.563 38.885 41.923 45.642 48.290 11.16012.198 12.19813.844 13.84415.379 15.37917.292 17.29235.563 35.56338.885 38.88541.923 41.92345.642 45.64248.290 48.290 11.160 11.808 12.879 14.573 16.151 18.114 36.741 40.113 43.194 46.963 49.645 11.80812.879 12.87914.573 14.57316.151 16.15118.114 18.11436.741 36.74140.113 40.11343.194 43.19446.963 46.96349.645 49.645 11.808 12.461 13.565 15.308 16.928 18.939 37.916 41.337 44.461 48.278 50.993 12.46113.565 13.56515.308 15.30816.928 16.92818.939 18.93937.916 37.91641.337 41.33744.461 44.46148.278 48.27850.993 50.993 12.461 13.121 13.12114.257 14.257 14.25716.047 16.047 16.04717.708 17.708 17.70819.768 19.768 19.76839.087 39.087 39.08742.557 42.557 42.55745.722 45.722 45.72249.588 49.588 49.58852.336 52.336 52.336 13.121 13.787 14.954 16.791 18.493 20.599 40.256 43.773 46.979 50.892 53.672 13.78714.954 14.95416.791 16.79118.493 18.49320.599 20.59940.256 40.25643.773 43.77346.979 46.97950.892 50.89253.672 53.672 13.787 20.707 22.164 24.433 26.509 29.051 51.805 55.758 59.342 63.691 66.766 20.70722.164 22.16424.433 24.43326.509 26.50929.051 29.05151.805 51.80555.758 55.75859.342 59.34263.691 63.69166.766 66.766 20.707 27.991 29.707 32.357 34.764 37.689 63.167 67.505 71.420 76.154 79.490 27.99129.707 29.70732.357 32.35734.764 34.76437.689 37.68963.167 63.16767.505 67.50571.420 71.42076.154 76.15479.490 79.490 27.991 35.534 37.485 40.482 43.188 46.459 74.397 79.082 83.298 88.379 91.952 35.53437.485 37.48540.482 40.48243.188 43.18846.459 46.45974.397 74.39779.082 79.08283.298 83.29888.379 88.37991.952 91.952 35.534 43.275 45.442 48.758 51.739 55.329 85.527 90.531 95.023 100.425 104.215 43.27545.442 45.44248.758 48.75851.739 51.73955.329 55.32985.527 85.52790.531 90.53195.023 95.023 100.425 104.215 43.275 100.425 104.215 51.172 51.17253.540 53.540 53.54057.153 57.153 57.15360.391 60.391 60.39164.278 64.278 64.27896.578 96.578 96.578 101.879 101.879 106.629 106.629 112.329 112.329 116.321 116.321 51.172 101.879 106.629 112.329 116.321 59.196 59.19661.754 61.754 61.75465.647 65.647 65.64769.126 69.126 69.12673.291 73.291 73.291 107.565 107.565 113.145 113.145 118.136 118.136 124.116 124.116 128.299 128.299 59.196 107.565 113.145 118.136 124.116 128.299 67.328 70.065 74.222 77.929 82.358 118.498 124.342 129.561 135.807 140.169 67.32870.065 70.06574.222 74.22277.929 77.92982.358 82.358 118.498 124.342 129.561 135.807 140.169 67.328 118.498 124.342 129.561 135.807 140.169 41831S4_INS p5-8 AM 11 1 41831S4_INS p5-8 11/8/07 10:03 AMAMPage Page 41831S4_INS p5-811/8/07 11/8/0710:03 10:03 Page Table Table 44— 4— Standard Standard Normal Normal Distribution Distribution (continued) (continued) Table — Standard Normal Distribution (continued) Table Table 44— — Standard Standard Normal Normal Distribution Distribution Table 4— Standard Normal Distribution Table Table Table 6— 6— 6— Chi-Square Chi-Square Chi-Square Distribution Distribution Distribution Area Area Area Area Area Area 11 1 αα α 22 2 zz z zz z .09 .09.09 .0002 .0002 .0002 .0003 .0003 .0003 .0005 .0005 .0005 .0007 .0007 .0007 .0010 .0010 .0010 .0014 .0014 .0014 .0019 .0019 .0019 .0026 .0026 .0026 .0036 .0036 .0036 .0048 .0048 .0048 .0064 .0064 .0064 .0084 .0084 .0084 .0110 .0110 .0110 .0143 .0143 .0143 .0183 .0183 .0183 .0233 .0233 .0233 .0294 .0294 .0294 .0367 .0367 .0367 .0455 .0455 .0455 .0559 .0559 .0559 .0681 .0681 .0681 .0823 .0823 .0823 .0985 .0985 .0985 .1170 .1170 .1170 .1379 .1379 .1379 .1611 .1611 .1611 .1867 .1867 .1867 .2148 .2148 .2148 .2451 .2451 .2451 .2776 .2776 .2776 .3121 .3121 .3121 .3483 .3483 .3483 .3859 .3859 .3859 .4247 .4247 .4247 .4641 .4641 .4641 −t −t −t 00 0 zz z zz z 00 0 zz z 3.4 � � 3.4 �3.4 3.3 � � 3.3 �3.3 3.2 � � 3.2 �3.2 3.1 � � 3.1 �3.1 3.0 � � 3.0 �3.0 2.9 � � 2.9 �2.9 2.8 � � 2.8 �2.8 2.7 � � 2.7 �2.7 2.6 � � 2.6 �2.6 2.5 � � 2.5 �2.5 2.4 � � 2.4 �2.4 2.3 � � 2.3 �2.3 2.2 � � 2.2 �2.2 2.1 � � 2.1 �2.1 2.0 � � 2.0 �2.0 1.9 � � 1.9 �1.9 1.8 � � 1.8 �1.8 1.7 � � 1.7 �1.7 1.6 � � 1.6 �1.6 1.5 � � 1.5 �1.5 1.4 � � 1.4 �1.4 1.3 � � 1.3 �1.3 1.2 � � 1.2 �1.2 1.1 � � 1.1 �1.1 1.0 � � 1.0 �1.0 0.9 � � 0.9 �0.9 0.8 � � 0.8 �0.8 0.7 � � 0.7 �0.7 0.6 � � 0.6 �0.6 0.5 � � 0.5 �0.5 0.4 � � 0.4 �0.4 0.3 � � 0.3 �0.3 0.2 � � 0.2 �0.2 0.1 � � 0.1 �0.1 �0.0 0.0 � � 0.0 Table Table Table 5— 5— 5— t-Distribution t-Distribution t-Distribution .08 .08.08 .0003 .0003 .0003 .0004 .0004 .0004 .0005 .0005 .0005 .0007 .0007 .0007 .0010 .0010 .0010 .0014 .0014 .0014 .0020 .0020 .0020 .0027 .0027 .0027 .0037 .0037 .0037 .0049 .0049 .0049 .0066 .0066 .0066 .0087 .0087 .0087 .0113 .0113 .0113 .0146 .0146 .0146 .0188 .0188 .0188 .0239 .0239 .0239 .0301 .0301 .0301 .0375 .0375 .0375 .0465 .0465 .0465 .0571 .0571 .0571 .0694 .0694 .0694 .0838 .0838 .0838 .1003 .1003 .1003 .1190 .1190 .1190 .1401 .1401 .1401 .1635 .1635 .1635 .1894 .1894 .1894 .2177 .2177 .2177 .2483 .2483 .2483 .2810 .2810 .2810 .3156 .3156 .3156 .3520 .3520 .3520 .3897 .3897 .3897 .4286 .4286 .4286 .4681 .4681 .4681 .07 .07.07 .0003 .0003 .0003 .0004 .0004 .0004 .0005 .0005 .0005 .0008 .0008 .0008 .0011 .0011 .0011 .0015 .0015 .0015 .0021 .0021 .0021 .0028 .0028 .0028 .0038 .0038 .0038 .0051 .0051 .0051 .0068 .0068 .0068 .0089 .0089 .0089 .0116 .0116 .0116 .0150 .0150 .0150 .0192 .0192 .0192 .0244 .0244 .0244 .0307 .0307 .0307 .0384 .0384 .0384 .0475 .0475 .0475 .0582 .0582 .0582 .0708 .0708 .0708 .0853 .0853 .0853 .1020 .1020 .1020 .1210 .1210 .1210 .1423 .1423 .1423 .1660 .1660 .1660 .1922 .1922 .1922 .2206 .2206 .2206 .2514 .2514 .2514 .2843 .2843 .2843 .3192 .3192 .3192 .3557 .3557 .3557 .3936 .3936 .3936 .4325 .4325 .4325 .4721 .4721 .4721 .06 .06.06 .0003 .0003 .0003 .0004 .0004 .0004 .0006 .0006 .0006 .0008 .0008 .0008 .0011 .0011 .0011 .0015 .0015 .0015 .0021 .0021 .0021 .0029 .0029 .0029 .0039 .0039 .0039 .0052 .0052 .0052 .0069 .0069 .0069 .0091 .0091 .0091 .0119 .0119 .0119 .0154 .0154 .0154 .0197 .0197 .0197 .0250 .0250 .0250 .0314 .0314 .0314 .0392 .0392 .0392 .0485 .0485 .0485 .0594 .0594 .0594 .0721 .0721 .0721 .0869 .0869 .0869 .1038 .1038 .1038 .1230 .1230 .1230 .1446 .1446 .1446 .1685 .1685 .1685 .1949 .1949 .1949 .2236 .2236 .2236 .2546 .2546 .2546 .2877 .2877 .2877 .3228 .3228 .3228 .3594 .3594 .3594 .3974 .3974 .3974 .4364 .4364 .4364 .4761 .4761 .4761 .05 .05.05 .0003 .0003 .0003 .0004 .0004 .0004 .0006 .0006 .0006 .0008 .0008 .0008 .0011 .0011 .0011 .0016 .0016 .0016 .0022 .0022 .0022 .0030 .0030 .0030 .0040 .0040 .0040 .0054 .0054 .0054 .0071 .0071 .0071 .0094 .0094 .0094 .0122 .0122 .0122 .0158 .0158 .0158 .0202 .0202 .0202 .0256 .0256 .0256 .0322 .0322 .0322 .0401 .0401 .0401 .0495 .0495 .0495 .0606 .0606 .0606 .0735 .0735 .0735 .0885 .0885 .0885 .1056 .1056 .1056 .1251 .1251 .1251 .1469 .1469 .1469 .1711 .1711 .1711 .1977 .1977 .1977 .2266 .2266 .2266 .2578 .2578 .2578 .2912 .2912 .2912 .3264 .3264 .3264 .3632 .3632 .3632 .4013 .4013 .4013 .4404 .4404 .4404 .4801 .4801 .4801 .04 .04.04 .03 .03.03 .0003 .0003 .0003 .0003 .0003 .0003 .0004 .0004 .0004 .0004 .0004 .0004 .0006 .0006 .0006 .0006 .0006 .0006 .0008 .0008 .0009 .0008 .0009 .0009 .0012 .0012 .0012 .0012 .0012 .0012 .0016 .0016 .0017 .0016 .0017 .0017 .0023 .0023 .0023 .0023 .0023 .0023 .0031 .0031 .0032 .0031 .0032 .0032 .0041 .0043 .0041 .0041 .0043 .0043 .0055 .0055 .0057 .0055 .0057 .0057 .0073 .0075 .0073 .0073 .0075 .0075 .0096 .0096 .0099 .0096 .0099 .0099 .0125 .0129 .0125 .0125 .0129 .0129 .0162 .0162 .0166 .0162 .0166 .0166 .0207 .0212 .0207 .0207 .0212 .0212 .0262 .0262 .0268 .0262 .0268 .0268 .0329 .0336 .0329 .0329 .0336 .0336 .0409 .0409 .0418 .0409 .0418 .0418 .0505 .0516 .0505 .0505 .0516 .0516 .0618 .0618 .0630 .0618 .0630 .0630 .0749 .0764 .0749 .0749 .0764 .0764 .0901 .0901 .0918 .0901 .0918 .0918 .1075 .1093 .1075 .1075 .1093 .1093 .1271 .1271 .1292 .1271 .1292 .1292 .1492 .1515 .1492 .1492 .1515 .1515 .1736 .1736 .1762 .1736 .1762 .1762 .2005 .2033 .2005 .2005 .2033 .2033 .2296 .2296 .2327 .2296 .2327 .2327 .2611 .2643 .2611 .2611 .2643 .2643 .2946 .2946 .2981 .2946 .2981 .2981 .3300 .3336 .3300 .3300 .3336 .3336 .3669 .3669 .3707 .3669 .3707 .3707 .4052 .4090 .4052 .4052 .4090 .4090 .4443 .4443 .4483 .4443 .4483 .4483 .4840 .4840 .4880 .4840 .4880 .4880 Critical Critical Values Values Critical Values cc c Level Level of ofConfidence Confidence Level of Confidence 0.80 0.80 0.80 0.90 0.90 0.90 0.95 0.95 0.95 0.99 0.99 0.99 zzcc zc 1.28 1.28 1.28 1.645 1.645 1.645 1.96 1.96 1.96 2.575 2.575 2.575 cc c −z −zcc−z c zz==0z0= 0 zzcc z c zz z .02 .02.02 .0003 .0003 .0003 .0005 .0005 .0005 .0006 .0006 .0006 .0009 .0009 .0009 .0013 .0013 .0013 .0018 .0018 .0018 .0024 .0024 .0024 .0033 .0033 .0033 .0044 .0044 .0044 .0059 .0059 .0059 .0078 .0078 .0078 .0102 .0102 .0102 .0132 .0132 .0132 .0170 .0170 .0170 .0217 .0217 .0217 .0274 .0274 .0274 .0344 .0344 .0344 .0427 .0427 .0427 .0526 .0526 .0526 .0643 .0643 .0643 .0778 .0778 .0778 .0934 .0934 .0934 .1112 .1112 .1112 .1314 .1314 .1314 .1539 .1539 .1539 .1788 .1788 .1788 .2061 .2061 .2061 .2358 .2358 .2358 .2676 .2676 .2676 .3015 .3015 .3015 .3372 .3372 .3372 .3745 .3745 .3745 .4129 .4129 .4129 .4522 .4522 .4522 .4920 .4920 .4920 .01 .01.01 .0003 .0003 .0003 .0005 .0005 .0005 .0007 .0007 .0007 .0009 .0009 .0009 .0013 .0013 .0013 .0018 .0018 .0018 .0025 .0025 .0025 .0034 .0034 .0034 .0045 .0045 .0045 .0060 .0060 .0060 .0080 .0080 .0080 .0104 .0104 .0104 .0136 .0136 .0136 .0174 .0174 .0174 .0222 .0222 .0222 .0281 .0281 .0281 .0351 .0351 .0351 .0436 .0436 .0436 .0537 .0537 .0537 .0655 .0655 .0655 .0793 .0793 .0793 .0951 .0951 .0951 .1131 .1131 .1131 .1335 .1335 .1335 .1562 .1562 .1562 .1814 .1814 .1814 .2090 .2090 .2090 .2389 .2389 .2389 .2709 .2709 .2709 .3050 .3050 .3050 .3409 .3409 .3409 .3783 .3783 .3783 .4168 .4168 .4168 .4562 .4562 .4562 .4960 .4960 .4960 .00 .00.00 .0003 .0003 .0003 .0005 .0005 .0005 .0007 .0007 .0007 .0010 .0010 .0010 .0013 .0013 .0013 .0019 .0019 .0019 .0026 .0026 .0026 .0035 .0035 .0035 .0047 .0047 .0047 .0062 .0062 .0062 .0082 .0082 .0082 .0107 .0107 .0107 .0139 .0139 .0139 .0179 .0179 .0179 .0228 .0228 .0228 .0287 .0287 .0287 .0359 .0359 .0359 .0446 .0446 .0446 .0548 .0548 .0548 .0668 .0668 .0668 .0808 .0808 .0808 .0968 .0968 .0968 .1151 .1151 .1151 .1357 .1357 .1357 .1587 .1587 .1587 .1841 .1841 .1841 .2119 .2119 .2119 .2420 .2420 .2420 .2743 .2743 .2743 .3085 .3085 .3085 .3446 .3446 .3446 .3821 .3821 .3821 .4207 .4207 .4207 .4602 .4602 .4602 .5000 .5000 .5000 zz z 0.0 0.00.0 0.1 0.10.1 0.20.2 0.2 0.3 0.30.3 0.40.4 0.4 0.5 0.50.5 0.60.6 0.6 0.7 0.70.7 0.80.8 0.8 0.9 0.90.9 1.01.0 1.0 1.1 1.11.1 1.21.2 1.2 1.3 1.31.3 1.41.4 1.4 1.5 1.51.5 1.61.6 1.6 1.7 1.71.7 1.81.8 1.8 1.9 1.91.9 2.02.0 2.0 2.1 2.12.1 2.22.2 2.2 2.3 2.32.3 2.42.4 2.4 2.5 2.52.5 2.62.6 2.6 2.7 2.72.7 2.82.8 2.8 2.9 2.92.9 3.03.0 3.0 3.1 3.13.1 3.23.2 3.2 3.3 3.33.3 3.4 3.43.4 .00 .00.00 .5000 .5000 .5000 .5398 .5398 .5398 .5793 .5793 .5793 .6179 .6179 .6179 .6554 .6554 .6554 .6915 .6915 .6915 .7257 .7257 .7257 .7580 .7580 .7580 .7881 .7881 .7881 .8159 .8159 .8159 .8413 .8413 .8413 .8643 .8643 .8643 .8849 .8849 .8849 .9032 .9032 .9032 .9192 .9192 .9192 .9332 .9332 .9332 .9452 .9452 .9452 .9554 .9554 .9554 .9641 .9641 .9641 .9713 .9713 .9713 .9772 .9772 .9772 .9821 .9821 .9821 .9861 .9861 .9861 .9893 .9893 .9893 .9918 .9918 .9918 .9938 .9938 .9938 .9953 .9953 .9953 .9965 .9965 .9965 .9974 .9974 .9974 .9981 .9981 .9981 .9987 .9987 .9987 .9990 .9990 .9990 .9993 .9993 .9993 .9995 .9995 .9995 .9997 .9997 .9997 .01 .01.01 .5040 .5040 .5040 .5438 .5438 .5438 .5832 .5832 .5832 .6217 .6217 .6217 .6591 .6591 .6591 .6950 .6950 .6950 .7291 .7291 .7291 .7611 .7611 .7611 .7910 .7910 .7910 .8186 .8186 .8186 .8438 .8438 .8438 .8665 .8665 .8665 .8869 .8869 .8869 .9049 .9049 .9049 .9207 .9207 .9207 .9345 .9345 .9345 .9463 .9463 .9463 .9564 .9564 .9564 .9649 .9649 .9649 .9719 .9719 .9719 .9778 .9778 .9778 .9826 .9826 .9826 .9864 .9864 .9864 .9896 .9896 .9896 .9920 .9920 .9920 .9940 .9940 .9940 .9955 .9955 .9955 .9966 .9966 .9966 .9975 .9975 .9975 .9982 .9982 .9982 .9987 .9987 .9987 .9991 .9991 .9991 .9993 .9993 .9993 .9995 .9995 .9995 .9997 .9997 .9997 tt t c-confidence c-confidence interval interval c-confidence interval .02 .02.02 .5080 .5080 .5080 .5478 .5478 .5478 .5871 .5871 .5871 .6255 .6255 .6255 .6628 .6628 .6628 .6985 .6985 .6985 .7324 .7324 .7324 .7642 .7642 .7642 .7939 .7939 .7939 .8212 .8212 .8212 .8461 .8461 .8461 .8686 .8686 .8686 .8888 .8888 .8888 .9066 .9066 .9066 .9222 .9222 .9222 .9357 .9357 .9357 .9474 .9474 .9474 .9573 .9573 .9573 .9656 .9656 .9656 .9726 .9726 .9726 .9783 .9783 .9783 .9830 .9830 .9830 .9868 .9868 .9868 .9898 .9898 .9898 .9922 .9922 .9922 .9941 .9941 .9941 .9956 .9956 .9956 .9967 .9967 .9967 .9976 .9976 .9976 .9982 .9982 .9982 .9987 .9987 .9987 .9991 .9991 .9991 .9994 .9994 .9994 .9995 .9995 .9995 .9997 .9997 .9997 .03 .03.03 .5120 .5120 .5120 .5517 .5517 .5517 .5910 .5910 .5910 .6293 .6293 .6293 .6664 .6664 .6664 .7019 .7019 .7019 .7357 .7357 .7357 .7673 .7673 .7673 .7967 .7967 .7967 .8238 .8238 .8238 .8485 .8485 .8485 .8708 .8708 .8708 .8907 .8907 .8907 .9082 .9082 .9082 .9236 .9236 .9236 .9370 .9370 .9370 .9484 .9484 .9484 .9582 .9582 .9582 .9664 .9664 .9664 .9732 .9732 .9732 .9788 .9788 .9788 .9834 .9834 .9834 .9871 .9871 .9871 .9901 .9901 .9901 .9925 .9925 .9925 .9943 .9943 .9943 .9957 .9957 .9957 .9968 .9968 .9968 .9977 .9977 .9977 .9983 .9983 .9983 .9988 .9988 .9988 .9991 .9991 .9991 .9994 .9994 .9994 .9996 .9996 .9996 .9997 .9997 .9997 .04 .04.04 .5160 .5160 .5160 .5557 .5557 .5557 .5948 .5948 .5948 .6331 .6331 .6331 .6700 .6700 .6700 .7054 .7054 .7054 .7389 .7389 .7389 .7704 .7704 .7704 .7995 .7995 .7995 .8264 .8264 .8264 .8508 .8508 .8508 .8729 .8729 .8729 .8925 .8925 .8925 .9099 .9099 .9099 .9251 .9251 .9251 .9382 .9382 .9382 .9495 .9495 .9495 .9591 .9591 .9591 .9671 .9671 .9671 .9738 .9738 .9738 .9793 .9793 .9793 .9838 .9838 .9838 .9875 .9875 .9875 .9904 .9904 .9904 .9927 .9927 .9927 .9945 .9945 .9945 .9959 .9959 .9959 .9969 .9969 .9969 .9977 .9977 .9977 .9984 .9984 .9984 .9988 .9988 .9988 .9992 .9992 .9992 .9994 .9994 .9994 .9996 .9996 .9996 .9997 .9997 .9997 .05 .05.05 .5199 .5199 .5199 .5596 .5596 .5596 .5987 .5987 .5987 .6368 .6368 .6368 .6736 .6736 .6736 .7088 .7088 .7088 .7422 .7422 .7422 .7734 .7734 .7734 .8023 .8023 .8023 .8289 .8289 .8289 .8531 .8531 .8531 .8749 .8749 .8749 .8944 .8944 .8944 .9115 .9115 .9115 .9265 .9265 .9265 .9394 .9394 .9394 .9505 .9505 .9505 .9599 .9599 .9599 .9678 .9678 .9678 .9744 .9744 .9744 .9798 .9798 .9798 .9842 .9842 .9842 .9878 .9878 .9878 .9906 .9906 .9906 .9929 .9929 .9929 .9946 .9946 .9946 .9960 .9960 .9960 .9970 .9970 .9970 .9978 .9978 .9978 .9984 .9984 .9984 .9989 .9989 .9989 .9992 .9992 .9992 .9994 .9994 .9994 .9996 .9996 .9996 .9997 .9997 .9997 .06 .06.06 .5239 .5239 .5239 .5636 .5636 .5636 .6026 .6026 .6026 .6406 .6406 .6406 .6772 .6772 .6772 .7123 .7123 .7123 .7454 .7454 .7454 .7764 .7764 .7764 .8051 .8051 .8051 .8315 .8315 .8315 .8554 .8554 .8554 .8770 .8770 .8770 .8962 .8962 .8962 .9131 .9131 .9131 .9279 .9279 .9279 .9406 .9406 .9406 .9515 .9515 .9515 .9608 .9608 .9608 .9686 .9686 .9686 .9750 .9750 .9750 .9803 .9803 .9803 .9846 .9846 .9846 .9881 .9881 .9881 .9909 .9909 .9909 .9931 .9931 .9931 .9948 .9948 .9948 .9961 .9961 .9961 .9971 .9971 .9971 .9979 .9979 .9979 .9985 .9985 .9985 .9989 .9989 .9989 .9992 .9992 .9992 .9994 .9994 .9994 .9996 .9996 .9996 .9997 .9997 .9997 .07 .07.07 .5279 .5279 .5279 .5675 .5675 .5675 .6064 .6064 .6064 .6443 .6443 .6443 .6808 .6808 .6808 .7157 .7157 .7157 .7486 .7486 .7486 .7794 .7794 .7794 .8078 .8078 .8078 .8340 .8340 .8340 .8577 .8577 .8577 .8790 .8790 .8790 .8980 .8980 .8980 .9147 .9147 .9147 .9292 .9292 .9292 .9418 .9418 .9418 .9525 .9525 .9525 .9616 .9616 .9616 .9693 .9693 .9693 .9756 .9756 .9756 .9808 .9808 .9808 .9850 .9850 .9850 .9884 .9884 .9884 .9911 .9911 .9911 .9932 .9932 .9932 .9949 .9949 .9949 .9962 .9962 .9962 .9972 .9972 .9972 .9979 .9979 .9979 .9985 .9985 .9985 .9989 .9989 .9989 .9992 .9992 .9992 .9995 .9995 .9995 .9996 .9996 .9996 .9997 .9997 .9997 .08 .08.08 .5319 .5319 .5319 .5714 .5714 .5714 .6103 .6103 .6103 .6480 .6480 .6480 .6844 .6844 .6844 .7190 .7190 .7190 .7517 .7517 .7517 .7823 .7823 .7823 .8106 .8106 .8106 .8365 .8365 .8365 .8599 .8599 .8599 .8810 .8810 .8810 .8997 .8997 .8997 .9162 .9162 .9162 .9306 .9306 .9306 .9429 .9429 .9429 .9535 .9535 .9535 .9625 .9625 .9625 .9699 .9699 .9699 .9761 .9761 .9761 .9812 .9812 .9812 .9854 .9854 .9854 .9887 .9887 .9887 .9913 .9913 .9913 .9934 .9934 .9934 .9951 .9951 .9951 .9963 .9963 .9963 .9973 .9973 .9973 .9980 .9980 .9980 .9986 .9986 .9986 .9990 .9990 .9990 .9993 .9993 .9993 .9995 .9995 .9995 .9996 .9996 .9996 .9997 .9997 .9997 .09 .09.09 .5359 .5359 .5359 .5753 .5753 .5753 .6141 .6141 .6141 .6517 .6517 .6517 .6879 .6879 .6879 .7224 .7224 .7224 .7549 .7549 .7549 .7852 .7852 .7852 .8133 .8133 .8133 .8389 .8389 .8389 .8621 .8621 .8621 .8830 .8830 .8830 .9015 .9015 .9015 .9177 .9177 .9177 .9319 .9319 .9319 .9441 .9441 .9441 .9545 .9545 .9545 .9633 .9633 .9633 .9706 .9706 .9706 .9767 .9767 .9767 .9817 .9817 .9817 .9857 .9857 .9857 .9890 .9890 .9890 .9916 .9916 .9916 .9936 .9936 .9936 .9952 .9952 .9952 .9964 .9964 .9964 .9974 .9974 .9974 .9981 .9981 .9981 .9986 .9986 .9986 .9990 .9990 .9990 .9993 .9993 .9993 .9995 .9995 .9995 .9997 .9997 .9997 .9998 .9998 .9998 d.f. d.f.d.f. 11 1 22 2 33 3 44 4 55 5 66 6 77 7 88 8 99 9 10 1010 11 1111 12 1212 13 1313 14 1414 15 1515 16 1616 17 1717 18 1818 19 1919 20 2020 21 2121 22 2222 23 2323 24 2424 25 2525 26 2626 27 2727 28 2828 29 2929 q qq tt t αα α tt −t −t −t Left-tailed Left-tailed test test Left-tailed test Level Level Level of of of confidence, confidence, 0.50 cc c 0.50 confidence, 0.50 One One tail, tail, 0.25 0.25 AA A One tail, 0.25 Two Two tails, tails, 0.50 AA A 0.50 Two tails, 0.50 1.000 1.000 1.000 .816 .816 .816 .765 .765 .765 .741 .741 .741 .727 .727 .727 .718 .718 .718 .711 .711 .711 .706 .706 .706 .703 .703 .703 .700 .700 .700 .697 .697 .697 .695 .695 .695 .694 .694 .694 .692 .692 .692 .691 .691 .691 .690 .690 .690 .689 .689 .689 .688 .688 .688 .688 .688 .688 .687 .687 .687 .686 .686 .686 .686 .686 .686 .685 .685 .685 .685 .685 .685 .684 .684 .684 .684 .684 .684 .684 .684 .684 .683 .683 .683 .683 .683 .683 .674 .674 .674 0.80 0.80 0.80 0.10 0.10 0.10 0.20 0.20 0.20 3.078 3.078 3.078 1.886 1.886 1.886 1.638 1.638 1.638 1.533 1.533 1.533 1.476 1.476 1.476 1.440 1.440 1.440 1.415 1.415 1.415 1.397 1.397 1.397 1.383 1.383 1.383 1.372 1.372 1.372 1.363 1.363 1.363 1.356 1.356 1.356 1.350 1.350 1.350 1.345 1.345 1.345 1.341 1.341 1.341 1.337 1.337 1.337 1.333 1.333 1.333 1.330 1.330 1.330 1.328 1.328 1.328 1.325 1.325 1.325 1.323 1.323 1.323 1.321 1.321 1.321 1.319 1.319 1.319 1.318 1.318 1.318 1.316 1.316 1.316 1.315 1.315 1.315 1.314 1.314 1.314 1.313 1.313 1.313 1.311 1.311 1.311 1.282 1.282 1.282 t αα α tt t Right-tailed Right-tailed test test Right-tailed test 0.90 0.90 0.95 0.98 0.99 0.90 0.95 0.95 0.98 0.98 0.99 0.99 0.05 0.05 0.025 0.01 0.005 0.05 0.025 0.025 0.01 0.01 0.005 0.005 0.10 0.10 0.05 0.02 0.01 0.10 0.05 0.05 0.02 0.02 0.01 0.01 6.31412.706 12.70631.821 31.82163.657 63.657 6.314 6.314 12.706 31.821 63.657 2.920 2.920 4.303 6.965 9.925 2.920 4.303 4.303 6.965 6.965 9.925 9.925 2.353 2.353 3.182 4.541 5.841 2.353 3.182 3.182 4.541 4.541 5.841 5.841 2.132 2.132 2.776 3.747 4.604 2.132 2.776 2.776 3.747 3.747 4.604 4.604 2.015 2.015 2.571 3.365 4.032 2.015 2.571 2.571 3.365 3.365 4.032 4.032 1.943 1.943 2.447 3.143 3.707 1.943 2.447 2.447 3.143 3.143 3.707 3.707 1.895 1.895 2.365 2.998 3.499 1.895 2.365 2.365 2.998 2.998 3.499 3.499 1.860 1.860 2.306 2.896 3.355 1.860 2.306 2.306 2.896 2.896 3.355 3.355 1.833 1.833 2.262 2.821 3.250 1.833 2.262 2.262 2.821 2.821 3.250 3.250 1.812 1.812 2.228 2.764 3.169 1.812 2.228 2.228 2.764 2.764 3.169 3.169 1.796 1.796 2.201 2.718 3.106 1.796 2.201 2.201 2.718 2.718 3.106 3.106 1.782 1.782 2.179 2.681 3.055 1.782 2.179 2.179 2.681 2.681 3.055 3.055 1.771 1.771 2.160 2.650 3.012 1.771 2.160 2.160 2.650 2.650 3.012 3.012 1.761 1.761 2.145 2.624 2.977 1.761 2.145 2.145 2.624 2.624 2.977 2.977 1.753 1.753 2.131 2.602 2.947 1.753 2.131 2.131 2.602 2.602 2.947 2.947 1.746 1.746 2.120 2.583 2.921 1.746 2.120 2.120 2.583 2.583 2.921 2.921 1.740 1.740 2.110 2.567 2.898 1.740 2.110 2.110 2.567 2.567 2.898 2.898 1.734 1.734 2.101 2.552 2.878 1.734 2.101 2.101 2.552 2.552 2.878 2.878 1.729 1.729 2.093 2.539 2.861 1.729 2.093 2.093 2.539 2.539 2.861 2.861 1.725 1.725 2.086 2.528 2.845 1.725 2.086 2.086 2.528 2.528 2.845 2.845 1.721 1.721 2.080 2.518 2.831 1.721 2.080 2.080 2.518 2.518 2.831 2.831 1.717 1.717 2.074 2.508 2.819 1.717 2.074 2.074 2.508 2.508 2.819 2.819 1.714 1.714 2.069 2.500 2.807 1.714 2.069 2.069 2.500 2.500 2.807 2.807 1.711 1.711 2.064 2.492 2.797 1.711 2.064 2.064 2.492 2.492 2.797 2.797 1.708 1.708 2.060 2.485 2.787 1.708 2.060 2.060 2.485 2.485 2.787 2.787 1.706 1.706 2.056 2.479 2.779 1.706 2.056 2.056 2.479 2.479 2.779 2.779 1.703 1.703 2.052 2.473 2.771 1.703 2.052 2.052 2.473 2.473 2.771 2.771 1.701 1.701 2.048 2.467 2.763 1.701 2.048 2.048 2.467 2.467 2.763 2.763 1.699 1.699 2.045 2.462 2.756 1.699 2.045 2.045 2.462 2.462 2.756 2.756 1.645 1.645 1.960 2.326 2.576 1.645 1.960 1.960 2.326 2.326 2.576 2.576 tt t −t −t −t 11 1 αα α 22 2 tt t Two-tailed Two-tailed test test Two-tailed test tt 11 1 αα α 22 2 αα α t χχ22 χ 2 χχ22 χ 2 Degrees Degrees of of of Degrees freedom freedom freedom 11 1 22 2 33 3 44 4 55 5 66 6 77 7 88 8 99 9 1010 10 1111 11 1212 12 1313 13 1414 14 1515 15 1616 16 1717 17 1818 18 1919 19 2020 20 2121 21 2222 22 2323 23 2424 24 2525 25 2626 26 2727 27 2828 28 2929 29 3030 30 4040 40 5050 50 6060 60 7070 70 8080 80 9090 90 100 100 100 χχ22 χ 2 LL Right tail Right tailtail Right 11 1 αα α 22 2 L χχ22 χ 2 RR χχ22 χ 2 R Two tails Two tails Two tails AA A 0.995 0.99 0.975 0.95 0.90 0.10 0.05 0.025 0.01 0.005 0.995 0.99 0.99 0.975 0.975 0.95 0.95 0.90 0.90 0.10 0.10 0.05 0.05 0.025 0.025 0.01 0.01 0.005 0.005 0.995 —— —— 0.001 0.001 0.004 0.016 2.706 3.841 5.024 6.635 7.879 0.001 0.004 0.004 0.016 0.016 2.706 2.706 3.841 3.841 5.024 5.024 6.635 6.635 7.879 7.879 — — 0.010 0.020 0.051 0.103 0.211 4.605 5.991 7.378 9.210 10.597 0.010 0.020 0.020 0.051 0.051 0.103 0.103 0.211 0.211 4.605 4.605 5.991 5.991 7.378 7.378 9.210 9.21010.597 10.597 0.010 0.072 0.115 0.216 0.352 0.584 6.251 7.815 9.348 11.345 12.838 0.072 0.115 0.115 0.216 0.216 0.352 0.352 0.584 0.584 6.251 6.251 7.815 7.815 9.348 9.34811.345 11.34512.838 12.838 0.072 0.207 0.297 0.484 0.711 1.064 7.779 9.488 11.143 13.277 14.860 0.207 0.297 0.297 0.484 0.484 0.711 0.711 1.064 1.064 7.779 7.779 9.488 9.48811.143 11.14313.277 13.27714.860 14.860 0.207 0.412 0.554 0.831 1.145 1.610 9.236 11.071 12.833 15.086 16.750 0.412 0.554 0.554 0.831 0.831 1.145 1.145 1.610 1.610 9.236 9.23611.071 11.07112.833 12.83315.086 15.08616.750 16.750 0.412 0.676 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.812 18.548 0.676 0.872 0.872 1.237 1.237 1.635 1.635 2.204 2.20410.645 10.64512.592 12.59214.449 14.44916.812 16.81218.548 18.548 0.676 0.989 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.475 20.278 0.989 1.239 1.239 1.690 1.690 2.167 2.167 2.833 2.83312.017 12.01714.067 14.06716.013 16.01318.475 18.47520.278 20.278 0.989 1.344 1.646 2.180 2.733 3.490 13.362 15.507 17.535 20.090 21.955 1.344 1.646 1.646 2.180 2.180 2.733 2.733 3.490 3.49013.362 13.36215.507 15.50717.535 17.53520.090 20.09021.955 21.955 1.344 1.735 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666 23.589 1.735 2.088 2.088 2.700 2.700 3.325 3.325 4.168 4.16814.684 14.68416.919 16.91919.023 19.02321.666 21.66623.589 23.589 1.735 2.156 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 25.188 2.156 2.558 2.558 3.247 3.247 3.940 3.940 4.865 4.86515.987 15.98718.307 18.30720.483 20.48323.209 23.20925.188 25.188 2.156 2.603 3.053 3.816 4.575 5.578 17.275 19.675 21.920 24.725 26.757 2.603 3.053 3.053 3.816 3.816 4.575 4.575 5.578 5.57817.275 17.27519.675 19.67521.920 21.92024.725 24.72526.757 26.757 2.603 3.074 3.571 4.404 5.226 6.304 18.549 21.026 23.337 26.217 28.299 3.074 3.571 3.571 4.404 4.404 5.226 5.226 6.304 6.30418.549 18.54921.026 21.02623.337 23.33726.217 26.21728.299 28.299 3.074 3.565 4.107 5.009 5.892 7.042 19.812 22.362 24.736 27.688 29.819 3.565 4.107 4.107 5.009 5.009 5.892 5.892 7.042 7.04219.812 19.81222.362 22.36224.736 24.73627.688 27.68829.819 29.819 3.565 4.075 4.660 5.629 6.571 7.790 21.064 23.685 26.119 29.141 31.319 4.075 4.660 4.660 5.629 5.629 6.571 6.571 7.790 7.79021.064 21.06423.685 23.68526.119 26.11929.141 29.14131.319 31.319 4.075 4.601 5.229 6.262 7.261 8.547 22.307 24.996 27.488 30.578 32.801 4.601 5.229 5.229 6.262 6.262 7.261 7.261 8.547 8.54722.307 22.30724.996 24.99627.488 27.48830.578 30.57832.801 32.801 4.601 5.142 5.812 6.908 7.962 9.312 23.542 26.296 28.845 32.000 34.267 5.142 5.812 5.812 6.908 6.908 7.962 7.962 9.312 9.31223.542 23.54226.296 26.29628.845 28.84532.000 32.00034.267 34.267 5.142 5.697 6.408 7.564 8.672 10.085 24.769 27.587 30.191 33.409 35.718 5.697 6.408 6.408 7.564 7.564 8.672 8.67210.085 10.08524.769 24.76927.587 27.58730.191 30.19133.409 33.40935.718 35.718 5.697 6.265 7.015 8.231 9.390 10.865 25.989 28.869 31.526 34.805 37.156 6.265 7.015 7.015 8.231 8.231 9.390 9.39010.865 10.86525.989 25.98928.869 28.86931.526 31.52634.805 34.80537.156 37.156 6.265 6.844 7.633 8.907 10.117 11.651 27.204 30.144 32.852 36.191 38.582 6.844 7.633 7.633 8.907 8.90710.117 10.11711.651 11.65127.204 27.20430.144 30.14432.852 32.85236.191 36.19138.582 38.582 6.844 7.434 8.260 9.591 10.851 12.443 28.412 31.410 34.170 37.566 39.997 7.434 8.260 8.260 9.591 9.59110.851 10.85112.443 12.44328.412 28.41231.410 31.41034.170 34.17037.566 37.56639.997 39.997 7.434 8.034 8.897 10.283 11.591 13.240 29.615 32.671 35.479 38.932 41.401 8.034 8.897 8.89710.283 10.28311.591 11.59113.240 13.24029.615 29.61532.671 32.67135.479 35.47938.932 38.93241.401 41.401 8.034 8.643 9.542 10.982 12.338 14.042 30.813 33.924 36.781 40.289 42.796 8.643 9.542 9.54210.982 10.98212.338 12.33814.042 14.04230.813 30.81333.924 33.92436.781 36.78140.289 40.28942.796 42.796 8.643 9.260 9.26010.196 10.196 10.19611.689 11.689 11.68913.091 13.091 13.09114.848 14.848 14.84832.007 32.007 32.00735.172 35.172 35.17238.076 38.076 38.07641.638 41.638 41.63844.181 44.181 44.181 9.260 9.886 10.856 12.401 13.848 15.659 33.196 36.415 39.364 42.980 45.559 9.88610.856 10.85612.401 12.40113.848 13.84815.659 15.65933.196 33.19636.415 36.41539.364 39.36442.980 42.98045.559 45.559 9.886 10.520 11.524 13.120 14.611 16.473 34.382 37.652 40.646 44.314 46.928 10.52011.524 11.52413.120 13.12014.611 14.61116.473 16.47334.382 34.38237.652 37.65240.646 40.64644.314 44.31446.928 46.928 10.520 11.160 12.198 13.844 15.379 17.292 35.563 38.885 41.923 45.642 48.290 11.16012.198 12.19813.844 13.84415.379 15.37917.292 17.29235.563 35.56338.885 38.88541.923 41.92345.642 45.64248.290 48.290 11.160 11.808 12.879 14.573 16.151 18.114 36.741 40.113 43.194 46.963 49.645 11.80812.879 12.87914.573 14.57316.151 16.15118.114 18.11436.741 36.74140.113 40.11343.194 43.19446.963 46.96349.645 49.645 11.808 12.461 13.565 15.308 16.928 18.939 37.916 41.337 44.461 48.278 50.993 12.46113.565 13.56515.308 15.30816.928 16.92818.939 18.93937.916 37.91641.337 41.33744.461 44.46148.278 48.27850.993 50.993 12.461 13.121 13.12114.257 14.257 14.25716.047 16.047 16.04717.708 17.708 17.70819.768 19.768 19.76839.087 39.087 39.08742.557 42.557 42.55745.722 45.722 45.72249.588 49.588 49.58852.336 52.336 52.336 13.121 13.787 14.954 16.791 18.493 20.599 40.256 43.773 46.979 50.892 53.672 13.78714.954 14.95416.791 16.79118.493 18.49320.599 20.59940.256 40.25643.773 43.77346.979 46.97950.892 50.89253.672 53.672 13.787 20.707 22.164 24.433 26.509 29.051 51.805 55.758 59.342 63.691 66.766 20.70722.164 22.16424.433 24.43326.509 26.50929.051 29.05151.805 51.80555.758 55.75859.342 59.34263.691 63.69166.766 66.766 20.707 27.991 29.707 32.357 34.764 37.689 63.167 67.505 71.420 76.154 79.490 27.99129.707 29.70732.357 32.35734.764 34.76437.689 37.68963.167 63.16767.505 67.50571.420 71.42076.154 76.15479.490 79.490 27.991 35.534 37.485 40.482 43.188 46.459 74.397 79.082 83.298 88.379 91.952 35.53437.485 37.48540.482 40.48243.188 43.18846.459 46.45974.397 74.39779.082 79.08283.298 83.29888.379 88.37991.952 91.952 35.534 43.275 45.442 48.758 51.739 55.329 85.527 90.531 95.023 100.425 104.215 43.27545.442 45.44248.758 48.75851.739 51.73955.329 55.32985.527 85.52790.531 90.53195.023 95.023 100.425 104.215 43.275 100.425 104.215 51.172 51.17253.540 53.540 53.54057.153 57.153 57.15360.391 60.391 60.39164.278 64.278 64.27896.578 96.578 96.578 101.879 101.879 106.629 106.629 112.329 112.329 116.321 116.321 51.172 101.879 106.629 112.329 116.321 59.196 59.19661.754 61.754 61.75465.647 65.647 65.64769.126 69.126 69.12673.291 73.291 73.291 107.565 107.565 113.145 113.145 118.136 118.136 124.116 124.116 128.299 128.299 59.196 107.565 113.145 118.136 124.116 128.299 67.328 70.065 74.222 77.929 82.358 118.498 124.342 129.561 135.807 140.169 67.32870.065 70.06574.222 74.22277.929 77.92982.358 82.358 118.498 124.342 129.561 135.807 140.169 67.328 118.498 124.342 129.561 135.807 140.169 Copyright © 2012 Pearson Education, Inc.
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