Download Hypothesis and Testing of Hypothesis

Hypothesis and Testing of
Hypothesis
Prof. KG Satheesh Kumar
Asian School of Business
Illustration 1: Court Verdict
• Under Indian legal system, an accused is
assumed innocent until proved guilty
“beyond a reasonable doubt”.
• This is called null hypothesis.
We may write: H0: Accused is innocent
• Court holds the null hypothesis as true
until it can be proved, based on evidence,
and beyond reasonable doubt, that it is
false
The Verdict
• If H0 is proved to be false, it is rejected
and an alternative hypothesis, H1, is
accepted
• We may write:
H1: Accused is not innocent, hence guilty.
• If H0 cannot be proved to be false, beyond
reasonable doubt, then it cannot be
rejected and is hence accepted
Four possible outcomes
Accused: Innocent
Verdict: Acquittal
Accused: Innocent
Verdict: Conviction
Type I Error
Accused: Guilty
Verdict: Acquittal
Type II Error
Accused: Guilty
Verdict: Conviction
Illustration 2: Bottling Cola
• Company claims 2 lit volume; consumer
advocate wants to test the claim
H0: Mean Volume >= 2 lit
H1: Mean Volume < 2 lit
• Consumers are happy, but company
suspects that there is overfilling
H0: Mean Volume <= 2 lit
H1: Mean Volume > 2 lit
Bottling Cola – Engineer’s View
• The plant engineer wants to take
corrective action if the average volume is
more than or less than 2 litres
H0: Mean Volume = 2 lit
H1: Mean Volume  2 lit
Prerequisites for this chapter
• Random variable and its probability (click)
distribution / probability density function
• The Normal Distribution (click)
• Sampling and Sampling Distribution (click)
• Estimation (click)
Hypothesis
• A thesis is something that has been proven to be true
• A hypothesis is something that has not yet been proven
to be true
• Hypothesis testing is the process of determining, through
statistical methods, whether or not a given hypothesis
may be accepted as true
• Hypothesis testing is an important part of statistical
inference – making decisions about the population based
on sample evidence
Setting up and testing hypotheses is an
essential part of statistical inference.
In order to formulate such a test, usually some
theory has been put forward, either because it is
believed to be true or because it is to be used as
a basis for argument, but has not been proved.
E.g.: Claiming that a new drug is better than the
current drug for treatment of the same
symptoms.
A Good Hypothesis
should
Be written as a
simple, clear and
precise statement
Be testable with
a straightforward
experiment
Predict the
anticipated results
in clear form
State relationship
between variables
Be consistent with
observation
and known facts
The question of interest is simplified into
two competing claims / hypotheses
between which we have a choice; the null
hypothesis, denoted H0, against the
alternative hypothesis, denoted H1.
These two competing claims / hypotheses
are not however treated on an equal basis:
special consideration is given to the null
hypothesis. In fact only the null hypothesis
is tested whether to reject or not.
Null and Alternative Hypothesis
• Hypothesis is a testable assertion about the
population (value of a parameter)
• Null hypothesis (Ho) is an assertion held true
unless we have sufficient statistical evidence to
conclude otherwise
• The alternative hypothesis (H1, Ha, or Hα) is the
negation of the null hypothesis.
• The two are mutually exclusive. One and only
one of the two can be true.
Determining the null hypothesis
• The null hypothesis is often a claim made by someone and
alternative hypothesis is the suspicion about that claim.
• There may be no claim; then what we wish to demonstrate is the
alternative hypothesis and its negation is the null hypothesis.
• H1 describes the situation we believe to be true and Ho describes
the situation contrary to what we believe about the population.
• Null hypothesis is the one which when true does not call for a
corrective action. If the alternative hypothesis is true, some
corrective action would be necessary.
• If the obtained statistics is unlikely to be true, what we reject is Ho
• Note: The equality sign always appear in Ho.
Examples of H0 and H1
• Ex 1: A pharmaceutical company claims that four out of
five doctors prescribe the pain medicine it produces. Set
up Ho and H1 to test this claim. (Answer)
• Ex 2: A medicine is effective only if the concentration of a
certain chemical is at least 200 ppm. At the same time,
the medicine would produce an undesirable side effect if
the concentration of the same chemical exceeds 200
ppm. Set up H0, H1. (Answer)
• Ex 3: A maker of golf balls claims that the variance of the
weights of the company’s golf balls is controlled to within
0.0028 oz2. Set up hypotheses to test this claim (Ans)
More examples
• Ex 4: The average cost of a traditional openheart surgery is claimed to be $49,160. If you
suspect that the claim exaggerates the cost, how
would you set up the hypotheses? (Ans)
• Ex 5: A vendor claims that he can fulfill an order
in at most six working days. You suspect that the
average is greater than six working days and
want to test the hypothesis. How will you set up
the hypotheses? (Ans)
More examples
• Ex 6: At least 20% of the visitors to a particular
store are said to end up placing an order. How
will you set up hypotheses to test the claim?
(Answer)
• Ex 7: Web surfers will lose interest if
downloading takes more than 12 seconds. If you
wish to test the effectiveness of a newly
designed web page in regard to download time,
how will you set up the null and alternative
hypotheses? (Answer)
Common types of hypothesis tests
•
Parametric test of hypotheses about population parameters:
–
Mean (); proportion (p) and variance (2) using z, t and chi-square distributions
–
Test of difference between two population means using t and z distributions
•
paid observations; independent observations
–
Test of difference between two population proportions using z distribution
–
Test of equality of two population variances using F-distribution
–
Analysis of variance for comparing several population means
•
Parametric tests are more powerful than non-parametric tests because the data are
derived from interval and ratio measurements
•
Non-parametric tests are used to test hypotheses with nominal and ordinal data
–
•
The Sign Test, The Runs Test, Wald-Wolfowitz Test, Mann-Whitney U Test, Kruskal-Wallis
Test, Chi-Square Test for Goodness of fit
An important assumption for parametric tests is that the population is approximately
normal (or sample size is large). No such assumptions are required for nonparametric tests, which are hence also called, distribution-free tests.
Steps in Hypothesis Testing
• Set up the null and alternative hypotheses
• Decide on the significance level, α (standard values:
10%, 5%, 1%)
• Using a random sample, get sample statistic and then
calculate test statistic
• Find the table value of test statistic corresponding to the
required α value
• Compare the calculated and table values of the test
statistic and interpret.
• Note: Only the null hypothesis is actually tested
Type I, Type II errors
• Four outcomes are possible
–
–
–
–
Ho is true and is not rejected (Not an error)
Ho is true, but is rejected (Type I error)
Ho is false, but not rejected (Type II error)
Ho is false and is rejected (Not an error)
• Type I error is when we reject a true null
hypothesis
• Type II error is when we do not reject a false null
hypothesis
One-tailed and two-tailed tests
• Left-tailed test: In case Ho makes a “>=“ claim,
then rejection occurs when the statistic is far
below, i.e. on the left tail.
• Right-tailed test: Ho makes a <= claim and
rejection occurs on the right tail
• Two-tailed test: Ho makes a “=“ claim and
rejection occurs on both tails.
• Rejection and non-rejection regions are marked
in the distribution of the sample statistic and the
test statistic for interpreting the test results.
The p-value
• The p-value
– is the probability of getting a sample evidence
at least as unfavorable as the sample statistic
when the null hypothesis is actually true.
– is a “credibility rating” for H0
– is the probability of Type I error
– is an approximate answer to the question,
“given the sample evidence, what is the
probability that Ho is true?”
Significance Level, α
• This is the maximum “set” probability of type I error.
Accordingly, α decides the policy to reject / accept H0.
• Policy: If p-value is less than α, reject H0
• If p-value is not less than α, we do not to reject H0, but
this does not mean that H0 is true. Only that we do not
have sufficient evidence to reject H0.
• The selected value of α indirectly decides the probability
of making a type II error. We use the symbol β for this
probability.
Confidence level
• The fraction, 1 – α is called the confidence
level. If α = 5%, the confidence level is
95%, which means we want to be at least
95% confident that Ho is false before we
reject it.
• Optimal α; compromise between Type I
and Type II errors; cost of each type of
error; producer’s risk and consumer’s risk.
Type II Error and Power of a Test
• Type II error, β is difficult to estimate; it
depends on α, the sample size, and the
actual population parameter.
• Power of a Test
– The complement of type II error, i.e. 1 - β is
called the power of the test. It is the
probability that a false null hypothesis will be
detected by the test.
Test Statistic
• Test Statistic
– A random variable, calculated from the
sample evidence, and having a well-known
probability distribution
– Mostly used are Z, t, χ2 and F. The
distributions of these random variables are
well-known and tables are available.
– See tables of Z and t distributions.
Test statistic used
• Test statistic for mean is z or t. (See next slide)
 Test statistic = (Sample mean – hypothesized population mean)/
SE; where SE is the Standard Error
• Test statistic for proportion (assuming large sample) is Z
 Z = (sample proportion – p)/ SE; where p is the hypothesized
population proportion and SE = (pq)/n); q = 1-p
• Test statistic for variance,
 χ2 = (n-1) S2/2 where S2 is the sample variance and 2 is the
hypothesized population variance
Test statistic for population mean
• When the null hypothesis is about the population
mean, the test statistic is:
 Z if population standard deviation,  is known
 t if sample standard deviation, S is known.
When Z is used, Z = (sample mean - )/(/n)
When t is used, t = (sample mean - )/(S/n)
In the latter case, use degrees of freedom as n-1
• It is necessary that either the population is
normal or the sample size is large enough
Examples on Hypothesis Testing
• Ex 8: A certain medicine is supposed to
contain an average of 247 ppm of a
chemical. If the concentration exceeds 247
ppm, the drug may cause undesirable side
effects. A random sample of 60 portions is
tested and the sample mean is found to be
250 ppm and sample standard deviation
12 ppm. Perform a statistical hypothesis
test at 1% and 5% significance. (Ans)
• Ex 9: In the above example, assume that
there are no side effects, but we are told
that the drug may be ineffective if the
concentration is below 247 ppm. The
sample evidence is the same as before.
Formulate and test the hypothesis. (Ans)
• Ex 10: In the above example, assume that
side effects and effectiveness are both to
be considered. The sample evidence is
the same. Formulate and test the
hypothesis. (Ans)
• Ex 11: Certain eggs are stated to have reduced
cholesterol content, with an average of only
2.5% cholesterol. A concerned health group
wants to test whether the claim is true. A random
sample of 100 eggs reveals a sample average
content of 3.0% cholesterol with a standard
deviation of 2.8%. Does the health group have
cause for action? (Ans)
• Ex 12: A survey of medical schools indicates that
16% of the faculty positions are vacant. A
placement agency conducts a survey to test this
claim, using a random sample of 300 faculty
positions and finds that 39 out of the 300 are
vacant. Test the claim at 5% level of significance
(Ans)
Thank you
Random Variable
• A variable associated with a random experiment like drawing a
random sample from the population – the variable may be mean,
proportion, variance
• A random variable is an uncertain quantity whose value depends on
chance
• A random variable (denoted by X) takes a range of discrete values
with some discrete probability distribution, P(X) or continuous values
with some probability density, f(X).
• P(X) or f(X), as the case may be, can be used to find the probability
that the random variable takes specific values or range of values
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The Normal Distribution
• If a random variable, X is affected by many independent
causes, none of which is overwhelmingly large, the
probability distribution of X closely follows normal
distribution. Then X is called normal variate and we write
X ~ N(, 2), where  is the mean and 2 is the variance
• A Normal pdf is completely defined by its mean,  and
variance, 2. The square root of variance is called
standard deviation .
• If several independent random variables are normally
distributed, their sum will also be normally distributed
with mean equal to the sum of individual means and
variance equal to the sum of individual variances.
The Normal pdf
The area under any pdf between two given
values of X is the probability that X falls
between these two values
Standard Normal Variate, Z
• SNV, Z is the normal random variable with
mean 0 and standard deviation 1
• Tables are available for Standard Normal
Probabilities
• X and Z are connected by:
Z = (X - ) / 
and X =  + Z
• The area under the X curve between X1
and X2 is equal to the area under Z curve
between Z1 and Z2.
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z
0.0
0.1
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0.7
0.8
0.9
1.0
1.1
1.2
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1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
0.00
0.0000
0.0398
0.0793
0.1179
0.1554
0.1915
0.2257
0.2580
0.2881
0.3159
0.3413
0.3643
0.3849
0.4032
0.4192
0.4332
0.4452
0.4554
0.4641
0.4713
0.4772
0.4821
0.4861
0.4893
0.4918
0.4938
0.4953
0.4965
0.4974
0.4981
0.4987
0.4990
0.4993
0.4995
0.4997
0.01
0.0040
0.0438
0.0832
0.1217
0.1591
0.1950
0.2291
0.2611
0.2910
0.3186
0.3438
0.3665
0.3869
0.4049
0.4207
0.4345
0.4463
0.4564
0.4649
0.4719
0.4778
0.4826
0.4864
0.4896
0.4920
0.4940
0.4955
0.4966
0.4975
0.4982
0.4987
0.4991
0.4993
0.4995
0.4997
0.02
0.0080
0.0478
0.0871
0.1255
0.1628
0.1985
0.2324
0.2642
0.2939
0.3212
0.3461
0.3686
0.3888
0.4066
0.4222
0.4357
0.4474
0.4573
0.4656
0.4726
0.4783
0.4830
0.4868
0.4898
0.4922
0.4941
0.4956
0.4967
0.4976
0.4982
0.4987
0.4991
0.4994
0.4995
0.4997
0.03
0.0120
0.0517
0.0910
0.1293
0.1664
0.2019
0.2357
0.2673
0.2967
0.3238
0.3485
0.3708
0.3907
0.4082
0.4236
0.4370
0.4484
0.4582
0.4664
0.4732
0.4788
0.4834
0.4871
0.4901
0.4925
0.4943
0.4957
0.4968
0.4977
0.4983
0.4988
0.4991
0.4994
0.4996
0.4997
0.04
0.0160
0.0557
0.0948
0.1331
0.1700
0.2054
0.2389
0.2704
0.2995
0.3264
0.3508
0.3729
0.3925
0.4099
0.4251
0.4382
0.4495
0.4591
0.4671
0.4738
0.4793
0.4838
0.4875
0.4904
0.4927
0.4945
0.4959
0.4969
0.4977
0.4984
0.4988
0.4992
0.4994
0.4996
0.4997
0.05
0.0199
0.0596
0.0987
0.1368
0.1736
0.2088
0.2422
0.2734
0.3023
0.3289
0.3531
0.3749
0.3944
0.4115
0.4265
0.4394
0.4505
0.4599
0.4678
0.4744
0.4798
0.4842
0.4878
0.4906
0.4929
0.4946
0.4960
0.4970
0.4978
0.4984
0.4989
0.4992
0.4994
0.4996
0.4997
0.06
0.0239
0.0636
0.1026
0.1406
0.1772
0.2123
0.2454
0.2764
0.3051
0.3315
0.3554
0.3770
0.3962
0.4131
0.4279
0.4406
0.4515
0.4608
0.4686
0.4750
0.4803
0.4846
0.4881
0.4909
0.4931
0.4948
0.4961
0.4971
0.4979
0.4985
0.4989
0.4992
0.4994
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0.07
0.0279
0.0675
0.1064
0.1443
0.1808
0.2157
0.2486
0.2794
0.3078
0.3340
0.3577
0.3790
0.3980
0.4147
0.4292
0.4418
0.4525
0.4616
0.4693
0.4756
0.4808
0.4850
0.4884
0.4911
0.4932
0.4949
0.4962
0.4972
0.4979
0.4985
0.4989
0.4992
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0.4997
0.08
0.0319
0.0714
0.1103
0.1480
0.1844
0.2190
0.2517
0.2823
0.3106
0.3365
0.3599
0.3810
0.3997
0.4162
0.4306
0.4429
0.4535
0.4625
0.4699
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0.4973
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0.4986
0.4990
0.4993
0.4995
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0.09
0.0359
0.0753
0.1141
0.1517
0.1879
0.2224
0.2549
0.2852
0.3133
0.3389
0.3621
0.3830
0.4015
0.4177
0.4319
0.4441
0.4545
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0.4706
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0.4890
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0.4952
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0.4974
0.4981
0.4986
0.4990
0.4993
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0.4997
0.4998
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Standard
Normal
Probabilities
(Table of z
distribution)
The z-value is
on the left and
top margins and
the probability
(shaded area in
the diagram) is
in the body of
the table
Sampling Distribution
• The sampling distribution of x is the probability
distribution of all possible values of x for a
given sample size n taken from the population.
• According to the Central Limit Theorem, for large
enough sample size, n, the sampling distribution
is approximately normal with mean  and
standard deviation /n. This standard deviation
is called standard error.
• CLT holds for non-normal populations also and
states: For large enough n, x ~ N(, 2/n)
Return
Estimation
• The value of an estimator (see next slide), obtained from a sample
can be used to estimate the value of the population parameter. Such
an estimate is called a point estimate.
• This is a 50:50 estimate, in the sense, the actual parameter value is
equally likely to be on either side of the point estimate.
• A more useful estimate is the interval estimate, where an interval is
specified along with a measure of confidence (90%, 95%, 99% etc)
• The interval estimate with its associated measure of confidence is
called a confidence interval.
• A confidence interval is a range of numbers believed to include the
unknown population parameter, with a certain level of confidence
Estimators
• Population parameters (, 2, p) and
Sample Statistics (x,s2, ps)
• An estimator of a population parameter is
a sample statistic used to estimate the
parameter
• Statistic,x is an estimator of parameter 
• Statistic, s2 is an estimator of parameter 2
• Statistic, ps is an estimator of parameter p
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z
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2.0
2.1
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3.0
3.1
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0.00
0.0000
0.0398
0.0793
0.1179
0.1554
0.1915
0.2257
0.2580
0.2881
0.3159
0.3413
0.3643
0.3849
0.4032
0.4192
0.4332
0.4452
0.4554
0.4641
0.4713
0.4772
0.4821
0.4861
0.4893
0.4918
0.4938
0.4953
0.4965
0.4974
0.4981
0.4987
0.4990
0.4993
0.4995
0.4997
0.01
0.0040
0.0438
0.0832
0.1217
0.1591
0.1950
0.2291
0.2611
0.2910
0.3186
0.3438
0.3665
0.3869
0.4049
0.4207
0.4345
0.4463
0.4564
0.4649
0.4719
0.4778
0.4826
0.4864
0.4896
0.4920
0.4940
0.4955
0.4966
0.4975
0.4982
0.4987
0.4991
0.4993
0.4995
0.4997
0.02
0.0080
0.0478
0.0871
0.1255
0.1628
0.1985
0.2324
0.2642
0.2939
0.3212
0.3461
0.3686
0.3888
0.4066
0.4222
0.4357
0.4474
0.4573
0.4656
0.4726
0.4783
0.4830
0.4868
0.4898
0.4922
0.4941
0.4956
0.4967
0.4976
0.4982
0.4987
0.4991
0.4994
0.4995
0.4997
0.03
0.0120
0.0517
0.0910
0.1293
0.1664
0.2019
0.2357
0.2673
0.2967
0.3238
0.3485
0.3708
0.3907
0.4082
0.4236
0.4370
0.4484
0.4582
0.4664
0.4732
0.4788
0.4834
0.4871
0.4901
0.4925
0.4943
0.4957
0.4968
0.4977
0.4983
0.4988
0.4991
0.4994
0.4996
0.4997
0.04
0.0160
0.0557
0.0948
0.1331
0.1700
0.2054
0.2389
0.2704
0.2995
0.3264
0.3508
0.3729
0.3925
0.4099
0.4251
0.4382
0.4495
0.4591
0.4671
0.4738
0.4793
0.4838
0.4875
0.4904
0.4927
0.4945
0.4959
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0.4977
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0.05
0.0199
0.0596
0.0987
0.1368
0.1736
0.2088
0.2422
0.2734
0.3023
0.3289
0.3531
0.3749
0.3944
0.4115
0.4265
0.4394
0.4505
0.4599
0.4678
0.4744
0.4798
0.4842
0.4878
0.4906
0.4929
0.4946
0.4960
0.4970
0.4978
0.4984
0.4989
0.4992
0.4994
0.4996
0.4997
0.06
0.0239
0.0636
0.1026
0.1406
0.1772
0.2123
0.2454
0.2764
0.3051
0.3315
0.3554
0.3770
0.3962
0.4131
0.4279
0.4406
0.4515
0.4608
0.4686
0.4750
0.4803
0.4846
0.4881
0.4909
0.4931
0.4948
0.4961
0.4971
0.4979
0.4985
0.4989
0.4992
0.4994
0.4996
0.4997
0.07
0.0279
0.0675
0.1064
0.1443
0.1808
0.2157
0.2486
0.2794
0.3078
0.3340
0.3577
0.3790
0.3980
0.4147
0.4292
0.4418
0.4525
0.4616
0.4693
0.4756
0.4808
0.4850
0.4884
0.4911
0.4932
0.4949
0.4962
0.4972
0.4979
0.4985
0.4989
0.4992
0.4995
0.4996
0.4997
0.08
0.0319
0.0714
0.1103
0.1480
0.1844
0.2190
0.2517
0.2823
0.3106
0.3365
0.3599
0.3810
0.3997
0.4162
0.4306
0.4429
0.4535
0.4625
0.4699
0.4761
0.4812
0.4854
0.4887
0.4913
0.4934
0.4951
0.4963
0.4973
0.4980
0.4986
0.4990
0.4993
0.4995
0.4996
0.4997
0.09
0.0359
0.0753
0.1141
0.1517
0.1879
0.2224
0.2549
0.2852
0.3133
0.3389
0.3621
0.3830
0.4015
0.4177
0.4319
0.4441
0.4545
0.4633
0.4706
0.4767
0.4817
0.4857
0.4890
0.4916
0.4936
0.4952
0.4964
0.4974
0.4981
0.4986
0.4990
0.4993
0.4995
0.4997
0.4998
Return
Standard
Normal
Probabilities
(Table of z
distribution)
The z-value is
on the left and
top margin and
the probability
(shaded area in
the diagram) is
in the body of
the table
Return
Ex 1
The claim is the null hypothesis and its
negation is the alternative hypothesis.
If p denotes the proportion of doctors
prescribing the medicine, we set the
hypotheses as:
Ho: p >= 0.8
H1: p < 0.8
Return
Ex 2
Null hypothesis is the one which calls for no
corrective action and the alternative hypothesis
is the one that calls for corrective action
If  denotes the concentration of the chemical,
we set up the hypotheses as:
Ho:  = 200 ppm
H1:  200 ppm
Return
Ex 3
The claim is the null hypothesis. Using 2
to denote variance, the hypotheses can be
set up as:
Ho: 2 <= 0.0028 oz2
H1: 2 > 0.0028 oz2
Return
Ex 4
The claim is the null hypothesis and your
suspicion (belief) is the alternative
hypothesis.
If  denotes the average cost, the
hypotheses are:
Ho:  >= $49,160
H1:  < $49,160
Return
Ex 5
The claim is the null hypothesis and your
suspicion is the alternative hypothesis.
If  denotes the average number of days
to fulfill an order, the hypotheses are:
Ho:  <= 6
H1:  > 6
Return
Ex 6
The claim becomes the null hypothesis.
Let p denote the proportion of visitors
placing an order. Then the hypotheses will
be set up as:
Ho: p >= 0.20
H1: p < 0.20
Return
Ex 7
Corrective action is needed if average
downloading time exceeds 12 seconds; so
this forms H1.
Let  denote the average download time.
Then:
Ho:  <= 12 s
H1:  > 12 s
Return
Ex 8
Let  denote the average ppm of the chemical. The hypotheses are:
Ho:  <= 247
H1:  > 247
Sample statistic, x = 250; sample SD, s = 12 and sample size n =
60 (large sample); standard error, SE = 12/60 = 1.55. Right-tailed
test
Since we know only sample SD, test statistic follows t-distribution
with degrees of freedom 59
Test statistic, t = (250-247)/1.55 = 1.936
From the table of t-distribution, one-tailed t-values for 59 df are:
t5% = 1.671 and t1% = 2.390
Comparing the calculated and table values of the test statistic, we
reject the null hypothesis at 5% level of significance (95%
confidence); but do not reject null hypothesis at 1% level of
significance (99% confidence level)
Return
Ex 9
Let  denote the average ppm of the
chemical. The hypotheses are:
Ho:  >= 247
H1:  < 247
Sample statistic, x = 250 does not go
against the null hypothesis and hence
there is no ground to reject it
Return
Ex 10
Let  denote the average ppm of the chemical. The
hypotheses are:
Ho:  = 247
H1:   247
Test statistic, t = (250-247)/1.55 = 1.936
From the table of t-distribution, two-tailed t-values for 59
df are:
t5% = 2.000 and t1% = 2.660
Comparing the calculated and table values of the test
statistic, we do not reject the null hypothesis either at 5%
level of significance (95% confidence); or at 1% level of
significance (99% confidence level)
Return
Ex 11
Let  denote the average % of cholesterol. The hypotheses are:
Ho:  <= 2.5
H1:  > 2.5
Sample statistic = 3; sample SD = 2.8 and sample size = 100 (large
sample); standard error, SE = 2.8/100 = 0.28. Right-tailed test
Test statistic, t = (3-2.5)/0.28 = 1.786
From the table of t-distribution, one-tailed t-values for 99 df are:
t5% = 1.660 and t1% = 2.364
Comparing the calculated and table values of the test statistic, we
reject the null hypothesis at 5% level of significance (95%
confidence); but do not reject null hypothesis at 1% level of
significance (99% confidence level)
Return
Ex 12
Let p denote the proportion of vacant positions. Then the
hypotheses are:
Ho: p >= 0.16
H1: p < 0.16
Left tailed test
Sample statistic = 39/300 = 0.13;
SE = (0.16x0.84/300) = 0.0212
Calculated test statistic, Z = (0.13 – 0.16)/0.0212 = -1.415
From the table of Z-distribution, one-tailed Z5% = -1.645
(Negative because left-tailed)
Comparing calculated and table values of the test statistic,
we do not reject the null hypothesis that 16% faculty
positions are vacant
Return