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Chapter 10
Statistical Inference: OneSample Hypothesis Test
I Scientific Hypothesis
 A testable supposition that is tentatively adopted
to account for certain facts and to guide in the
investigation of others.
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II Statistical Hypotheses
 Statements about one of more parameters of a
population distribution
A. Null Hypothesis (H0)
 Hypothesis that the researcher tests
B. Alternative Hypothesis (H1)
 Hypothesis that corresponds to the researcher’s
scientific hunch
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C. Examples of Null and Alternative Hypotheses
H 0 :   0
H1 :   0
H 0 :   0
H 0 :   0
H1 :   0
H1 :   0
1.  denotes the unknown mean of a population;
0 denotes a hypothesized value of the population
mean that is to be tested.
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D. Characteristics of the Null Hypothesis, H0,
and the Alternative Hypothesis, H1
1. H0 contains an exact statement,  = 0, that can
be tested. H1 is an inexact statement:  > 0,
 < 0, or  ≠ 0.
2. The null and alternative hypotheses are mutually
exclusive and exhaustive. If the null hypothesis is
rejected, the alternative is probably true.
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E. Statistical Hypotheses for An Experiment
1. The dean at Idle-On-In College believes that a
new registration procedure will reduce the time
required for students to register. For the past
several years, the current procedure has required
3.10 hours. The statistical hypotheses are
H 0 :   3.10
H1 :   3.10,
where denotes the unknown population mean
for the new procedure and 0 = 3.10 denotes the
mean of the current procedure.
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3. To test the null hypothesis, the dean conducts a
trial run with a random sample of n = 27 students
and obtains the following data:
X  2.90 and ˆ  0.3013
4. The new procedure appears to require less time
than the old procedure. But the dean worries
because she knows that sample means vary from
sample to sample. She wonders, “Is it possible that
the population mean of the new procedure is really
equal to or greater than the old procedure?”
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4. To determine if the null hypothesis is credible or
not credible, the dean computes a t test statistic
for her data and determines the probability of
obtaining a sample mean of 2.90 if the true
population mean is really equal to or greater than
3.10.
5. She discovers that the probability of obtaining a
a sample mean of 2.90 if 0 = 3.10 is .001. She
decides to reject the null hypothesis and concludes
that the new procedure is better.
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F. Role of Logic in Hypothesis Testing
Scientific
hypothesis
Deductive inference
Inductive
inference
Statistical
hypotheses
Random sampling
and estimation of
population parameter
Statistical
test
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III Two Statistics for Testing H0:  = 0
A. t Test Statistic
X  0
t
ˆ n
X   Xi / n
0  hypothesized pop. mean
ˆ 
 X
X
2
i
n 1
n  sample size
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B. Sampling Distribution of the t Statistic
(same as normal distribution)
12
4
f(t)
0
t
Figure 1. Graph of the distribution of t for 4, 12, and ∞
degrees of freedom. When n = , the t distribution is
identical to the z (standard normal) distribution.
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1. Concept of degree of freedom, df or 
 Degrees of freedom for the t statistic:  = n – 1
3. Variance of the t statistic:
Var(t) 

2
4. A bit of history: 1908, the beginning of a new
era in statistics
5. Contribution of William Sealey Gossett (Student)
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Table 1. Percentage Points of Student’s t Distribution
Level of Significance for a One-Tailed Test
.05
.025
.01
.005
Degrees of
Freedom, 
10
19
20
26
30
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∞
Level of Significance for a Two-Tailed Test
.10
.05
.02
.01
1.812
1.729
1.725
1.706
1.697
1.686
1.645
2.228
2.093
2.086
2.056
2.042
2.024
1.960
2.764
2.539
2.528
2.479
2.457
2.429
2.326
3.169
2.861
2.845
2.779
2.750
2.712
2.576
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C. z Test Statistic
z
X  0
/ n
X   Xi / n
0  hypothesized pop. mean
  known population
standard deviation
n  sample size
1. The z test statistic is rarely used because the
population standard deviation is usually not
known.
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2. The sampling distribution of z is the standard
normal distribution.
3. Var(z) = 1
D. Comparison of t and z
X  0 Random variable Š− Constant

t
Random variable
ˆ n
z
X  0
/
Random variable Š− Constant

Constant
n
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IV Steps in Testing a Null Hypothesis
A. Registration Example at Idle-on-in College
1. Is a new registration procedure better than the
current procedure? For the past several years, the
current procedure has required 3.10 hours.
2. To evaluate the new procedure, the dean conducts
a trial run with a random sample of n = 27
students and obtains the following data:
X  2.90 and ˆ  0.3013
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3. Is the new procedure, X  2.90, really better than
the current procedure, 0  3.10,
4. Because of sampling variability, it is unlikely that
the sample mean is equal to the population mean
of the new procedure.
5. It seems reasonable to reject the null hypothesis if
the observed sample mean, X  2.90, is very
improbable given that 0 is equal to 3.10.
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X = 2.62
n = 27
X = 3.24
n = 27
X = 2.86
n = 27
=?
X = 2.98
n = 27
N = 12,660
X = 3.21
n = 27
X = 2.90
n = 27
Figure 2. Illustration of sampling variability of sample means
for six random samples from a population
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6. A sample mean that would be observed 5 or fewer
times in a hundred replications of an experiment
(Probability = 5/100 = .05) is considered very
improbable.
7. A sample mean with probability 5/100 = .05 is
reasonable grounds for rejecting a null hypothesis.
This criterion is called a significance level and
denoted by  05
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B. Five-Steps in Testing a Null Hypothesis for
the Registration Example
Step 1. State the statistical
hypotheses:
H 0 :   3.10
H1 :   3.10
Step 2. Specify the test
statistic:
t statistic because she wants
to test  ≥ 3.10,  is unknown,
the sample is random, and the
population distribution of X is
probably approximately normal
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Step 3. Specify the sample
size:
n = 27
and the sampling
distribution:
t distribution with  = n – 1 = 26,
because  is unknown and must
be estimated, and the population
distribution of X is probably
approximately normal
Step 4. Specify the
significance level:
 = .05
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Step 5. Obtain a random
sample of size n,
compute t, and
make a decision
Decision Rule: Reject the null hypothesis if t falls in the lower
5% of the sampling distribution of t; otherwise, do not
reject the null hypothesis. If the null hypothesis is
rejected, conclude that the new registration procedure
reduces the time to register; if the null hypothesis is not
rejected, do not draw this conclusion.
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C. Selecting a Level of Significance, 
1. Common conventions in specifying 
D. Specifying the Critical Region for Rejecting
H0
1. The location of the critical region is determined by
the alternative hypothesis.
2. The size of the critical region is determined by the
level of significance, .
3. The critical region identifies values of X that
are very unlikely if the null hypothesis is true.
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Critical region
f ( t ) for  = .05
–3
–2
–1
0
1
2
3
t
Critical
value = –1.706
X
2.926
2.984
Reject H 0
3.042
3.100
3.158
3.216
3.274
Don't reject H0
Figure 3. Critical region for rejecting the null hypothesis
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V Computation of t Statistic
Student
Registration
Time, Xi
1
2
3
.
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2.9
2.7
2.4
.
2.5
0.00
0.04
0.25
.
0.16
 ( X i  X )2  2.36
 X i  78.3
 X i 78.3
X

 2.90
n
27
( X i  X )2
ˆ 
2
(
X

X
)
 i
n 1
2.36

 0.3013
27  1
24
t
X  0
2.90  3.10
 0.20


 3.449
ˆ n 0.3013 27 0.0580
  n  1  27  1  26
t.05,26  1.706
A. Reject H0:   3.10 because
t  3.449  t.05, 26  1.706
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Critical region
f ( t ) for  = .05
–3
–2
–1
0
1
2
3
t
Critical
value = –1.706
X
2.926
2.984
Reject H 0
3.042
3.100
3.158
3.216
3.274
Don't reject H0
Figure 4. X  2.90 and t  3.449 fall in the  = 05
critical region
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VI Some Experimental Design Considerations
A. John Henry Effect
B. Importance of Control Groups
VII One- and Two-Tailed Tests
A. One-Tailed Test
1. Critical region is in either the upper or
lower tail of the sampling distribution.
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Critical region
 = .05
f (t )
t
–3
–2
–1
0
1
3
2
t.05, 26= 1.706
Don't reject H 0
Reject H 0
Figure 5. Critical region for H0:  ≤ 3.10
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B. Two Tailed Test
f (t )
Critical region
 /2 = .025
Critical region
 /2 = .025
t
–3
– t.05/2, 26= – 2.056
Reject
H0
–2
–1
0
Don't reject H 0
1
2
3
t.05/2, 26= 2.056
Reject
H0
Figure 6. For H0:  = 3.10, the critical region is divided
between the upper and lower tails of the t distribution.
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C. Denoting Critical Values of t
1. Upper and lower two-tailed critical values are
denoted by, respectively, t/2,  and –t/2,  ,
where  is the significance level and  is the
degrees of freedom.
2.
t,  denotes the upper one-tailed critical value
3.
–t,  denotes the lower one-tailed critical value
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D. Merits of One- and Two-Tailed Tests
VIII Type I and Type II Errors
A. Type I Error, corresponds to , (Rejecting a
True Null Hypothesis)
B. Type II Error, corresponds to , (Failing to
Reject a False Null Hypothesis)
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C. Correct Acceptance and Correct Rejection
D. Power, 1 – ,
for the Registration Example
1. Value of the sample mean that cuts off the lower
.05 region of the t sampling distribution
X .05  0  t.05, 26ˆ / n
 3.10  (1.706) / (0.3013) / 27  3.001
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2. Minimum reduction in registration time that would
warrant changing to the new procedure
0    3.10  2.95  nine minutes
where  = 2.95 corresponds to the new
registration time.
3. The size of the region corresponding to a Type II
error can be determined by computing a t statistic
for the difference X.05    3.001 2.950
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4. t statistic for computing the probability of a Type II
error, , and power, 1 – 
X .05   
t
ˆ n
3.001 2.950
0.051
 0.880


0.3013 / 27 0.058
5. Area above t = 0.880 is .19 (Prob. of a Type II error)
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Sampling distribution
under H 0
 = .05
f ( t)
1 –  = .95
 0 = 3.1
Sampling distribution
under H when ' = 2.95 X = 3.001
1
.05
t
1 – ^ = .81
^
=
t
' = 2.95
Reject H 0
Don’t reject H0
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Table 2. Probabilities Associated with the Decision Process
True Situation
 0
' 5
≤
Š 3.10
Researcher’s
Decision
 0
Correct acceptance
1 –  5
Type I error
 05
Type II error
^
 
Correct rejection
^
1 –  
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E. Factors that determine the probability of a
Type II error,  and power, 1 – 
1. Level of significance, 
2. Size of the sample, n
3. Size of the population standard deviation, 
4. Magnitude of the difference between  and 0.
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5. Simple way to increase power: increase the
sample size.
IX Determining the Required Sample Size (n)
A. Cohen’s Effect Size, d
d  |   0 | /

d = 0.2 is a small effect

d = 0.5 is a medium effect

d = 0.8 is a large effect
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Table 3. Approximate n Required for Testing Hypotheses
One-Sample Test
One-Sided
Hypothesis,
1 
Effect
Size,
d
0.2
0.5
0.8
Two-Sided
Hypothesis,
1 

0
0
5
0
0
5
.05
.01
.05
.01
.05
.01
156
253
27
43
12
19
215
328
36
55
15
24
272
396
45
66
19
28
198
294
34
51
15
22
264
374
44
63
19
27
326
447
54
75
22
32
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B. Statistical Significance Versus Practical
Significance
C. Reporting p Values
1. Obtaining p values from Excel
 Select INSERT from the Excel menu
 Select Function
 Select the TDIST function
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2. TDIST function
TDIST(x,deg_freedom,tails)
 Replace x with the t value
 Replace deg_freedom with the df for t
 Replace tails with 1 for one-tail and 2 for twotail
TDIST(3.449,26,1)
The p value is .0009652, or < .001
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