ST 361: Ch8 Testing Statistical Hypotheses:
Testing Hypotheses about Means (§8.2-2) : Two-Sample t Test
Topics: Hypothesis testing with population means
► One-sample problem: Testing for a Population mean 
1. Assume population SD is known: use a z test
2. Assume population SD is not known: use a t test
► Two-sample problem: : Testing for 2 population means  1 and 2 
► A Special Case: the Paired t test
-------------------------------------------------------------------------------------------------------------------► TWO-sample problem: Testing for 2 population means  1 and 2 
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Motivating Example
Is there a difference between the life of batteries made by Duracell and Eveready? Let 1 be the
mean lifetime for Duracell batteries, and  2 be that of Eveready batteries. Perform a 5% level of
test.
Duracell
Eveready
n (batteries)
80
100
x
4.1
4.5
Sample SDs
1.8
2.0
Step 1: Specify the hypotheses
parameter of interest =
H0 :
Ha
Step 2: significance level  =
Step 3: test statistic (???? Pages 2-3)
t* =  1.410, df = 175
Step 4: p-value
Step 5: conclusion:
1
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Calculating the Test Statistics for Testing Two Means
 Need
x1  x2  ~ Normal
 We will focus on the case that  1 and  2 are not known
 Test statistic is _________________________________________________________
____________________________________________________________________
 When the population SD’s  1 and  2 are NOT known, use ______________________
_____________ and hence work with a _____ test.
 SE1 2   SE2 2 


4
4
SE1
SE2

n1  1
n2  1
And df =

 
2

(round down!!!) where SE1 
s1
s
and SE2  2
n1
n2
 Note that the test statistic should be consistent with your hypotheses. That is, if your
hypotheses are stated in terms of H 0 : 2  1 = 0 (as opposed to 1  2 ) , then the
corresponding test statistic should be
t* 
(Back to the battery example)
Step 3: test statistic
Note that df 
 SE1 2   SE2 2 


4
4
SE1
SE2

n1  1
n2  1

 

2
 175.5
2
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Summary of the testing procedure for two population means:
Step (1) Hypotheses
H 0 : 1  2  _____ vs. H a : 1  2 _______ (lower-tail test)
H a : 1  2 _______ (upper-tail test)
H a : 1  2 _______ (two-sided test)
Step (2) Significance level
Step (3) Test statistic
t* 
With df =
Step (4) P-value =
x1  x2   1   2   x1  x2   c
s12 s 22

n1 n2
 SE1 2   SE2 2 


4
4
SE1
SE2

n1  1
n2  1

 
2

_______________
_______________
_______________
(a)
s12 s 22

n1 n2
where SE1 
s1
s
and SE2  2
n1
n2
if H a : 1   2  c
if H a : 1   2  c
if H a : 1   2  c
Conclusion: Reject H 0 if p-value <  , and draw conclusion according to H a
Otherwise do not reject H 0 , and draw conclusion according to H 0
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Ex. Mary can take either a scenic route to work or a non-scenic route. She decides that use of nonscenic route can be justified only if it reduces true average travel time by more than 10 min.
(a) If 1 refers to the scenic route and  2 to the non-scenic route, what hypotheses should be tested?
(b) What should be the test statistic for testing your hypothesis?
x  x   1   2 
x  x   1   2 
(1) t *  1 2
(2) t *  2 1
2
s1

n1
2
s2
2
s1
n2
n1

2
s2
n2
x  x    2  1 
x  x    2  1 
(3) t *  1 2
(4) t *  2 1
2
s1

n1
2
s2
2
s1
n2
n1

2
s2
n2
(c) If 1 refers to the non-scenic route and  2 to the scenic route, what hypothesis should be tested?
(d) What should be the test statistic for testing your hypothesis?
x  x   1   2 
x  x   1   2 
(1) t *  1 2
(2) t *  2 1
2
s1

n1
2
s2
2
s1
n2
n1

2
s2
n2
x  x    2  1 
x  x    2  1 
(3) t *  1 2
(4) t *  2 1
2
s1
n1

2
s2
2
s1
n2
n1

2
s2
n2
4
Ex. Many people take ginkgo supplements advertised to improve memory. Are these over-the-counter
supplements effective? In a study, elderly adults were assigned to the treatment group or control
group. The 104 participants who were assigned to the treatment group took 40 mg of ginkgo 3 times
a day for 6 weeks. The 115 participants assigned to the control group took a placebo pill 3 times a
day for 6 weeks. At the end of 6 weeks, the Wechsler Memory Scale was administered. Higher
scores indicate better memory function. Summary values are given in the following table:
N
x
s
Ginkgo
104
5.7
0.6
Placebo
115
5.5
0.6
Based on these results, is there evidence that taking 40mg of ginkgo 3 times a day is effective in
increasing mean performance on the Wechsler Memory Scale?
Note that df 
 SE1  2   SE2  2 


4
4
SE1
SE2

n1  1
n2  1




2
 214.81
Step 1: parameter of interest =
H0 :
Ha
Step 2: significance level  =
Step 3: test statistic =
Step 4: p-value =
Step 5: Conclusion:
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Ex. Consider an experiment involving the comparison of the mean heart rate following 30 minutes of aerobic
exercise among females aged 20 to 24 years as compared to females aged 30-34 years. For this
experiment, 10-second heart rates are recorded on each participant following 30 minutes of intense
aerobic exercise and converted to beats per minute (i.e., heart rate per 60 seconds). The sample data are
given below:
Age 20-24
Age 30-34
Sample size
15
10
Sample mean
150.22
141.10
Sample SD
40.0
10.0
SE   SE   = 16.522
Here df 
2 2
2
1
2
n1  1
n2  1
SE1 2  SE2 2
(a) What assumptions do we need in to have the mean difference follow a normal distribution?
(b) Is the mean heart rate for the age group 20-24 more than that for the age group 30-34 by 5 beats per
minute? Use =0.05.
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