ST 361: Ch8 Testing Statistical Hypotheses:
Testing Hypotheses about Means (§8.2-3) : Paired 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
------------------------------------------------------------------------------------------------------------------- Motivating example: A nutrition expert is examining a weight loss program to evaluate its
effectiveness (i.e., if participants lose weight on the program). Ten subjects are randomly selected for
the investigation. Each subject’s initial weight is recorded, they follow the program for 6 weeks, and
they are again weighed. The data are given below:
Subject
1
2
3
4
5
6
7
8
9
10
Initial Weight
180
142
126
138
175
205
116
142
157
136
Final Weight
165
138
128
136
170
197
115
128
144
130
Assume the weights follow a normal distribution. Is the program effective in reducing the weight?
Conduct a test using =0.05.
KEY THOGHT:

The two samples (initial weights and final weights) cannot be analyzed separately as we did in
the case of 2 independent samples because they are __________________________
____________________________. Such data type is referred to ________________.

When the samples are paired, we instead analyze ____________________________. The
difference, denoted by ______, can be computed by subtracting the “before” value by the
“after” value or vice versa, depending on the question of interest.

Then the question becomes to test if the true population mean of this d variable (denoted by
________) is greater/less/different from 0.

Indeed if d =X1-X2, then d = _______________________

Therefore this is essentially a “one-sample” problem, and the only difference is that here we
are interested in a variable called d instead of X (and hence the parameter of interest is
denoted by _______ instead of .
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 Inference about d :
A. What would be a good estimator of  d ?
B. The sampling distribution of d :
1. What is the mean and SD of the sampling distribution of d ?
2. When will d follow a Normal distribution?
C. Hypothesis testing for d :
In reality, because most of the time the population SD  d will not known and the sample SD sd
will be used, we will use a _____ test statistic, and such test is referred to as ______________
(Back to the weight example)
Subject
1
2
3
4
5
6
7
8
9
10
Initial Weight
180
142
126
138
175
205
116
142
157
136
Here d  6.6 and sd 
Final Weight
165
138
128
136
170
197
115
128
144
130
d
i
d 
n 1
Difference (d) = Initial - Final
15
4
-2
2
5
8
1
14
13
6
2
 5.82
2
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. How does energy intake compare to energy expenditure? One aspect of this issue was considered in
a 2002 study, which contained the data as displayed below (measured in MJ/day):
Player
1
2
3
4
5
6
7
Expenditure 14.4
12.1
14.3
14.2
15.2
15.5
17.8
Intake
9.2
11.8
11.6
12.7
15.0
16.3
14.6
Perform a statistical test to see whether the energy intake is smaller than energy expenditure at a
significance level of 0.01.
Note: Some summary statistics of the 7 players: xE  14.786, sE  1.718. xI  13.029, sI  2.428.
Define d  E  I , then d  1.757, sd  1.197
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