Download Independent Samples: Comparing Means

Paired Samples
Lecture 39
Section 11.3
Tue, Nov 15, 2005
Independent Samples
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In a paired study, two observations are made on
each subject, producing one sample of bivariate
data.
Or we could think of it as two samples of
paired data.
Often these are “before” and “after”
observations.
By comparing the “before” mean to the “after”
mean, we can determine whether the intervening
treatment had an effect.
Independent Samples
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On the other hand, with independent samples,
there is no logical way to “pair” the data.
One sample might be from a population of
males and the other from a population of
females.
Or one might be the treatment group and the
other the control group.
The samples could be of different sizes.
Paired Samples
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We have two samples of the same size n.
For each value x in the first sample, there is a
corresponding value y in the second sample.
We are interested in the differences D = Y – X.
See Example 11.1, p. 679 – Weight Change.
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The two lists must be in the same order!
Example
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Example 11.1 on the TI-83:
Enter {148, 176, 153, …, 132} into list L1 (before)
 Enter {154, 181, 151, …, 128} into list L2 (after)
 Compute the differences:
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L2 – L1  L3 (after – before)
 L3 = {6, 5, -2, …, -4}.
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Call this variable D.
 Use 1-Var Stats to find the meand and standard
deviation sD of D.
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d = 1.75, sD = 3.412.
The Distribution of D
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Let n be the sample size.
If the samples are large, then D has (approx.) a
normal distribution.
However, if the samples are small, then we will
need an additional assumption.
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The populations are normal.
In that case, D will have a normal distribution,
too.
The Distribution of D
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To keep this section simple and to follow the
book’s example, we will use only the t
distribution.
The Paired t-Test
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The hypotheses.
H0: D = 0
 H1: D  0 or D < 0 or D > 0
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The test statistic.
d 0
t
sD / n
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The other steps are the same.
Example
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Conduct the paired t-test for the data in Example
11.1.
 H0: D = 0
 H1: D > 0
  = 0.05
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t
d 0
sD / n
Example
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Compute t:
The differences are 6, 5, -2, 4, 2, 2, 1, -4.
 d = 1.75.
 sD = 3.412.
 t = 1.75/(3.412/8) = 1.451.
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Compute the p-value:
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p-value = P(t > 1.451) = 0.0951. (df = 7)
Example
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Since p-value > , do not reject H0.
At the 5% level of significance, those who stop
smoking do not gain weight.
Practice
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Work Example 11.2 on the TI-83 as a
hypothesis-testing problem.
Enter the data into L1 and L2.
Store the differences in L3.
Use T-Test with the data in L3.
Confidence Intervals
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A confidence for D is
 sD 
d  t

 n
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Use the t table or the TI-83.
Example
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Find a 90% confidence interval for D based on
the data in Example 11.1.
The 90% confidence interval is 1.75  2.286.
On the TI-83, use TInterval.
Practice
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Find a 99% confidence interval for D based on
the data in Example 11.2.