Comparing the Means of Two Dependent Populations

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Comparing the Means of Two Dependent Populations

• The Paired T-test .

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Do males earn higher average starting salaries
than females?
(in 1,000s) Males Females 34
28 32 30 29 22 ?
? 35 32 Sample Mean 33
28
Real question is whether males and females in the
same job earn different average salaries. Better
then to compare the difference in salaries in
pairs of males and females.
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Same example, but now a Paired Study
Job Males Females DifferenceM-F Non-Profit
22 20.5 1.5 Education 29
28 1.0 Doctor 80 78 2.0 ?
? ? Scientist 35 32 3.0 (in
1,000s) Sample Average Difference 1.9
How likely is it that a paired sample would have
a difference as large as 1,900 if the true
difference were 0?
Problem reduces to a One-Sample T-test on
differences!!!!
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Hypotheses for Paired T-test
Does the average difference of the population,
?D, differ from 0?
Null hypothesis H0 ?D ?1 - ?2 0
Alternative hypotheses HA ?D ?1 - ?2 ? 0
HA ?D ?1 - ?2 gt 0 HA ?D ?1 - ?2 lt 0
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The Paired-T Test Statistic

• If
• there are n pairs
• and the differences are normally distributed

Then The following test statistic, which follows
a t-distribution with n-1 d.f., gives us our
P-value
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The Paired-T Confidence Interval

• If
• there are n pairs
• and the differences are normally distributed

Then The following confidence interval, with t
following t-distribution with n-1 d.f. estimates
the actual population difference
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Paired T in Minitab

• Enter your paired data in two columns.
• Select Stat gtgt Basic Statistics gtgt Paired T
• In the boxes labeled First (Second) Sample,
  select the variables that contains your first
  (second) samples.
• Under Options, specify the confidence level, the
  null mean value (usually 0), and the alternative
  hypothesis. Select OK.
• Select OK.

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Ways Pairing Can Occur

• When, in before and after studies, the same
  subjects are measured twice.
• When subjects serve as their own control by
  receiving both of two different treatments.
• When subjects in one group are matched with a
  similar subject in the second group.

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What is the effect of exercise on pulse rates?
H0 ?D ?Before - ?After 0 vs. HA ?D lt 0
Paired T for Before - After N
Mean StDev SE Mean Before 64
69.41 10.44 1.30 After 64
80.84 11.40 1.43 Difference 64
-11.44 8.07 1.01 95 CI for mean
difference (-13.45, -9.42) T-Test of mean
difference 0 (vs lt 0) T-Value -11.34
P-Value 0.000
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What is the effect of alcohol on useful
consciousness?
H0 ?D ?No alcohol - ?Alcohol 0 vs. HA ?D gt 0
Paired T for NoAlcohol - Alcohol
N Mean StDev SE Mean NoAlcohol
10 546.6 238.8 75.5 Alcohol
10 351.0 210.9 66.7 Difference
10 195.6 230.5 72.9 95 CI for
mean difference (30.7, 360.5) T-Test of mean
difference 0 (vs gt 0) T-Value 2.68
P-Value 0.013
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For males, does average actual and ideal weights
differ from 0?
H0 ?D ?Actual - ?Ideal 0 vs. HA ?D ? 0
Paired T for M_Actual - M_Ideal
N Mean StDev SE Mean M_Actual
83 172.72 28.65 3.14 M_Ideal
83 169.60 22.09 2.43 Difference
83 3.12 17.01 1.87 95 CI for mean
difference (-0.59, 6.83) T-Test of mean
difference 0 (vs not 0)T-Value 1.67
P-Value 0.098
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For females, does average actual and ideal
weights differ from 0?
H0 ?D ?Actual - ?Ideal 0 vs. HA ?D ? 0
Paired T for F_Actual - F_Ideal
N Mean StDev SE Mean F_Actual
88 138.20 28.00 2.98 F_Ideal
88 125.64 14.01 1.49 Difference
88 12.57 18.68 1.99 95 CI for mean
difference (8.61, 16.53) T-Test of mean
difference 0 (vs not 0) T-Value
6.31 P-Value 0.000