Chapter 19
Two Sample
Problems
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Two Sample Problems
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Conditions
• Two populations, independent samples
• Population means and standard
deviations not known
Population 1: μ1, σ1
Population 2: μ2, σ2
• Use t procedures in inferential methods
• In theory, both populations Normal, but in
practice, not necessary when n large -see Ch 18
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Two Sample Problems
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Sampling Distribution of
Mean Difference
• Figure shows
sampling distribution
of xbar1 − xbar2
• Standard error of
xbar1 − xbar2 is:
2
1

2
2
s
s
SE 

n1 n2
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s12 s22

n1 n2
x1  x 2
m 1−m 2
Two Sample Problems

4
Two-Sample t-test
• H0: μ1 = μ2 versus
Ha: μ1 ≠ μ2 (two-sided) or
Ha: μ1 > μ2 (one-sided, right) or
Ha: μ1 < μ2 (one-sided, left)
• Test statistic
x1  x2
s12 s22
t
where SEx1  x2 

SEx1  x2
n1 n2
• Conservative estimate for df = the smaller of
(n1−1) or (n2−1)
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Two Sample Problems
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Example: “Smoking Moms”
• Does parental smoking
damage kid’s lungs?
• Forced Vital Capacity
(FVC) = physiologic
measure of lung
capacity (volume of air
Group xbar s
n
exhaled in 6 seconds) Exposed 75.5 9.3 30
• Measure FVC in
exposed and
Non-exp 88.2 15.1 30
nonexposed children
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Hypothesis Test (Smoking Moms)
Hypotheses
H0: μ1 = μ2 versus
Ha: μ1 < μ2 (one-sided) or
Ha: μ1 ≠ μ2 (two-sided)
Test statistic
t
x1  x2
s12 s22

n1 n2

75.5  88.2
12.7

  3.922
3.238
9.32 15.12

30
30
df  n  1  30  1  29
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Two Sample Problems
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P-value (Smoking Moms)
•
•
•
•
Use absolute value of t statistic (3.922)
Table C, row for df = 29
Largest t* in that row = 3.659 (P = 0.0005)
Therefore:
P < 0.0005 (one-sided)
P < 0.001 (two-sided)
• Highly significant evidence against H0
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Two Sample Problems
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C Level Conf. Int. for μ1 − μ2

( x1  x2 )  t  SE x1  x2
where
SE x1  x2 
2
s1
n1

2
s2
n2
and
df  the smaller of (n1  1) or (n2  1)
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Two Sample Problems
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CI for “Smoking Mom’s” Example
95% confidence for example, use t* = 2.045
SE 
2
2
s12 s22
9.3 15 .1



 3.238
n1 n2
30
30
( x1  x2 )  (t  )( SE)  (75 .5  88 .2)  (2.045 )(3.238 )
 12 .7  6.6
 (19 .3,6.1)
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Two Sample Problems
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CI for “Smoking Mom’s” Example
90% confidence for example, use t* = 1.699
( x1  x2 )  t 
s12 s 22

n1 n2
9.32 15 .12
 (75 .5  88 .2)  (1.699 )

30
30
 12 .7  5.5
 (18 .2,7.2)
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Two Sample Problems
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Which type of test? One-sample? paired
samples? two samples?
• Comparing vitamin content of bread,
• Is blood pressure altered by use
immediately after baking versus 3
of an oral contraceptive?
days later (the same loaves are used
Comparing a group of women
on day one and 3 days later).
not using an oral contraceptive
• Comparing vitamin content of bread,
with a group taking it.
immediately after baking versus 3
• Review insurance records for
days later (tests made on
dollar amount paid after fire
independent loaves).
damage in houses equipped
• Average fuel efficiency for 2005
vehicles is 21 miles per gallon. Is
average fuel efficiency higher in the
new generation “green vehicles?”
with a fire extinguisher versus
houses without one. Was there
a difference in the average
dollar amount paid?
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Two Sample Problems
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