Two-Sample Testing: Small Samples

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Two-Sample Testing Small Samples
Problem 9.15 Bear gallbladder is used in Chinese
medicine to treat inflammation. Due to the
difficulty of obtaining bear gallbladder,
researchers are searching for a more readily
available source of animal bile. A study was
performed to determine if pig gallbladder is an
effective substitute for bear gallbladder. Twenty
male mice were divided randomly into two groups
10 were given a dosage of bear bile and 10 were
given a dosage of pig bile. All mice receive an
injection of croton oil in the left ear lobe to
induce inflammation. Four hours later, both the
left and right ear lobes were weighed, with the
difference (in milligrams) representing the
degree of swelling. Summary statistics are
provided in the following table.
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Summary Statistics
Question What conclusion can we make about
whether pig bile is an effective substitute for
bear bile?
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Comparing Means of Several Populations
Problem 10.21 Studies conducted at the
University of Melbourne indicate that there may
be a difference between the pain thresholds of
blondes and brunettes. Men and women of various
ages were divided into four categories according
to hair color light blond, dark blond, light
brunette, and dark brunette. Each person in the
experiment was given a pain threshold score based
on his/her performance in a pain sensitivity test
(higher scores mean higher pain tolerance). The
data is provided in the following table.
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Data from Experiment
Question Based on this data set, could we
conclude that there are differences in the mean
pain threshold of blondes and brunettes?
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Descriptive Statistics
Variable N Mean Median
StDev SE Mean LightBlo 5
59.20 60.00 8.53 3.81 DarkBlon
5 51.20 52.00 9.28
4.15 LightBru 4 42.50 41.50
5.45 2.72 DarkBrun 5
37.40 35.00 8.32 3.72
Note Would usually accompany this summary by a
comparative box plots, but since we only have a
few observations per group in this example, it is
not very appropriate in this case.
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Comparative DotPlots of the Four Groups
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Computations of Sum of Squares
SS due to Treatment (SSTr) ? ni(LMeani -
OMean)2 5(59.2 - 47.84)2 5(51.2 - 47.84)2
4(42.5 - 47.84)2 5(37.4-47.84)2
1360.7264 Lmeani sample mean of the
observations in sample i Omean overall sample
mean of all observations
SS due to Error (SSE) ?(ni-1)(Si)2
(5-1)(8.53)2 (5-1)(9.28)2 (4-1)(5.45)2 (5-
1)2(8.32)2 1001.5143 Si sample standard
deviation of the observations in sample i
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Test Procedure (ANOVA)
To test the null H0 m1 m2 mp versus the
alternative H1 at least two means are different,
we use the F-test which rejects H0 whenever
where Fap-1,N-p is the tabular value from the
F-distribution with (p-1, N-p) degrees-of-freedom
p is the number of groups, and N n1 n2
np is the total number of observations. Or, one
may simply compare the p-value (observed
significance level) to the nominal level (usually
.05).
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Analysis of Variance(Using Minitab)
Test Statistic
One-way Analysis of Variance Analysis of
Variance Source DF SS MS
F P Factor 3 1360.7 453.6
6.79 0.004 Error 15 1001.8
66.8 Total 18 2362.5
Individual 95 CIs For Mean
Based on Pooled
StDev Level N Mean StDev
--------------------------------- LightBlo
5 59.200 8.526
(-----------) DarkBlon 5 51.200 9.284
(-----------) LightBru 4
42.500 5.447 (-------------) DarkBrun
5 37.400 8.325 (-----------)
---------------------
------------ Pooled StDev 8.172
36 48 60 72
p-value
Conclusion Since the p-value is very small, we
can conclude that at least two of the population
means are different.
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Interpretations
If the F-test (ANOVA) fails to reject H0, then
you conclude that there are no differences among
the population means of the p populations. If
the F-test rejects H0, then you conclude that at
least two of the population means are different
(but not necessarily all of them). You then
proceed to examine the individual confidence
intervals, or you could perform pairwise t-tests
to determine which population means are
different. The use of the pairwise t-test will
entail an increase in the Type I error, so a more
appropriate analysis to detect which means are
different uses multiple comparisons procedures.