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Chapter 14: Hypothesis Tests – Small Samples
population 2, respectively.
:
(
)
:
(
)
̅
̅
√
, where
is the pooled standard deviation.
D.R
: For a specified significance level
, reject the null hypothesis if
the computed test statistic value
-
or if
where
+
– 2.
Conclusion
: ……….
Note
: This is a two-tailed test because of the not-equal-to symbol in the
alternative hypothesis. Also, note that the level of significance is shared
equally when finding the critical
value (
⁄
).
Example 14-8
:A random sample of size
= 16 selected from a normal
distribution yielded a standard deviation of
= 14 and a mean
̅
1
= 75. A
second random sample of size
= 12 selected from a different normal
distribution yielded a standard deviation of
= 16 and a mean
̅
2
= 85.
Assuming that the population variances are equal, is there a significant
difference between the population means at the 5% level of significance?
Solution:
Let
be equal to the population average for population 1.
Let
be equal to the population average for population 2.
Since the distributions for both populations are normally distributed and
both sample sizes are less than 30, then we can apply the small sample
-test
for the difference between two population means.
To minimize the computations in the problem, we will use the
Small
Sample Test for the Difference betweenTwo Independent Means
workbook to help with the computations. A portion of the workbook is
shown in
Figure 14-17
in which the population variances were assumed to
be equal.