This section briefly covers statistical inference for two samples and is divided into the following subsections:
Inference on the Difference in Population Means When Variances Are Known
Inference on the Difference in Population Means When Variances Are Unknown
The test statistic used here is based on the standard normal distribution.
Let 
and 
represent the means of two populations,
and 
and 
their variances, respectively. Let
be the hypothesized difference in
the population means and 
and 
be the sample means obtained from
two samples of sizes 
and 
drawn randomly from the two populations,
respectively. The test statistic can be obtained as:
(12)
The statements for the hypothesis test are:
If 
, then the hypothesis will test for
the equality of the two population means.
If the population variances can be assumed to be equal then the following
test statistic based on the 
distribution can be used. Let 
, 
, 
and 
be the sample means and variances
obtained from randomly drawn samples of sizes 
and 
from the two populations, respectively.
The weighted average, 
, of the two sample variances is:
has (
+ 
-- 2) degrees of freedom. The test
statistic can be calculated as:
(13)
follows the 
distribution with (
+ 
-- 2) degrees of freedom. This test
is also referred to as the two-sample pooled 
test.
If the population variances cannot be assumed to be equal then the following test statistic is used:
(14)
follows the 
distribution with 
degrees of freedom. 
is defined as follows:
The test statistic used here is based on the 
distribution. If 
and 
are the sample variances drawn randomly
from the two populations and 
and 
are the two sample sizes, respectively,
then the test statistic that can be used to test the equality of the population
variances is:
(15)
The test statistic follows the 
distribution with (
-- 1) degrees of freedom in the numerator
and (
-- 1) degrees of freedom in the denominator.
Example 3.5
Assume that an analyst wants to know if the variances of two normal
populations are equal at a significance level of 0.05. Random samples
drawn from the two populations give the sample standard deviations as
1.84 and 2, respectively. Both the sample sizes are 20. The hypothesis
test may be conducted as follows:
It is clear that this is a two-sided hypothesis and the critical region will be located on both sides of the probability distribution.
. Here the test statistic is based
on the 
distribution. For the two-sided hypothesis
the critical values are obtained as:
and
These values and the critical regions are
shown in Figure 3.16. The analyst would fail to reject 
if the test statistic 
is such that:
or
|
Figure 3.16: Critical values and rejection region for
Example 3.5 marked on the |
corresponding to the given data is:
Since 
lies in the acceptance region, the
analyst fails to reject 
at a significance level of 0.05.
See Also:
Statistical Inference for a Single Sample
Simple Linear Regression Analysis
distribution.