Database Reference
In-Depth Information
A common hypothesis test is to compare the means of two populations. Two such
hypothesis tests are discussed in Section 3.3.2.
3.3.2 Difference of Means
Hypothesis testing is a common approach to draw inferences on whether or not
the two populations, denoted and , are different from each other. This
section provides two hypothesis tests to compare the means of the respective
populations based on samples randomly drawn from each population. Specifically,
the two hypothesis tests in this section consider the following null and alternative
hypotheses.
•
:
•
:
The
and
denote the population means of
and
, respectively.
The basic testing approach is to compare the observed sample means, and
, corresponding to each population. If the values of and are approximately
equal to each other, the distributions of and overlap substantially (
Figure
the sample means indicates that the null hypothesis should be rejected. Formally,
the difference in means can be tested using Student's
t
-test or the Welch's
t
-test.
Figure 3.23
Overlap of the two distributions is large if
Student's t-test
Student's
t
-test assumes that distributions of the two populations have equal but
unknown variances. Suppose
and
samples are randomly and independently
selected from two populations,
and
, respectively. If each population is
normally distributed with the same mean (
) and with the same variance,
