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The means of two populations based on a simple random paired sample.

The basic strategy for comparing the means of two populations based on a simple random paired sample is given in the following steps:

1. Independently and randomly take a paired sample.

2. The variable analyzed is the difference between the paired values of the variable under study.

3. By the use of a paired sample, we can remove an extraneous source of variation.

4. The sampling error thus made in estimating the difference between the populations means will generally smaller and, therefore, more likely to detect difference between the population means when such differences exist.

Suppose that $x$is a variable on each of two populations whose members can be paired. For each pair, we calculate the difference between the values of the variable $x$on the members of the pair denoted by $d$and is called as paired-difference variable.

${\mathrm{μ}}_d={\mathrm{μ}}_1-{\mathrm{μ}}_2$

The paired difference variable is normally distributed.

Test Statistic:

The test statistic for comparing two population means in a paired sample of size $n$is defined as follows:

$t=\frac{\stackrel{¯}{d}-{\mathrm{μ}}_d}{s_d/\sqrt{n}}~t_{(n-1)}$