# 9.2: Comparison of Two Population Means - Small, Independent Samples

Learning Objectives

• To learn how to construct a confidence interval for the difference in the means of two distinct populations using small, independent samples.
• To learn how to perform a test of hypotheses concerning the difference between the means of two distinct populations using small, independent samples.

When one or the other of the sample sizes is small, as is often the case in practice, the Central Limit Theorem does not apply. We must then impose conditions on the population to give statistical validity to the test procedure. We will assume that both populations from which the samples are taken have a normal probability distribution and that their standard deviations are equal.

## Confidence Intervals

When the two populations are normally distributed and have equal standard deviations, the following formula for a confidence interval for $$\mu _1-\mu _2$$ is valid.

$$100(1-\alpha )\%$$ Confidence Interval for the Difference Between Two Population Means: Small, Independent Samples

$(\bar{x_1}-\bar{x_2})\pm t_{\alpha /2}\sqrt{s_{p}^{2}\left ( \dfrac{1}{n_1}+\dfrac{1}{n_2} \right )} \label{eq1}$

where

$s_{p}^{2}=\dfrac{(n_1-1)s_{1}^{2}+(n_2-1)s_{2}^{2}}{n_1+n_2-2}$

The number of degrees of freedom is

$df=n_1+n_2-2.$

The samples must be independent, the populations must be normal, and the population standard deviations must be equal. “Small” samples means that either