## Pooled variance procedure

Let and be the sample standard deviations constructed from and , respectively. When it is reasonable to assume  ,'' we can construct the pooled sample variance

The test statistic

has the -distribution with degrees of freedom under the null hypothesis . Thus, we reject the null hypothesis with significant level when the observed value of satisfies . Or, equivalently we can compute the -value

with having a -distribution with degrees of freedom, and reject when .

Confidence interval. The following table shows the corresponding confidence interval of the population mean difference , when your null hypothesis is rejected.

 Hypothesis testing -level confidence interval vs. . vs. . vs. .

Example. Suppose that we consider the significant level , and that we have obtained and from the control group of size , and and from the experimental group of size . Here we have assumed that . Then we can compute the square root of the pooled sample variance , and the test statistic

Thus, we can obtain , and reject . We conclude that the two population means are significantly different. And the 99% confidence interval for the mean difference is .

Generated by MATH GO: 2006-03-21