## Comparison of Two Means

Data are collected from two groups, say ``Group 1'' and ``Group 2,'' concerning with how Group 1 and Group 2 differ in terms of their respective population means and . Then the hypothesis test can be stated with the alternative hypothesis

The test statistic is calculated from the sample means and and the sample standard deviations and from Group 1 and Group 2 with the respective sample sizes and

In general procedure the variance estimate of is given by with the sample variances and of Group 1 and 2. Then the test statistic is likely observed around zero, positive, or negative, under the respective null hypothesis `` ,'' `` ,'' or `` .'' The opposite of such an observation is expressed by the p-value smaller than , and it suggests an evidence against the null hypothesis .

When it is reasonable to assume that the two population variances and of Group 1 and 2 are equal, the variance estimate is given by via pooled sample variance .

If the null hypothesis is rejected, it would be preferable to construct the confidence interval for the population mean difference .

Here the choices of confidence level are 90%, 95%, or 99%.