## t-Test

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 .

Data must be arranged either (a) in two columns each of which contains data for the respective group, or (b) all in a single column with another categorical variable identifying "Group 1" and "Group 2." Then data analysis begins with the calculation of the respective sample means and , and the sample standard deviations and from Group 1 and Group 2 with the respective sample sizes and .

The hypothesis test must be described by the alternative hypothesis

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, not toward negatively extreme, or not toward positively extreme, under the null hypothesis against the respective alternative hypotheses `` ,'' `` ,'' or `` .'' The opposite of such an observation is expressed by the p-value smaller than , and it suggests evidence to support the alternative 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
.
In pooled t-test,
is called the Cohen's *d*,
aiming at the estimate of standardized mean difference.

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%.