If we have two independent populations with unknown variances, then we can use the t – distribution to test hypotheses about the means of the two populations – either that they are equal or that they differ by a certain amount. The procedure is:

Write down the null hypothesisfor exampleor

Write down the alternative hypothesisfor exampleor

Specify the significance level

Write down the number of degrees of freedom

Write down the critical value (one tailed test), values (two tailed test) or the critical region (one tailed test) or critical regions (two tailed test).

Calculate the sample meansand variances,and

Calculated the pooled variance

Calculate the value of the test statistic

Interpret the test statistic. If the value of the test statistic is in the critical region,can be rejected at the significance level of the test.
Example: Conduct a hypothesis test at the 10 % level for the equality of the means of the populations from which the following samples are drawn.
Population 1: 4, 2, 4, 8, 2, 6
Population 2: 7, 3, 6, 1, 9, 9, 10
For a significance level of 10% with 6+72 =11 degrees of freedom, (we are conducting a two tailed test since we are testing for equality) the critical values are
This is not in the critical region since