If we have two populations with population statisticsandthen the mean difference between the means of two samples isand if the sample sizes areand (from populations 1 and 2 respectively) are large, then difference between the sample means is normally distributed:(from the central limit theorem).

When the sample sizes are small we need to make the additional assumptions

- andare normally distributed
- The samples are independent
- The variances of the populations are equal

In practice the sample variances can be very dissimilar, but the equality of the population variances can be tested using the F – test.

In general we do not know the population variances and must calculate estimates for the population variances,andIf we assumeandare normally distributed then we can use an estimator for common variance and the difference between the means of the two samples is has a t – distribution withdegrees of freedom

We can then construct confidence intervals for some significance level %alpha using

Example: A sample of the heights of boys and girls is taken and the following results are obtained. Conduct a 90% confidence interval for the mean difference between the heights of boys and girls for the sample sizes given.

Boy’s heights: 153, 149, 148, 158, 159, 141, 142, 145

Girl’s heights: 143, 147, 133, 126, 139, 132, 143

and

The confidence interval is then