## Constructing Confidence Intervals For the Difference of Two Means From Two Normal Populations

If we have two populations with population statisticsandthenthe mean difference between the means of two samples isandif the sample sizes areand(frompopulations 1 and 2 respectively) are large, then difference betweenthe sample means is normally distributed:(fromthe central limit theorem).

When the sample sizes are small we need to make the additionalassumptions

andarenormally distributed

The samples are independent

The variances of the populations are equal

In practice the sample variances can be very dissimilar, but theequality of the population variances can be tested using the F â€“test.

In general we do not know the population variances and mustcalculate estimates for the population variances,andIfwe assumeandarenormally distributed then we can use an estimator for commonvarianceand the difference between the means of the two samples is has a t â€“distribution withdegreesof freedom

We can then construct confidence intervals for some significancelevel %alpha using

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