Introduction

This chapter is a continuation of the last chapter. As in the last chapter, we have two populations X and Y but we now want to estimate the difference between the populations. We do need to make one important assumption:

The populations differ by at most a shift  in locations (centers).

Fortunately we have at least visual checks for this assumption in comparison dotplots, boxplots and back-to-back stem leaf plots based on the samples we obtain. For example, if the lengths of the boxes in the comparison boxplots are much different then this is an indication that scale (or noise) level is also different between the populations. Or, if, provided the sample sizes are large enough, the shapes of the back-to-back stem leaf plots are quite different then this would indicate that the populations differ by more than a shift in locations.

Under this assumption, the problem is easily parameterized. Let $\Delta$  be the difference in locations  of the populations. In many problems, we think of $\Delta$ as the effect  between the populations. If $\mu_1$ is the mean of the first population and $\mu_2$ is the mean of the second population then $\Delta = \mu_2 - \mu_1$. But $\Delta$ is also the difference in population medians, shift is shift. Hence, if $\theta_1$ is the median of the first population and $\theta_2$ is the median of the second population then $\Delta = \theta_2 - \theta_1$. So we want to estimate $\Delta$ and we will be done. What's that? Louder, I can't hear you. Right! We must also estimate the error of estimation. We want a confidence interval for $\Delta$, too. How much did our estimate of $\Delta$ miss $\Delta$ by?

One final word. The value to check for in the confidence interval is 0. For if 0 is in the confidence interval then there may be no difference between the populations. Note this is another way of testing for a difference between populations. In particular, consider the hypotheses:

• H₀: Populations X and Y are the same.

• H_A: A measurement from population Y is either typically smaller than a measurement from population X or typically larger than a measurement from population X.

We are now dealing with a location problem, so we recast these hypotheses as:

• H₀: $\Delta = 0$.

• H_A: $\Delta \neq 0$.

We reject H₀ in favor of H_A, if 0 is not in the confidence interval.