## Estimating a Difference Score

Imagine that instead of estimating a single population mean μ, you wanted to estimate the difference between two population means μ _{1} and μ _{2}, such as the difference between the mean weights of two football teams. The statistic has a sampling distribution just as the individual means do, and the rules of statistical inference can be used to calculate either a point estimate or a confidence interval for the difference between the two population means.

Suppose you wanted to know which was greater, the mean weight of Landers College's football team or the mean weight of Ingram College's team. You already have a point estimate of 198 pounds for Landers's team. Suppose that you draw a random sample of players from Ingram's team, and the sample mean is 195. The point estimate for the difference between the mean weights of Landers's team (μ _{1}) and Ingram's team (μ _{2}) is 198 – 195 = 3.

But how accurate is that estimate? You can use the sampling distribution of the difference score to construct a confidence interval for μ _{1} – μ _{2}. Suppose that when you do so, you find that the confidence interval limits are (–3, 9), which means that you are 90 percent certain that the mean for the Landers team is between 3 pounds lighter and 9 pounds heavier than the mean for the Ingram team (see Figure 1).

Figure 1.The relationship between point estimate, confidence interval, and *z*‐score, for a test of the difference of two means.

Suppose that instead of a confidence interval, you want to test the two‐tailed hypothesis that the two team weights have different means. Your null hypothesis would be:

*H* _{0}: μ _{1} = μ _{2}

or

*H* _{0}: μ _{1} – μ _{2}= 0

To reject the null hypothesis of equal means, the test statistic—in this example, *z‐*score—for a difference in mean weights of 0 would have to fall in the rejection region at either end of the distribution. But you have already seen that it does not—only difference scores less than –3 or greater than 9 fall in the rejection region. For this reason, you would be unable to reject the null hypothesis that the two population means are equal.

This characteristic is a simple but important one of confidence intervals for difference scores. If the interval contains 0, you would be unable to reject the null hypothesis that the means are equal at the same significance level.