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.