Illuminating Statistical Analysis Using Scenarios and Simulations

25Judging Mean Differences

Statistical Scenario—Sample Mean Difference

Say you want to see if there is a difference of opinion between females and males in your community regarding a new proposed policy. You decide to survey 100 random women and 100 random men and record their responses using the following scale:

Strongly agree

1

2

3

4

5

6

7

Strongly disagree

This is your evidence:

Your sample mean for females is 3.0 with a sample variance of 3.1.

And your sample mean for males is 3.5 with a sample variance of 2.9.

How likely is it to get a sample mean difference as extreme as −0.5 (and, two-tail, also +0.5) by chance when males and females have the same population mean?

Null hypothesis: The female population mean minus the male population mean equals zero. In other words, the female and male population means are equal.

This statistical scenario is analogous to what we saw with the difference between two sample proportions. For the simulation, we'll assume the null hypothesis of equal population means for females and males is true, so the difference in the population means is zero. We'll make both the males' and females' population means equal 3.0. We want to see how frequently we should expect to get two sample means that are 0.5 apart or more simply by chance when the two population means are in fact equal. The uncertainty, as usual, is driven by the variances and sample sizes. We'll use the sample variances of 3.1 and 2.9 as estimates for the female and male ...