Illuminating Statistical Analysis Using Scenarios and Simulations

22Homing in on the Population Mean I

Statistical Scenario—Sample Mean I

Say you want to see how strongly, on average, members of your community agree or disagree with a new proposed policy. You decide to survey 100 random community members and record their responses using the following scale.

Strongly agree

1

2

3

4

5

6

7

Strongly disagree

Let's say the true, unknown population mean is 4.0 and the unknown population variance is 1.0.

What are the chances you would get a sample mean of 4.5?

Let's look at some simulation results that mimic 1000 surveyors who each survey 100 randomly selected people in the community—Figure 22.1. Only omniscient beings know that the overall community average is 4.0 and the overall community variance is 1.0.

What are the chances that a surveyor would get a sample mean of 4.5?

Figure 22.1

It looks like it is almost impossible for a surveyor to get a sample that has a sample mean of 4.5. In other words, with a sample size of 100 it seems that sample means will almost never be that far from the true population mean; random samples will almost never be that out of whack. About 95% of the simulation's sample means are within 0.2 of the true population mean.

Next, let's look at the formulaic method. There is a “trick” formula that has the same form as the one we used for proportions. Standard Error in both cases is the square root of ...