Illuminating Statistical Analysis Using Scenarios and Simulations

23Homing in on the Population Mean II

Let's look at this from the perspective of one of the surveyors (you).

Statistical Scenario—Sample Mean II

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 members and record their responses using the following scale:

Strongly agree

1

2

3

4

5

6

7

Strongly disagree

Say your sample mean is 4.15 and your sample variance is 1.0.

Could the true community mean equal 4.0?

In other words, does the 95% confidence interval centered on the sample mean of 4.15 contain the value 4.0?

As the lone surveyor, you will want to construct a 95% confidence interval around your sample mean of 4.15 to check whether it contains 4.0. Since the true population variance in this scenario is unknown—as is usually the case in practice—the sample variance is used instead.

Note: Using the sample variance as an estimate of the population variance introduces additional uncertainty. We won't address this source of uncertainty until Chapter 27.

Let's simulate the uncertainty surrounding our sample mean of 4.15. We'll simulate 1000 random samples of size 100 based on the scenario's mean of 4.15 and variance of 1.0. Given the results shown in Figure 23.1, it appears that 4.0 is not especially unlikely. Eyeballing, the 95% confidence interval appears to be about 3.95–4.35. Given this, we would not rule out the hypothesis that the true population mean could be 4.0. ...