EXAMPLE 2 Lots and lots of samples

Here’s another big idea of statistics: to see how trustworthy one sample is likely to be, ask what would happen if we took many samples from the same population. Let’s try it and see. Suppose that, in fact (unknown to Gallup), exactly 50% of all American adults feel childhood vaccines are extremely important. That is, the truth about the population is that p = 0.5. What if Gallup used the sample proportion from an SRS of size 100 to estimate the unknown value of the population proportion p?

42

Figure 3.1 illustrates the process of choosing many samples and finding for each one. In the first sample, 56 of the 100 people felt childhood vaccines are extremely important, so . Only 36 in the next sample felt childhood vaccines are extremely important, so for that sample = 0.36. Choose 1000 samples and make a plot of the 1000 values of like the graph (called a histogram) at the right of Figure 3.1. The different values of the sample proportion run along the horizontal axis. The height of each bar shows how many of our 1000 samples gave the group of values on the horizontal axis covered by the bar. For example, in Figure 3.1 the bar covering the values between 0.40 and 0.42 has a height of slightly over 50. Thus, more than 50 of our 1000 samples had values between 0.40 and 0.42.

Of course, Gallup interviewed 1015 people, not just 100. Figure 3.2 shows the results of 1000 SRSs, each of size 1015, drawn from a population in which the true sample proportion is p = 0.5. Figures 3.1 and 3.2 are drawn on the same scale. Comparing them shows what happens when we increase the size of our samples from 100 to 1015.