Example 15

EXAMPLE 15 What Happens in Many Samples?

We start with a scenario in which we know that the population proportion responding “Agree” to some statement is . Using computer simulation, we repeatedly take random samples, first of size 100 and later of size 1500. For each sample, we calculate the sample proportion .

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Results from Samples of Size 100

A histogram of the 1000 values of from our computer-simulated data appears in Figure 7.6. This histogram gives us an idea of the shape, center, and spread of the distribution of the sample proportion for samples of size 100 drawn from a population in which .

Figure 7.6: Figure 7.6 Draw 1000 SRSs of size 100 from a population with proportion of “Agrees.” The histogram shows the distribution of the 1000 sample proportions .

The USA Today Pew Research Center poll interviewed around 1500 people, not just 100. Again, we use computer simulation to generate 1000 samples of size 1500 and record the value of the sample proportion for each sample. A histogram of the based on these 1000 samples is shown in Figure 7.7.

Figure 7.7: Figure 7.7 Draw 1000 SRSs of size 1500 from a population with proportion of Agrees. The histogram shows the distribution of the 1000 sample proportions .

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For comparative purposes, Figures 7.6 and 7.7 are drawn using the same horizontal scale. This allows us to compare what happens when we increase the size of our samples from 100 to 1500. These histograms display the sampling distribution of the statistic for two sample sizes. Notice that for both situations, the histograms are centered at , the known value of the parameter. The histograms are single-peaked and roughly symmetric—what we would expect for a normal distribution (refer to Section 5.8, page 209). However, the variability is much smaller for the situation in which the sample size is 1500 compared to only 100.