Example 18.3

EXAMPLE 3 A sampling distribution

Take a simple random sample of 1015 adults. Ask each whether they feel childhood vaccinations are extremely important. The proportion who say Yes

is the sample proportion . Do this 1000 times and collect the 1000 sample proportions from the 1000 samples. The histogram in Figure 18.2 shows the distribution of 1000 sample proportions when the truth about the population is that 54% would favor such an amendment. The results of random sampling are of course random: we can’t predict the outcome of one sample, but the figure shows that the outcomes of many samples have a regular pattern.

Figure 18.2: Figure 18.2 The sampling distribution of a sample proportion from SRSs of size 1015 drawn from a population in which 54% of the members would give positive answers, Examples 3. The histogram shows the distribution from 1000 samples. The Normal curve is the ideal pattern that describes the results of a very large number of samples.

This repetition reminds us that the regular pattern of repeated random samples is one of the big ideas of statistics. The Normal curve in the figure is a good approximation to the histogram. The histogram is the result of these particular 1000 SRSs. Think of the Normal curve as the idealized pattern we would get if we kept on taking SRSs from this population forever. That’s exactly the idea of probability—the pattern we would see in the very long run. The Normal curve assigns probabilities to sample proportions computed from random samples.

433

This Normal curve has mean 0.540 and standard deviation about 0.016. The “95” part of the 68–95–99.7 rule says that 95% of all samples will give a falling within 2 standard deviations of the mean. That’s within 0.032 of 0.540, or between 0.508 and 0.572. We now have more concise language for this fact: the probability is 0.95 that between 50.8% and 57.2% of the people in a sample will say Yes. The word “probability” says we are talking about what would happen in the long run, in very many samples.

We note that of the 1000 SRSs, 95% of the sample proportions were between 0.509 and 0.575, which agrees quite well with the calculations based on the Normal curve. This confirms our assertion that the Normal curve is a good approximation to the histogram in Figure 18.2.