EXAMPLE 17 The 68–95–99.7 Rule Again

In Example 16, the population parameter, the proportion of adults who agreed with some statement, is $p=0.6$. if we take SRSs of size 1500, the sample proportions $\hat{p}$ follow the normal distribution with a mean of 0.6 and a standard deviation of 0.0126. The “95” part of the 68–95–99.7 rule from Section 5.9 (page 216) says that 95% of all samples give a $\hat{p}$ within 2 standard deviations of the truth about the population. So in this example, 95% of all samples have $\hat{p}$ within $2\times 0.0126$ of 0.60, that is, between 0. 57 and 0.63 (rounded to two decimal places). Figure 7.10 illustrates this use of the 68–95–99.7 rule.