Question

If a hurricane were headed your way, would you evacuate? The headline of a press release issued January \(21,2009,\) by the survey research company International Communications Research (icrsurvey.com) states, "Thirty-one Percent of People on High-Risk Coast Will Refuse Evacuation Order, Survey of Hurricane Preparedness Finds." This headline was based on a survey of 5,046 adults who live within 20 miles of the coast in high hurricane risk counties of eight southern states. The sample was selected to be representative of the population of coastal residents in these states, so assume that it is reasonable to regard the sample as if it were a random sample. a. Suppose you are interested in learning about the value of \(p\), the proportion of adults who would refuse to evacuate. This proportion can be estimated using the sample proportion, \(\hat{p} .\) What is the value of \(\hat{p}\) for this sample? b. Based on what you know about the sampling distribution of \(\hat{p}\), is it reasonable to think that the estimate is within 0.03 of the actual value of the population proportion? Explain why or why not.

Short answer

\(\hat{p} = 0.31\) and it is indeed reasonable to think that the estimate is within \(0.03\) of the actual value of the population proportion based on the computed standard error.

Step-by-step solution

Calculate the Sample Proportion

The sample proportion, \(\hat{p}\), is calculated by dividing the number of successes, in this case, the number of adults refusing to evacuate, by the total number of individuals in the sample. The press release mentions that 31 percent of people, or \(0.31\) of the population, would refuse an evacuation order. Therefore, \(\hat{p} = 0.31\).

Understand Sampling Distributions of Proportions

The sampling distribution of \(\hat{p}\) can be approximated by a normal distribution if the conditions for a binomial distribution are met. These conditions are: 1) The sample size is large enough such that \(n\hat{p} \geq 10\) and \(n(1 - \hat{p}) \geq 10\) where \(n\) denotes the sample size, and 2) The sampling is with replacement or the population is at least 20 times larger than the sample. Given these conditions, the standard error for a sample proportion is given by \( \sqrt{\hat{p}(1 - \hat{p})/n}\). With \(n = 5046, \hat{p} = 0.31\), the standard error is approximately \(0.007\).

Assess the Interval Estimate for the Population Proportion

Considering a 95% confidence interval, which involves about 2 standard errors on each side of \(\hat{p}\), we would expect \(\hat{p}\) to fall within \(0.03\) of the population proportion about 95% of the time. Given the computed standard error, an interval of \(2 * 0.007 = 0.014\) is less than \(0.03\). Therefore, it is reasonable to think that the estimate is within \(0.03\) of the actual value of the population proportion.