Question

A Sampling Distribution for Performers in the Rock and Roll Hall of Fame Exercise 3.38 tells us that 206 of the 303 inductees to the Rock and Roll Hall of Fame have been performers. The data are given in RockandRoll. Using all inductees as your population: (a) Use StatKey or other technology to take many random samples of size \(n=10\) and compute the sample proportion that are performers. What is the standard error of the sample proportions? What is the value of the sample proportion farthest from the population proportion of \(p=0.68 ?\) How far away is it? (b) Repeat part (a) using samples of size \(n=20\). (c) Repeat part (a) using samples of size \(n=50\). (d) Use your answers to parts (a), (b), and (c) to comment on the effect of increasing the sample size on the accuracy of using a sample proportion to estimate the population proportion.

Short answer

Without the specific results from a statistical calculator or software, the short answer can't be definitively provided. However, it is expected that as the sample size increases, the standard error will decrease. This would mean that the accuracy of the sample proportion as an estimate of the population proportion improves (the sample proportions should get closer to \(p=0.68\)). Therefore, the size of the sample contributes significantly to the accuracy of the estimates.

Step-by-step solution

Sampling Distribution for \(n=10\)

Take many random samples from the population of size \(n=10\). For each sample, calculate the sample proportion that are performers and calculate the standard error. The standard error can be calculated using the formula \(SE = \sqrt{\frac{p(1 - p)}{n}}\), where \(p\) is the population proportion and \(n\) is the sample size. Identify the value of the sample proportion farthest from the population proportion of \(p=0.68\).

Sampling Distribution for \(n=20\)

Repeat the same process as in Step 1, but this time with a sample size of \(n=20\). Calculate the sample proportion and standard error for each sample and identify the sample proportion that deviates the most from \(p=0.68\).

Sampling Distribution for \(n=50\)

Follow the same procedure as in the previous steps, but with a larger sample size of \(n=50\). Again, calculate the sample proportion and standard error for each sample and find the sample proportion that differs the most from \(p=0.68\).

Analyzing the Impact of Increasing Sample size on Accuracy

Using the results obtained in the previous steps, analyze how the sample size affects the accuracy of estimating the population proportion. Pay attention to how the standard error decreases as the sample size increases - a smaller standard error indicates a more accurate estimate of the population proportion. Also, note if the sample proportions that are the farthest from \(p=0.68\) get closer to the population proportion as the sample size increases. This would also indicate an increase in accuracy.