Your market research firm has been hired to conduct a survey. The goal is to produce estimates of both average annual income and proportion with a post-secondary credential (PSC) in a target population. A simple random sample from the population will be used. Informed pre-study guesses are that (i) 34% of this population has a PSC, and (ii) the standard deviation of annual income in this population is around $ 21000. Your client desires a 95% confidence interval with width no more than 0.05 for the proportion with a post-secondary credential, and a 95% confidence interval with width no more than $5,000 for the average income in the population.

(a)Using the pre-study guesses, and being mindful of not proposing a more costly study than necessary, what sample size do you recommend? Provide your answer as an integer.

(b)Regardless of what sample size is ultimately chosen, what is the worst-case scenario for the 95% confidence interval for the proportion with a PSC, in terms of how much wider than anticipated it could turn out to be? Give your answer to two decimal places as a percentage of the width of the confidence interval based on the pre-study estimate.