1. Provide a complete response for the following:

A large university provides housing for 10% of its graduate students to live on campus. The university's housing office thinks that the percentage of graduate students looking for housing on campus may be more than 10%. The housing office decides to survey a random sample of graduate students, and 62 of the 481 respondents say that they are looking for housing on campus. 

a. On the basis of the survey data, would you recommend that the housing office consider increasing the amount of housing on campus available to graduate students? Give appropriate evidence to support your recommendation. 

b. In addition to the 481 graduate students who responded to the survey, there were 19 who did not respond. If these 19 had responded, is it possible that your recommendation would have changed? Explain. 

2. Land's Beginning is a company that sells its merchandise through the mail. It is considering buying a list of addresses from a magazine. The magazine claims that at least 25% of its subscribers have high incomes (they define this to be household income in excess of $100,000). Land's Beginning would like to estimate the proportion of high-income people on the list. Checking income is very difficult and expensive but another company offers this service. Land's Beginning will pay to find incomes for an SRS of people on the magazine's list. They would like the margin of error of the 95% confidence interval for the proportion to be 0.05 or less. Use the guessed value p = 0.25 to find the required sample size. 

3. For a single proportion the margin of error of a confidence interval is largest for any given sample size n and confidence level C when p-hat = 0.5. This led us to use p-hat = 0.5 for planning purposes. Use these conservative values in the following calculations, and assume that the sample sizes of n. Calculate the margins of error of the 99% confidence intervals for the following choices of n: 10, 30, 50, 100, 200, and 500. Present the results in a table. Summarize your conclusions.

4. What assumptions must we make in determining the necessary sample size for a study of categorical data (using proportions)?