a. Suppose a student studies a random sample of data on a categorical variable and calculates a 95% confidence interval for the population proportion to be (0.546, 0.674). Determine what the sample proportion must have been, and explain why.

b. Suppose two students plan to collect separate random samples, with the first student using a sample size of 500 and the second student using a sample size of 1500. If the first student plans to construct a 99% confidence interval for the population proportion and the second student plans to construct a 90% confidence interval, who is more likely to obtain an interval that succeeds in capturing the population proportion? Explain.

c. Suppose three students conduct a group project to estimate the proportion of students at their university who are from a different state. They take a random sample of students and find that 20% of their sample of students are from a different state. They then determine the following confidence intervals for the population proportion who are from a different state. One of these is a 90% confidence interval, one is a 99% confidence interval, and one is incorrect. Identify which is which. (In other words, write "90%" below the appropriate interval, "99%" below another, and "incorrect" below the third.)

• (.116, .284), (176,.344), (.146,.254)