a. Suppose Alejandro studies a random sample of
data on a categorical variable and calculates a 95% confidence
interval for the population proportion to be (.846, .974).
Determine what the sample proportion must have been, and explain
why.

b. Suppose Brad and Carly plan to collect
separate random samples, with Brad using a sample size of 1500 and
Carly using a sample size of 500. If Brad plans to construct a 90%
confidence interval for the population proportion and Carly plans
to construct a 99% 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.)

(.156, .244)
                
(.196, .244)
                
(.146, .254)