Question

These next few questions ask about confidence intervals.

a) Bruce Wayne 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 Clark and Lois plan to collect separate random samples, with Clark using a sample size of 500 and Lois using a sample size of 1500. Brad plans to construct a 99% confidence interval for the population proportion and Lois plans to construct a 90% confidence interval. Who is more likely to obtain an interval that succeeds in capturing the population proportion? Explain.

c) Three students conduct a group project to estimate the proportion of students at their college that 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 try to 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. Label one “90%”, label one “99%”, and label one “incorrect.”

i. (0.116, 0.284)

ii. (0.176, 0.344)

iii. (0.146, 0.254)

Answer 1

a)

confidence interval is                  
lower limit =    0.546       upper limit=   0.674  
sample proportion = (lower limit+upper limit)/2= (   0.546   +   0.674   ) / 2 =   0.61

b)

Brad  is more likely to obtain an interval that succeeds in capturing the population proportion

because his CI would be wider as sample size is small causing margin of eror to be higher, also, confidence level is higher, which also causes wider CI

c)

i. (0.116, 0.284) = 99%

ii. (0.176, 0.344) =incorrect

iii. (0.146, 0.254) = 90%