Question

A researcher wants to estimate the proportion of students who have attended at least one football, basketball or baseball games during their first semester on campus. She surveys 500 students and 350 reported that they have attended at least one game.

a) What is the standard error of the sample proportion of students who have attended at least one game during their first semester?

b) What is the margin of error for the proportion of students who have attended at least one game.

c) Construct a 90% confidence interval for the proportion of students who have attended at least on game

d) Describe the sampling distribution of the proportion of students who have attended at least on game

Answer 1

a)
sample proportion, pcap = 350/500 = 0.7
sample size, n = 500
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.7 * (1 - 0.7)/500) = 0.0205

b)
Given CI level is 90%, hence α = 1 - 0.9 = 0.1
α/2 = 0.1/2 = 0.05, Zc = Z(α/2) = 1.64

Margin of Error, ME = zc * SE
ME = 1.64 * 0.0205
ME = 0.0336

c)
CI = (pcap - z*SE, pcap + z*SE)
CI = (0.7 - 1.64 * 0.0205 , 0.7 + 1.64 * 0.0205)
CI = (0.6664 , 0.7336)

d)
sampling distribution is N(0.7, 0.0205)