Question

In a survey of 1000 randomly selected adults in the United States, participants were asked what their most favorite and what their least favorite subject was when they were in school (Associated Press, August 17, 2005). In what might seem like a contradiction, math was chosen more often than any other subject in both categories! Math was chosen by 226 of the 1000 as the favorite subject, and it was also chosen by 380 of the 1000 as the least favorite subject.

(a) Construct a 95% confidence interval for the proportion of U.S. adults for whom math was the favorite subject in school. (Give the answers to four decimal places.)
(  ,  )

(b) Construct a 95% confidence interval for the proportion of U.S. adults for whom math was the least favorite subject. (Give the answers to four decimal places.)

Answer 1

a)
sample proportion, pcap = 226/1000 = 0.226
sample size, n = 1000
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.226 * (1 - 0.226)/1000) = 0.0132

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, Zc = Z(α/2) = 1.96

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.226 - 1.96 * 0.0132 , 0.226 + 1.96 * 0.0132)
CI = (0.2001 , 0.2519)

b)
sample proportion, pcap = 380/1000 = 0.38
sample size, n = 1000
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.38 * (1 - 0.38)/1000) = 0.0153

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, Zc = Z(α/2) = 1.96

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.38 - 1.96 * 0.0153 , 0.38 + 1.96 * 0.0153)
CI = (0.3500 , 0.4100)