Question

At the movies: A movie theater is considering a showing of Puppet Master for a 80's thowback night. In order to ensure the success of the evening, they've asked a random sample of 53 patrons whether they would come to the showing or not. Of the 53 patrons, 30 said that they would come to see the film. Construct a 95% confidence interval to determine the true proportion of all patrons who would be interested in attending the showing.

a) What is the point estimate for the true proportion of interested patrons? (please input a proportion accurate to four decimal places)

b) Complete the interpretation of the confidence interval. Please provide the bounds for the confidence interval in decimal form, accurate to four decimal places, and list the lower bound first.
"We are ... % confident that the true proportion of patrons interested in attending the showing of Puppet Master is between ... and ... "

c) The theater is only willing to show Puppet Master if they are confident that at least 40% of their patrons would be interested in seeing the film. Should they show it?

Answer 1

a)

Point estimate = sample proportion, = 0.5660

b)

sample size, n = 53
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.566 * (1 - 0.566)/53) = 0.0681

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

Margin of Error, ME = zc * SE
ME = 1.96 * 0.0681
ME = 0.1335

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.566 - 1.96 * 0.0681 , 0.566 + 1.96 * 0.0681)
CI = (0.4325 , 0.6995)

We are 95 % confident that the true proportion of patrons interested in attending the showing of Puppet Master is between 0.4325 and 0.6995

c)

yes, because confidenc einterval does not contain 40%