Question

1. In 2010, TIME and the Pew Research Center collaborated on a survey of 2622 U.S. adults asking them, “Is marriage becoming obsolete?” The number of people who responded with a “yes” was 1018. We want to use these data to estimate the true proportion of U.S. adults who believe marriage is becoming obsolete.
a. Calculate a 95% confidence interval for this problem.
b. Interpret the 95% confidence interval.
c. If the researcher wanted the margin of error to be no more than 0.01, what would the necessary sample size be for a 95% confidence interval with the same sample proportion?

Answer 1

Solution :

Given that,

n = 2622

x = 1018

a) Point estimate = sample proportion = $\hat p$ = x / n = 1018 / 2622 = 0.388

1 - $\hat p$ = 1 - 0.388 = 0.612

At 95% confidence level

$\alpha$ = 1 - 95%

$\alpha$ =1 - 0.95 =0.05

$\alpha$/2 = 0.025

Z$\alpha$/2 = Z0.025 = 1.960

Margin of error = E = Z$\alpha$ / 2 * (($\hat p$ * (1 - $\hat p$ ))$\sqrt$ / n)

= 1.96 ($\sqrt$((0.388 * 0.612) / 2622)

= 0.019

A 95% confidence interval for population proportion p is ,

$\hat p$ ± E   

= 0.388 ± 0.019

= ( 0.369, 0.407 )

b) We are 95% confident that the true proportion of U.S. adults who believe marriage is becoming obsolete between 0.369 and 0.407.

c) margin of error = E = 0.01

sample size = n = (Z$\alpha$ / 2 / E )2 * $\hat p$ * (1 - $\hat p$ )

= (1.96 / 0.01)2 * 0.388 * 0.612

= 9122.10

sample size = n = 9123