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 230 of the 1000 as the favorite subject, and it was also chosen by 370 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. b. Construct a \(95 \%\) confidence interval for the proportion of U.S. adults for whom math was the least favorite subject.

Short answer

a. The 95% confidence interval for the proportion of U.S. adults for whom math was the favorite subject in school is approximately (0.203, 0.257). b. The 95% confidence interval for the proportion of U.S. adults for whom math was the least favorite subject is approximately (0.343, 0.397).

Step-by-step solution

Calculate proportions

First, need to calculate the proportions of U.S adults whose favorite and least favorite subject was math. For favorites, the proportion ( \( p_1 \) ) is 230 out of 1000, or \( p_1 = 230/1000 = 0.23 \). For least favorites, the proportion (\( p_2 \)) is 370 out of 1000, or \( p_2 = 370/1000 = 0.37 \).

Construct the confidence interval for favorite subject

Plug \( p_1 \), the z-value (1.96), and the sample size (1000) into the formula for the confidence interval: \( p_1 \pm 1.96 \sqrt{p_1(1-p_1)/1000} \). After solving, this gives \( 0.23 \pm 1.96 \sqrt{0.23(1-0.23)/1000} \). Calculating the square root and carrying out the operation, find the confidence interval to be approximately (0.203, 0.257).

Construct the confidence interval for least favorite subject

Using \( p_2 \), the z-value (1.96), and the sample size (1000) into the formula for the confidence interval: \( p_2 \pm 1.96 \sqrt{p_2(1-p_2)/1000} \). After solving, this gives \( 0.37 \pm 1.96 \sqrt{0.37(1-0.37)/1000} \). Calculating the square root and carrying out the operation, find the confidence interval to be approximately (0.343, 0.397).