Question

A researcher wishes to​ estimate, with 99 ​% ​confidence, the population proportion of adults who think the president of their country can control the price of gasoline. Her estimate must be accurate within 4 ​% of the true proportion. ​a) No preliminary estimate is available. Find the minimum sample size needed. ​b) Find the minimum sample size​ needed, using a prior study that found that 28 ​% of the respondents said they think their president can control the price of gasoline. ​c) Compare the results from parts​ (a) and​ (b). ​(a) What is the minimum sample size needed assuming that no prior information is​ available? nequals nothing ​(Round up to the nearest whole number as​ needed.) ​(b) What is the minimum sample size needed using a prior study that found that 28 ​% of the respondents said they think their president can control the price of​ gasoline? nequals nothing ​(Round up to the nearest whole number as​ needed.) ​(c) How do the results from ​(a) and ​(b)​ compare? A. Having an estimate of the population proportion has no effect on the minimum sample size needed. B. Having an estimate of the population proportion raises the minimum sample size needed. C. Having an estimate of the population proportion reduces the minimum sample size needed.

Answer 1

a) At 99% confidence level, the critical value is z_0.005 = 2.575

Margin of error = 0.04

or, z_0.005 * sqrt(p(1 - p)/n) = 0.04

or, 2.575 * sqrt(0.5(1 - 0.5)/n) = 0.04

or, n = (2.575 * sqrt(0.5 * 0.5)/0.04)^2

or, n = 1037

b) Margin of error = 0.04

or, z_0.005 * sqrt(p(1 - p)/n) = 0.04

or, 2.575 * sqrt(0.28(1 - 0.28)/n) = 0.04

or, n = (2.575 * sqrt(0.28 * 0.72)/0.04)^2

or, n = 836

c) Option - C) Having an estimate of the population proportion reduces the minimum sample size needed.