A researcher wishes to estimate the percentage of adults who

A researcher wishes to estimate the percentage of adults who support abolishing the penny What size sample should be obtained if he wishes the estimate to be within 3 percentage points with 95% confidence if he uses a previous estimate of 38%? he does not use any prior estimates?

Solution

a) p = 22% = 0.22

Std error = sqrt[p *(1-p)/n] = sqrt[0.22 * (1-0.22)/n] = sqrt[0.1716/n]

Margin of error = +4% = +0.04

alpha = 1 - 0.95 = 0.05

Critical probability = 1 - alpha/2 = 0.975

Critical value for cumulative probability 0.975 is +1.96

Critical value for 95% confidence is +1.96

Margin of Error = Critical value * Std Error

0.04 = 1.96 * sqrt(0.1716/n)

n = 412.011

Thus n = 412

b) No prior estimates implies p = 0.5

Std error = sqrt[p *(1-p)/n] = sqrt[0.5 * (1-0.5)/n] = sqrt[0.25/n]

Margin of error = +4% = +0.04

alpha = 1 - 0.95 = 0.05

Critical probability = 1 - alpha/2 = 0.975

Critical value for cumulative probability 0.975 is +1.96

Critical value for 95% confidence is +1.96

Margin of Error = Critical value * Std Error

0.04 = 1.96 * sqrt(0.25/n)

n = 600.25

Thus n = 600