Question

A politician wants to know his chances of winning in the next election. He has tasked a researcher to interview a random sample of his contituents about their perceptions of his leadership. Previous polls have revealed that approximately 62.5% are in favor of his leadership. The politician has adopted the 90% confidence level and of the population value by at most 0.02. How many respondents must be interviewed?

Answer 1

$p\:=62.5\%=\:0.625$

$Margin\:of\:error,E=0.02$

$The\:critical\:value\:for\:the\:significance\:level\:\alpha \:=\:0.1\:is\:z_c\:=\:1.645,$

$which\:is\:obtained\:by\:looking\:into\:a\:standard\:normal\:probability\:table.$

$Sample size, n​\:\:\ge ​\:\:p\left(1−p\right)\left(\:\frac{Zc}{E}\:​\:\right)^2$

$=0.625\left(1−0.625\right)\left(\:\frac{1.645}{0.02}\:​\:\right)^2\:$

$=1585.28\:\approx 1586$

$Required\:sample\:size,\:n=1586$