Question

A previous random sample of 4000 U.S. citizens yielded 2250 who are in favor of gun control legislation. How many citizens would need to be sampled for a 95% confidence interval to estimate the true proportion within 2%

Answer 1

Solution :

Given that,

n = 4000

x = 2250

$\hat p$ = 2250 / 4000 = 0.5625

1 - $\hat p$ = 1 - 0.5625 = 0.4375

margin of error = E = 2% = 0.02

At 95% confidence level the z is ,

$\alpha$ = 1 - 95% = 1 - 0.95 = 0.05

$\alpha$ / 2 = 0.05 / 2 = 0.025

Z_$\alpha$/2 = Z_0.025 = 1.96

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

= (1.96 / 0.02)² * 0.5625 * 0.4375

= 2363.48

= 2364

2364 citizens would need to be sampled .