Question

What percentage of college students owns a cellular phone? Let p be the proportion of college students that owns a cellular phone. a) If no preliminary study is made to estimate p, how large a sample is needed to be 95% sure that a point estimate will be within a distance of 0.08 from p? b) A preliminary study shows that approximately 38% of college students own cellular phones. How large a sample is needed to be 95% sure that a point estimate will be within a distance of 0.08 from p?

Answer 1

Solution :

Given that,

margin of error = E = 0.08

Z_$\alpha$/2 = 1.96

(a)

$\hat p$ = 0.5

1 - $\hat p$ = 0.5

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

= (1.96 / 0.08)² * 0.5 * 0.5

= 151

sample size = n = 151

(b)

$\hat p$ = 0.38

1 - $\hat p$ = 0.62

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

= (1.96 / 0.08)² * 0.38 * 0.62

= 142

sample size = n = 142