Question

The brand manager for a brand of toothpaste must plan a campaign designed to increase brand recognition. He wants to first date the percentage of adults who have heard of the brand Howitany adults must be survey orto e o confident that is estimate is within percentage points of the tre population percentage? Complete parts (a) through e below. a) Asume that nothing is known about the percentage of adults who have heard of the brand, (Round up to the more integer b) Assume that a recent survey is that about % of als have heard of the brand Round up to the nearest Integer

Answer 1

Solution,

Given that,

a) $\hat p$ =  1 - $\hat p$ = 0.5  

margin of error = E = 0.08

At 80% confidence level

$\alpha$ = 1 - 80%

$\alpha$ = 1 - 0.80 =0.20

$\alpha$/2 = 0.10

Z_$\alpha$/2 = Z_0.10 = 1.282

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

= (1.282 / 0.08)² * 0.5 * 0.5

= 64.20

sample size = n = 65

b) $\hat p$ = 0.86

1 - $\hat p$ = 1 - 0.86 = 0.14

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

= (1.282 / 0.08)² * 0.86 * 0.14

= 30.91

sample size = n = 31