Question

The brand manager for a brand of toothpaste must plan a campaign designed to increase brand recognition. He wants to first determine the percentage of adults who have heard of the brand. How many adults must he survey in order to be 80​% confident that his estimate is within seven percentage points of the true population​ percentage? Complete parts​ (a) through​ (c) below

​a) Assume that nothing is known about the percentage of adults who have heard of the brand.

n=

(Round up to the nearest​ integer.)

​b) Assume that a recent survey suggests that about 86​% of adults have heard of the brand.

n=

​(Round up to the nearest​ integer.)

Answer 1

Solution :

Given that,

margin of error = E = 7% = 0.07

At 80% confidence level the z is ,

= 1 - 80% = 1 - 0.80 = 0.20

/ 2 = 0.20 / 2 = 0.10

Z_/2 = Z_0.10 = 1.28

(a)

= 0.5

1 - = 1 - 0.5 = 0.5  

sample size = n = (Z _{/ 2} / E )² * * (1 - )

= (1.28 / 0.07)² * 0.5 * 0.5

= 83.59 = 84

sample size = 84

(b)

= 86% = 0.86

1 - = 1 - 0.86 = 0.14

sample size = n = (Z _{/ 2} / E )² * * (1 - )

= (1.28 / 0.07)² * 0.86 * 0.14

= 40.25 = 40

sample size = 40