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 five percentage points of the true population percentago? Complete parts (a) through (c) below a) Assume that nothing is known about the percentage of adults who have heard of the brand Round up to the most integer) b) Asume that a recent survey suggests that about 77% of adults have heard of the brand Round up to the nearest ego)

Answer 1

Solution:
Given that,
Margin of error= E=5% = 0.05,
(1–α)%=80%
(1–α)=0.80
α=0.20
α/2=0.10
Zα/2=1.28 ....... from standard normal table.
a)Since prior information is not known for the proportion so we can take it as 0.5
p^=0.5
Sample size = n=(Zα/2)^2 ×{[p^ ×(1-p^)]/(E^2)}
=(1.28)^2 ×{[0.5×(1-0.5)]/(0.05^2)}
=163.84
= 164 .... nearest integer number.

n =164
b) Here , prior information is known i.e
p^ =77%=0.77
Sample size = n=(Zα/2)^2 ×{[p^ ×(1-p^)]/(E^2)}
=(1.28)^2 ×{[0.77×(1-0.77)]/(0.05^2)}
=116.0642
= 116 .... nearest integer number.

n=116