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 six 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 85​% of adults have heard of the brand.

n=

​(Round up to the nearest​ integer.)

​c) Given that the required sample size is relatively​ small, could he simply survey the adults at the nearest​ college?

A.

​Yes, a sample of students at the nearest college is a simple random​ sample, so the results should be representative of the population of adults.

B.

​No, a sample of students at the nearest college is a stratified​ sample, not a simple random​ sample, so it is very possible that the results would not be representative of the population of adults.

C.

​No, a sample of students at the nearest college is a cluster​ sample, not a simple random​ sample, so it is very possible that the results would not be representative of the population of adults.

D.

​No, a sample of students at the nearest college is a convenience​ sample, not a simple random​ sample, so it is very possible that the results would not be representative of the population of adults.

Answer 1

Solution :

The sample size required to estimate the population percentage with 80% confidence is given as follows :

Where, Z_(0.20/2) is critical z-value at 80% confidence level, E is margin of error and p̂ is point estimate of the percentage.

a) Since, manager wants to estimate the percentage within six percentage points of the true population​ percentage therefore,

Margin of error (E) = 6%

When nothing is known about the percentage we assume that, p̂ = 50%

Using Z-table we get, Z_(0.20/2) = 1.2816

Hence, the required sample size is,

b) Since, manager wants to estimate the percentage within six percentage points of the true population​ percentage therefore,

Margin of error (E) = 6%

The point estimate of percentage is, p̂ = 85%

Using Z-table we get, Z_(0.20/2) = 1.2816

Hence, the required sample size is,

c) No, a sample of students at the nearest college is a convenience​ sample, not a simple random​ sample, so it is very possible that the results would not be representative of the population of adults.

Option (D) is correct.

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