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.

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 90​% confident that his estimate is within eight 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=enter your response here

​(Round up to the nearest​ integer.)

Answer 1

To determine the sample size needed for the brand manager to be 90% confident that his estimate is within eight percentage points of the true population percentage, we can use the following formula:

n = (Z^2 * p * (1-p)) / E^2

where:

n = sample size

Z = Z-value corresponding to the desired level of confidence (in this case, 90% confidence)

p = estimated proportion (since nothing is known about the percentage, we can use 0.5 as a conservative estimate)

E = margin of error (in this case, 8 percentage points)

Substituting the values into the formula:

n = (Z^2 * p * (1-p)) / E^2

n = (Z^2 * 0.5 * (1-0.5)) / (0.08)^2

To determine the Z-value for a 90% confidence level, we can refer to the standard normal distribution table or use a calculator. The Z-value for a 90% confidence level is approximately 1.645.

Substituting the values into the formula:

n = (1.645^2 * 0.5 * (1-0.5)) / (0.08)^2

n = (2.705 * 0.5 * 0.5) / 0.0064

n = 0.67625 / 0.0064

n ≈ 105.78

Since the sample size needs to be a whole number, we round up to the nearest integer.

Therefore, the brand manager must survey at least 106 adults in order to be 90% confident that his estimate is within eight percentage points of the true population percentage.