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 = 84
(round up to the nearest integer.)
b) assume that a recent survey suggests that about 80% of adults have heard of the brand.
n =
(round up to the nearest integer.)

Explanation:

Step1: Find the z - score for 80% confidence level

For an 80% confidence level, the significance level \(\alpha=1 - 0.80 = 0.20\). Then \(\alpha/2=0.10\). Using the standard normal distribution table or a calculator, \(z_{\alpha/2}=z_{0.10}\approx1.28\)

Step2: Use the formula for sample size when estimating a proportion

The formula for sample size \(n\) when estimating a proportion is \(n=\dfrac{z_{\alpha/2}^{2}\cdot p(1 - p)}{E^{2}}\)

Given that \(E = 0.07\) (7 percentage points) and \(p = 0.80\) (from part b)

Substitute the values into the formula:

$$
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$$

Since we round up to the nearest integer.

Answer:

\(n = 54\)