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.)

Explanation:

Step1: Find the z - score

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: Determine the value of \(p\) and \(q\)

Since nothing is known about the percentage of adults who have heard of the brand, we use \(p = 0.5\) and \(q=1 - p = 0.5\) (this gives the maximum value of \(pq\)).

Step3: Calculate the sample size \(n\)

The formula for the sample size \(n\) when estimating a proportion is \(n=\frac{z_{\alpha/2}^{2}pq}{E^{2}}\), where \(E = 0.07\) (7 percentage points).
Substitute the values: \(z_{\alpha/2}=1.28\), \(p = 0.5\), \(q = 0.5\), and \(E=0.07\) into the formula.

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Answer:

\(n = 84\)