Question

The brand manager for a brand toothpaste must plan a campaign designed to increase brand recognition. He wants to first determine the percentage of adults Wii have heard of the brand. How many adults must he survey in order to be 90%confident that hiss estimate is within five percentage points off the true population percentage?

Answer 1

We need to find out how many adults must the brand manager survey in order to be 90% confident that his estimate is within five percentage points of the true population percentage.

From the given data we know that our confidence level is 90%. From Standard Normal Table we know that the critical level at 90% confidence level is 1.645. In other words, $Z_{Critical}=1.645$.

We also know that E=5% or E=0.05

Also, since, $\hat{p}$ is not given, we will assume that $\hat{p}$=0.5. This is because, the formula that we use will have $\hat{p}(1-\hat{p})$ in the expression and that will be maximum only when $\hat{p}$=0.5. (For any other value of $\hat{p}$, we will get a value less than 0.25. For example if, $\hat{p}$ is 0.4, then $1-\hat{p}=0.6$ and thus, $\hat{p}(1-\hat{p})=0.24$.).

We will now use the formula

$n=(\frac{Z_{Critical}}{E})^2\hat{p}(1-\hat{p})$

We will now substitute all the data that we have and we will get

$n=(\frac{1.645}{0.05})^2\times0.5(1-0.5)$

$n=(32.9)^2\times0.25$

$n=270.6025$

which can approximated to n=271.

So, the brand manager needs a sample size of 271