Question

As a manager for an advertising company, you must plan a campaign designed to increase Twitter usage. A recent survey suggests that 85% of adults know what Twitter is. How many adults should you survey in order to be 90% confident that your estimate is within 5% of the true population proportion?

Answer 1

Given:
Sample proportion = 85% = 0.85.
Confidence level (CL) = 90%.
Confidence interval (CI) $\hat{p} \pm z^{*} \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} } \\ where \\ n=sample\, size \\ z^{*}=1.645 \, at \, 90\% \, CL$

We want the confidence interval to be 5% or 0.05. Therefore
$0.85 - 1.645 \sqrt{ \frac{0.85(1-0.85)}{n} }=0.05 \\ 0.85 - \frac{0.5874}{ \sqrt{n} } =0.05 \\ \frac{0.5874}{ \sqrt{n} }=0.8 \\ \sqrt{n} = .7342 \\ n=0.54$

Because this number is less than 1, a default sample size of 30 (for the normal distribution) is recommended.

Answer: 30