Question

Most polls (performed by market research companies) are given with a margin of error of about 3% (if you read poll results in a national newspaper, you will often see something like these results are given plus or minus 3 percentage points"). 1. If you wish to poll Vancouver residents to determine their level of support for a plan to increase green space in the city, and you wish to use a 80% confidence level, what is the minimum number of residents you should poll to obtain results that have a margin of error of 3%? 因回 2. Suppose you wanted to obtain the same margin of error, but at a 90% confidence level, how many more residents would you need to sample (how many more than you gave in your answer to question 1)?

Answer 1

Given : Margin of errorE=3%=0.03

Since , the estimate of the proportion is unknown.

Therefore , assume that $\hat{p}=0.5$

a) The critical value is , $Z_{\alpha/2}=Z_{0.20/2}=1.28$ ; From Z-table

Therefore , the required sample size is ,

$n=\frac{Z_{\alpha/2}^2*\hat{p}(1-\hat{p})}{E^2}=\frac{1.28^2*0.5(1-0.5)}{0.03^2}\approx 455$

b) The critical value is , $Z_{\alpha/2}=Z_{0.10/2}=1.64$ ; From Z-table

Therefore , the required sample size is ,

$n=\frac{Z_{\alpha/2}^2*\hat{p}(1-\hat{p})}{E^2}=\frac{1.64^2*0.5(1-0.5)}{0.03^2}\approx 747$