Question 8
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A political candidate has asked you to conduct a poll to determine what percentage of people support her.
If the candidate only wants a $1 \%$ margin of error at a $90 \%$ confidence level, what size of sample is needed?
Give your answer in whole people.
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Step 1 ：We are given that the Z-score for a 90% confidence level is 1.645, the estimated proportion of the population is 0.5, and the margin of error is 0.01.

Step 2 ：We use these values in the formula for calculating the sample size: \(n = \frac{{Z^2 \cdot p \cdot (1-p)}}{{E^2}}\).

Step 3 ：Substituting the given values into the formula, we get \(n = \frac{{(1.645)^2 \cdot 0.5 \cdot (1-0.5)}}{{(0.01)^2}}\).

Step 4 ：Solving this equation gives us \(n = 6765.50625\).

Step 5 ：Since we can't have a fraction of a person, we round up to the nearest whole number to get \(n = 6766\).

Step 6 ：Final Answer: The required sample size is \(\boxed{6766}\).