what size of sample is needed? When finding the z-value, round it to four decimal places.

Question

A political candidate has asked you to conduct a poll to determine what percentage of people support her. If the candidate only wants a 10% margin of error at a 99% confidence level, what size of sample is needed? When finding the z-value, round it to four decimal places.

Answer 1

To determine the sample size needed for a poll with a 10% margin of error at a 99% confidence level, we use the formula:

\[ n = \frac{{Z^2 \cdot p \cdot (1 - p)}}{{E^2}} \]

Where:

- \( n \) = sample size needed

- \( Z \) = Z-value (corresponding to the desired confidence level)

- \( p \) = estimated proportion of the population supporting the candidate (usually taken as 0.5 if unknown)

- \( E \) = margin of error (proportion)

Given:

- Margin of error (\( E \)) = 0.10

- Confidence level = 99%

First, we find the Z-value corresponding to a 99% confidence level. This can be found using a standard normal distribution table or a calculator. Let's use the value of 2.576, rounded to four decimal places.

Now, we plug in the values into the formula:

\[ n = \frac{{2.576^2 \cdot 0.5 \cdot (1 - 0.5)}}{{0.10^2}} \]

\[ n = \frac{{6.635776 \cdot 0.25}}{{0.01}} \]

\[ n = \frac{{1.658944}}{{0.01}} \]

\[ n ≈ 165.89 \]

Rounded up to the nearest whole number, the sample size needed is 166.