To determine the margin of error for the given poll at a 99% confidence level, we will use the provided formula for margin of error:
[tex]\[
E = z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}
\][/tex]
Given values:
- \(\hat{p} = 0.38\) (proportion of Californians who believe they are doing all they can to conserve water)
- \(n = 80\) (sample size)
- \(z^* = 2.58\) (z-score for 99% confidence level)
Step-by-Step Solution:
1. Calculate the standard error:
[tex]\[
\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.38 \times (1 - 0.38)}{80}}
\][/tex]
2. Simplify inside the square root:
[tex]\[
0.38 \times (1 - 0.38) = 0.38 \times 0.62 = 0.2356
\][/tex]
[tex]\[
\frac{0.2356}{80} = 0.002945
\][/tex]
3. Find the square root:
[tex]\[
\sqrt{0.002945} \approx 0.05426
\][/tex]
4. Multiply by the z-score to find the margin of error:
[tex]\[
E = 2.58 \times 0.05426 \approx 0.14001
\][/tex]
5. Convert the margin of error to a percentage:
[tex]\[
0.14001 \times 100 \approx 14.001
\][/tex]
6. Round to the nearest whole percent:
[tex]\[
14.001 \approx 14
\][/tex]
Therefore, to the nearest whole percent, the margin of error for the poll is [tex]\(14\%\)[/tex].
