To determine the margin of error for Sameer's survey, we need to use the formula for margin of error in proportion problems. Here is a step-by-step approach:
1. Identify the given information:
- Number of workers polled, [tex]\( n = 120 \)[/tex]
- Number of satisfied workers, [tex]\( x = 90 \)[/tex]
- Desired confidence level, [tex]\( 99\% \)[/tex]
- Corresponding critical value, [tex]\( z^* = 2.58 \)[/tex]
2. Calculate the sample proportion ([tex]\(\hat{p}\)[/tex]):
[tex]\[
\hat{p} = \frac{x}{n} = \frac{90}{120} = 0.75
\][/tex]
3. Use the formula for the margin of error [tex]\(E\)[/tex]:
[tex]\[
E = z^* \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\][/tex]
4. Substitute the known values into the formula:
[tex]\[
E = 2.58 \sqrt{\frac{0.75(1 - 0.75)}{120}}
\][/tex]
5. Simplify inside the square root first:
[tex]\[
\hat{p}(1 - \hat{p}) = 0.75 \times 0.25 = 0.1875
\][/tex]
[tex]\[
\frac{\hat{p}(1 - \hat{p})}{n} = \frac{0.1875}{120} \approx 0.0015625
\][/tex]
6. Take the square root of the above result:
[tex]\[
\sqrt{0.0015625} \approx 0.03954
\][/tex]
7. Multiply this result by the critical value [tex]\(z^*\)[/tex]:
[tex]\[
E = 2.58 \times 0.03954 \approx 0.10198
\][/tex]
8. Convert the margin of error into a percentage:
[tex]\[
E \times 100 = 0.10198 \times 100 \approx 10.198\%
\][/tex]
Thus, the margin of error for Sameer's survey, when rounded to the nearest whole number, is approximately [tex]\(10\%\)[/tex]. Therefore, the correct answer is:
[tex]\[
\boxed{10\%}
\][/tex]
