To determine the margin of error for the polling question, we will follow several steps involving statistics and probability calculations.
### Step-by-Step Solution:
1. Sample Size and Positive Responses:
- Sample size ([tex]\( n \)[/tex]) = 400
- Number of positive responses = 288
2. Calculate the Sample Proportion ([tex]\( \hat{p} \)[/tex]):
The sample proportion is the number of positive responses divided by the total sample size:
[tex]\[
\hat{p} = \frac{\text{positive responses}}{\text{sample size}} = \frac{288}{400} = 0.72
\][/tex]
3. Determine the Critical Value (z[tex]\(^\)[/tex]):
For a confidence level of 95%, the critical value ([tex]\( z^ \)[/tex]) is 1.96.
4. Calculate the Margin of Error (E):
We use the margin of error formula:
[tex]\[
E = z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}
\][/tex]
Substituting the given values:
[tex]\[
E = 1.96 \sqrt{\frac{0.72 \times (1 - 0.72)}{400}}
\][/tex]
5. Simplify Inside the Square Root:
[tex]\[
1 - 0.72 = 0.28
\][/tex]
[tex]\[
\frac{0.72 \times 0.28}{400} = \frac{0.2016}{400} = 0.000504
\][/tex]
6. Calculate the Square Root:
[tex]\[
\sqrt{0.000504} \approx 0.02243
\][/tex]
7. Multiply by the Critical Value:
[tex]\[
E = 1.96 \times 0.02243 \approx 0.044
\][/tex]
8. Convert the Margin of Error to Percentage:
[tex]\[
0.044 \times 100 = 4.4\%
\][/tex]
### Conclusion:
The approximate margin of error for this polling question is 4.4%.
Out of the options given:
- 4%
- 5%
- 7%
- 8%
The closest answer to our calculated margin of error is 4%. So, the correct choice is 4% (although it would be more precise to say 4.4%).
