Let's break down the problem step-by-step to find the approximate margin of error for the given polling question:
1. Identify the given values:
- The sample size \( n = 400 \)
- The number of favorable responses \( x = 288 \)
- The desired confidence interval of \( 95\% \) corresponds to a \( z^* \)-score of \( 1.96 \).
2. Calculate the sample proportion (\(\hat{p}\)):
[tex]\[
\hat{p} = \frac{x}{n} = \frac{288}{400} = 0.72
\][/tex]
3. Use the formula for the margin of error (\(E\)):
[tex]\[
E = z^* \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
\][/tex]
Plugging in the values, we get:
[tex]\[
E = 1.96 \sqrt{\frac{0.72 (1 - 0.72)}{400}}
\][/tex]
4. Calculate the quantity inside the square root:
[tex]\[
0.72 (1 - 0.72) = 0.72 \times 0.28 = 0.2016
\][/tex]
[tex]\[
\frac{0.2016}{400} = 0.000504
\][/tex]
5. Take the square root:
[tex]\[
\sqrt{0.000504} \approx 0.0224
\][/tex]
6. Multiply by the \( z^* \)-score:
[tex]\[
E = 1.96 \times 0.0224 \approx 0.0440
\][/tex]
7. Convert the margin of error to a percentage:
[tex]\[
E \times 100 = 0.0440 \times 100 = 4.40\%
\][/tex]
Hence, the approximate margin of error for this polling question is approximately \( 4.4\% \). Therefore, the closest answer among the given options is:
[tex]\[
4\%
\][/tex]
So, the answer is [tex]\( 4\% \)[/tex].
