Answer :
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].
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].
