To calculate the margin of error for a proportion, we can use the formula:
\[ \text{Margin of Error} = z \times \sqrt{\frac{p(1-p)}{n}} \]
Where:
- \( z \) is the z-score (1.96 for 95% confidence),
- \( p \) is the proportion of respondents who said they would benefit from summer school,
- \( n \) is the total number of respondents.
First, we need to calculate \( p \): \[ p = \frac{\text{number of favorable responses}}{\text{total number of responses}} = \frac{18}{75} = 0.24 \]
Now, we can substitute \( p \) and \( n \) into the margin of error formula:
\[ \text{Margin of Error} = 1.96 \times \sqrt{\frac{0.24(1-0.24)}{75}} \]
Calculating the inside of the square root:
\[ 1-0.24 = 0.76 \]
\[ 0.24 \times 0.76 = 0.1824 \]
Now, divide by \( n = 75 \):
\[ \frac{0.1824}{75} = 0.002432 \]
Next, take the square root:
\[ \sqrt{0.002432} \approx 0.0493 \]
Now we can calculate the margin of error:
\[ \text{Margin of Error} = 1.96 \times 0.0493 \approx 0.0967 \]
Converting this to a percentage:
\[ 0.0967 \approx 9.67% \]
Rounded to the nearest whole number, the margin of error is approximately 10%.
So, the correct response is:
10%