To calculate the margin of error for the proportion of students who would benefit from summer school, you can use the following formula:
\[ \text{Margin of Error (ME)} = z \times \sqrt{\frac{p(1 - p)}{n}} \]
where:
- \( z \) is the z-score (1.96 for a 95% confidence level),
- \( p \) is the sample proportion,
- \( n \) is the sample size.
First, let's calculate the sample proportion \( p \):
\[ p = \frac{18}{75} = 0.24 \]
Now we can plug the values into the margin of error formula:
- Calculate \( 1 - p \):
\[ 1 - p = 1 - 0.24 = 0.76 \]
- Now substitute \( p \), \( 1 - p \), \( n \), and \( z \) into the formula:
\[ \text{ME} = 1.96 \times \sqrt{\frac{0.24 \times 0.76}{75}} \]
- Calculate \( 0.24 \times 0.76 \):
\[ 0.24 \times 0.76 = 0.1824 \]
- Then calculate \( \frac{0.1824}{75} \):
\[ \frac{0.1824}{75} = 0.002432 \]
- Now calculate the square root:
\[ \sqrt{0.002432} \approx 0.04932 \]
- Finally, calculate the margin of error:
\[ \text{ME} = 1.96 \times 0.04932 \approx 0.0967 \]
Thus, the margin of error (rounded) is approximately \( 0.097 \) or \( 9.7% \).
Given your options: 10%, 47%, 24%, 0.096%, the closest choice is 10%.