Question

In a survey of 75 students, 18 said they would benefit from summer school.

Given a 95% level of confidence with a z-score of 1.96, what is the margin of error?

(1 point)
Responses

24%
24%

0.096%
0.096%

47%
47%

10%
10%

Answers

GPT-4o mini
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%**