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%
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%**
\[
\text{Margin of Error} = z \times \sqrt{\frac{p(1-p)}{n}}
\]
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%**