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 0.096% 0.096% 47% 47% 24% 24% 10%
Answers
GPT-4o mini
To calculate the margin of error (ME) for a proportion at a 95% confidence level, you can use the following formula:
\[
ME = z \times \sqrt{\frac{p(1 - p)}{n}}
\]
Where:
- \( z \) is the z-score (1.96 for a 95% confidence level),
- \( p \) is the sample proportion (the number of students who would benefit from summer school divided by the total number of students),
- \( n \) is the total number of students surveyed.
1. Calculate the sample proportion \( p \):
\[
p = \frac{18}{75} = 0.24
\]
2. Calculate the margin of error:
\[
ME = 1.96 \times \sqrt{\frac{0.24(1 - 0.24)}{75}}
\]
First, calculate \( 1 - p \):
\[
1 - p = 1 - 0.24 = 0.76
\]
Next, compute \( p(1 - p) \):
\[
p(1 - p) = 0.24 \times 0.76 = 0.1824
\]
Then, calculate \( \frac{p(1 - p)}{n} \):
\[
\frac{0.1824}{75} = 0.002432
\]
Now take the square root:
\[
\sqrt{0.002432} \approx 0.04932
\]
Finally, compute the margin of error:
\[
ME = 1.96 \times 0.04932 \approx 0.0966
\]
So, the margin of error is approximately 0.0966. In percentage terms, this is:
\[
0.0966 \times 100 \approx 9.66\%
\]
Note that the options provided in your question do not match this rather calculated number; however, the margin of error calculated as a percentage is expressed correctly. If we round to the nearest hundredth, the margin of error is approximately **9.7%**. If you need to choose from the provided options, it appears they may have a miscalculation or misrepresentation in percentage should reflect "10%."
\[
ME = z \times \sqrt{\frac{p(1 - p)}{n}}
\]
Where:
- \( z \) is the z-score (1.96 for a 95% confidence level),
- \( p \) is the sample proportion (the number of students who would benefit from summer school divided by the total number of students),
- \( n \) is the total number of students surveyed.
1. Calculate the sample proportion \( p \):
\[
p = \frac{18}{75} = 0.24
\]
2. Calculate the margin of error:
\[
ME = 1.96 \times \sqrt{\frac{0.24(1 - 0.24)}{75}}
\]
First, calculate \( 1 - p \):
\[
1 - p = 1 - 0.24 = 0.76
\]
Next, compute \( p(1 - p) \):
\[
p(1 - p) = 0.24 \times 0.76 = 0.1824
\]
Then, calculate \( \frac{p(1 - p)}{n} \):
\[
\frac{0.1824}{75} = 0.002432
\]
Now take the square root:
\[
\sqrt{0.002432} \approx 0.04932
\]
Finally, compute the margin of error:
\[
ME = 1.96 \times 0.04932 \approx 0.0966
\]
So, the margin of error is approximately 0.0966. In percentage terms, this is:
\[
0.0966 \times 100 \approx 9.66\%
\]
Note that the options provided in your question do not match this rather calculated number; however, the margin of error calculated as a percentage is expressed correctly. If we round to the nearest hundredth, the margin of error is approximately **9.7%**. If you need to choose from the provided options, it appears they may have a miscalculation or misrepresentation in percentage should reflect "10%."