Question
A survey is done to determine if students prefer bike riding, skating, or neither. If 102 participants are surveyed and 72% prefer bike riding, interpret the margin of error. (1 point) ± 10.09% ± 9.9% ± 0.09% ± 10%
Answers
GPT-4o mini
To determine the margin of error for the survey results, you can use the formula for margin of error (ME) in a proportion, which is given by:
\[
ME = Z \times \sqrt{\frac{p(1 - p)}{n}}
\]
Where:
- \(Z\) is the Z-score corresponding to the confidence level (commonly 1.96 for a 95% confidence level).
- \(p\) is the sample proportion (0.72 in this case).
- \(n\) is the sample size (102 participants).
Calculating the margin of error:
1. Calculate \(p(1 - p)\):
\[
p(1 - p) = 0.72 \times (1 - 0.72) = 0.72 \times 0.28 = 0.2016
\]
2. Calculate \(\frac{p(1 - p)}{n}\):
\[
\frac{0.2016}{102} \approx 0.001976
\]
3. Calculate the square root:
\[
\sqrt{0.001976} \approx 0.0444
\]
4. For a 95% confidence level, \(Z \approx 1.96\):
\[
ME = 1.96 \times 0.0444 \approx 0.0870
\]
5. Convert this margin of error from a proportion to a percentage:
\[
ME \approx 0.0870 \times 100 \approx 8.70\%
\]
The margin of error for 102 participants with 72% preferring bike riding is approximately ± 8.70%. However, none of the provided options directly match this percentage.
Given the provided options:
- ± 10.09%
- ± 9.9%
- ± 0.09%
- ± 10%
The most conservative matching option would be **± 10%**, as it is the best approximation given the calculated margin of error.
\[
ME = Z \times \sqrt{\frac{p(1 - p)}{n}}
\]
Where:
- \(Z\) is the Z-score corresponding to the confidence level (commonly 1.96 for a 95% confidence level).
- \(p\) is the sample proportion (0.72 in this case).
- \(n\) is the sample size (102 participants).
Calculating the margin of error:
1. Calculate \(p(1 - p)\):
\[
p(1 - p) = 0.72 \times (1 - 0.72) = 0.72 \times 0.28 = 0.2016
\]
2. Calculate \(\frac{p(1 - p)}{n}\):
\[
\frac{0.2016}{102} \approx 0.001976
\]
3. Calculate the square root:
\[
\sqrt{0.001976} \approx 0.0444
\]
4. For a 95% confidence level, \(Z \approx 1.96\):
\[
ME = 1.96 \times 0.0444 \approx 0.0870
\]
5. Convert this margin of error from a proportion to a percentage:
\[
ME \approx 0.0870 \times 100 \approx 8.70\%
\]
The margin of error for 102 participants with 72% preferring bike riding is approximately ± 8.70%. However, none of the provided options directly match this percentage.
Given the provided options:
- ± 10.09%
- ± 9.9%
- ± 0.09%
- ± 10%
The most conservative matching option would be **± 10%**, as it is the best approximation given the calculated margin of error.