Question
A survey of 650 people reported that 32% favored the reelection of the current mayor. Which of the following correctly interprets the margin of error for a 90% confidence interval?(1 point) Responses With 99% confidence, the sample proportion of people who favor the reelection of the current mayor falls between (0.29,0.35) . With 99% confidence, the sample proportion of people who favor the reelection of the current mayor falls between left parenthesis 0.29 comma 0.35 right parenthesis . With 90% confidence, the sample proportion of people who favor the reelection of the current mayor falls between (0.29,0.35) . With 90% confidence, the sample proportion of people who favor the reelection of the current mayor falls between left parenthesis 0.29 comma 0.35 right parenthesis . With 90% confidence, the sample proportion of people who favor the reelection of the current mayor falls between (0.58,0.64) . With 90% confidence, the sample proportion of people who favor the reelection of the current mayor falls between left parenthesis 0.58 comma 0.64 right parenthesis . With 90% confidence, the sample proportion of people who favor the reelection of the current mayor falls between (0.32,0.36) .
Answers
GPT-4o mini
To interpret the margin of error for a 90% confidence interval based on the information provided, we first need to calculate the margin of error using the formula for the confidence interval of a proportion.
1. **Sample Proportion (p)**: Given that 32% of the respondents favor the reelection of the mayor, the sample proportion p = 0.32.
2. **Sample Size (n)**: The total number of respondents is 650.
3. **Standard Error (SE)**: The formula for the standard error of a proportion is:
\[
SE = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.32(1-0.32)}{650}}
\]
First, calculate \(0.32(1 - 0.32) = 0.32 \times 0.68 = 0.2176\).
Now, calculate the standard error:
\[
SE = \sqrt{\frac{0.2176}{650}} \approx \sqrt{0.000334} \approx 0.0183
\]
4. **Z-score for 90% Confidence**: The Z-score for a 90% confidence level is approximately 1.645.
5. **Margin of Error (ME)**: This is calculated as:
\[
ME = Z \times SE = 1.645 \times 0.0183 \approx 0.0301
\]
6. **Confidence Interval**: The 90% confidence interval for the proportion who favor reelection can be calculated using:
\[
p \pm ME = 0.32 \pm 0.0301
\]
This results in:
\[
(0.32 - 0.0301, 0.32 + 0.0301) = (0.2899, 0.3501)
\]
Rounding to two decimal places gives us approximately (0.29, 0.35).
Based on this calculation, the correct interpretation of the margin of error for a 90% confidence interval is:
**With 90% confidence, the sample proportion of people who favor the reelection of the current mayor falls between (0.29, 0.35).**
1. **Sample Proportion (p)**: Given that 32% of the respondents favor the reelection of the mayor, the sample proportion p = 0.32.
2. **Sample Size (n)**: The total number of respondents is 650.
3. **Standard Error (SE)**: The formula for the standard error of a proportion is:
\[
SE = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.32(1-0.32)}{650}}
\]
First, calculate \(0.32(1 - 0.32) = 0.32 \times 0.68 = 0.2176\).
Now, calculate the standard error:
\[
SE = \sqrt{\frac{0.2176}{650}} \approx \sqrt{0.000334} \approx 0.0183
\]
4. **Z-score for 90% Confidence**: The Z-score for a 90% confidence level is approximately 1.645.
5. **Margin of Error (ME)**: This is calculated as:
\[
ME = Z \times SE = 1.645 \times 0.0183 \approx 0.0301
\]
6. **Confidence Interval**: The 90% confidence interval for the proportion who favor reelection can be calculated using:
\[
p \pm ME = 0.32 \pm 0.0301
\]
This results in:
\[
(0.32 - 0.0301, 0.32 + 0.0301) = (0.2899, 0.3501)
\]
Rounding to two decimal places gives us approximately (0.29, 0.35).
Based on this calculation, the correct interpretation of the margin of error for a 90% confidence interval is:
**With 90% confidence, the sample proportion of people who favor the reelection of the current mayor falls between (0.29, 0.35).**