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?
A. With 99% confidence, the sample proportion of people who favor the reelection of the current mayor falls between (0.29, 0.35).
B. With 90% confidence, the sample proportion of people who favor the reelection of the current mayor falls between (0.58, 0.64).
C. With 90% confidence, the sample proportion of people who favor the reelection of the current mayor falls between (0.29, 0.35).
D. With 90% confidence, the sample proportion of people who favor the reelection of the current mayor falls between (0.99, 0.26).
A. With 99% confidence, the sample proportion of people who favor the reelection of the current mayor falls between (0.29, 0.35).
B. With 90% confidence, the sample proportion of people who favor the reelection of the current mayor falls between (0.58, 0.64).
C. With 90% confidence, the sample proportion of people who favor the reelection of the current mayor falls between (0.29, 0.35).
D. With 90% confidence, the sample proportion of people who favor the reelection of the current mayor falls between (0.99, 0.26).
Answers
GPT-4o mini
To interpret the margin of error and calculate the confidence interval for the proportion of people who favor the reelection of the current mayor, we start by identifying the sample proportion and the margin of error.
1. **Sample proportion (p)**:
The survey found that 32% of the 650 people favored the reelection. Thus,
\[
p = 0.32
\]
2. **Sample size (n)**:
The sample size is
\[
n = 650
\]
3. **Calculation of standard error (SE)**:
The standard error for a proportion can be calculated using the formula:
\[
SE = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.32(1-0.32)}{650}} = \sqrt{\frac{0.32 \times 0.68}{650}} \approx \sqrt{\frac{0.2176}{650}} \approx \sqrt{0.000334} \approx 0.0183
\]
4. **Finding the z-score for a 90% confidence interval**:
For a 90% confidence level, the z-score (critical value) is approximately 1.645.
5. **Margin of error (ME)**:
The margin of error is calculated as:
\[
ME = z \times SE \approx 1.645 \times 0.0183 \approx 0.0301
\]
6. **Confidence interval**:
The confidence interval for the proportion is:
\[
(p - ME, p + ME) = (0.32 - 0.0301, 0.32 + 0.0301) = (0.2899, 0.3501)
\]
Rounding, this gives us approximately:
\[
(0.29, 0.35)
\]
Now we can interpret the result and evaluate the provided options:
- A. With 99% confidence, the sample proportion of people who favor the reelection of the current mayor falls between (0.29, 0.35). → Incorrect confidence level
- B. With 90% confidence, the sample proportion of people who favor the reelection of the current mayor falls between (0.58, 0.64). → Incorrect interval
- C. With 90% confidence, the sample proportion of people who favor the reelection of the current mayor falls between (0.29, 0.35). → Correct interval
- D. With 90% confidence, the sample proportion of people who favor the reelection of the current mayor falls between (0.99, 0.26). → Incorrect interval
The correct answer is **C**.
1. **Sample proportion (p)**:
The survey found that 32% of the 650 people favored the reelection. Thus,
\[
p = 0.32
\]
2. **Sample size (n)**:
The sample size is
\[
n = 650
\]
3. **Calculation of standard error (SE)**:
The standard error for a proportion can be calculated using the formula:
\[
SE = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.32(1-0.32)}{650}} = \sqrt{\frac{0.32 \times 0.68}{650}} \approx \sqrt{\frac{0.2176}{650}} \approx \sqrt{0.000334} \approx 0.0183
\]
4. **Finding the z-score for a 90% confidence interval**:
For a 90% confidence level, the z-score (critical value) is approximately 1.645.
5. **Margin of error (ME)**:
The margin of error is calculated as:
\[
ME = z \times SE \approx 1.645 \times 0.0183 \approx 0.0301
\]
6. **Confidence interval**:
The confidence interval for the proportion is:
\[
(p - ME, p + ME) = (0.32 - 0.0301, 0.32 + 0.0301) = (0.2899, 0.3501)
\]
Rounding, this gives us approximately:
\[
(0.29, 0.35)
\]
Now we can interpret the result and evaluate the provided options:
- A. With 99% confidence, the sample proportion of people who favor the reelection of the current mayor falls between (0.29, 0.35). → Incorrect confidence level
- B. With 90% confidence, the sample proportion of people who favor the reelection of the current mayor falls between (0.58, 0.64). → Incorrect interval
- C. With 90% confidence, the sample proportion of people who favor the reelection of the current mayor falls between (0.29, 0.35). → Correct interval
- D. With 90% confidence, the sample proportion of people who favor the reelection of the current mayor falls between (0.99, 0.26). → Incorrect interval
The correct answer is **C**.