To address the question, we first calculate how many eligible voters in the city disapprove of the mayor based on the percentage provided.
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Calculate the number of people who disapprove of the mayor: \[ \text{Number of disapproving voters} = 33% \times 29,352 \] \[ = 0.33 \times 29,352 \approx 9,757.16 \] So approximately 9,757 eligible voters do not approve of the job the mayor is doing.
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Construct a confidence interval: The sample size provided (515 eligible voters) allows us to calculate a standard error for the estimated proportion.
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Sample Proportion \( p \): \[ p = 0.33 \]
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Standard Error (SE): \[ SE = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.33(1-0.33)}{515}} = \sqrt{\frac{0.33 \times 0.67}{515}} \] \[ SE \approx \sqrt{\frac{0.2211}{515}} \approx \sqrt{0.000428} \approx 0.0207 \]
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Margin of Error (ME) at 95% confidence: \[ ME = z \times SE = 1.96 \times 0.0207 \approx 0.0406 \]
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Confidence Interval for the proportion \( p \): \[ \text{Lower limit} = p - ME = 0.33 - 0.0406 \approx 0.2894 \] \[ \text{Upper limit} = p + ME = 0.33 + 0.0406 \approx 0.3706 \]
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Convert the confidence interval for the proportion back to the number of voters: Multiply the limits of the proportion by the total number of eligible voters (29,352):
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Lower limit number of disapproving voters: \[ = 0.2894 \times 29,352 \approx 8,502.14 \]
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Upper limit number of disapproving voters: \[ = 0.3706 \times 29,352 \approx 10,888.86 \]
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So, the confidence interval for the number of voters who do not support the mayor is approximately between 8,502 and 10,889.
- Analyze the provided options:
- $1,891 eligible voters do not support the mayor. (This is too low.)
- $149 and 191 eligible voters do not support the mayor. (This is also too low.)
- $8,000 and 10,000 eligible voters do not support the mayor. (This range is reasonable.)
- $28,837 and 29,867 eligible voters do not support the mayor. (This is way too high.)
The most reasonable statement is between 8,000 and 10,000 eligible voters do not support the mayor.