To find the number of eligible voters who do not approve of the mayor, we need to calculate the sample proportion and then estimate the confidence interval for the population proportion.
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Calculate the proportion of disapproval among the sample:
- Sample size \( n = 515 \)
- Proportion of disapproval \( \hat{p} = 0.33 \)
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Calculate the number of voters who do not approve from the sample:
- Number of voters who do not approve in the sample \( = 0.33 \times 515 \approx 173.95 \)
- Rounding gives us approximately \( 174 \) voters in the sample.
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Calculate the standard deviation of the sample proportion: \[ \text{Standard deviation (SD)} = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.33 \times (1 - 0.33)}{515}} \approx \sqrt{\frac{0.33 \times 0.67}{515}} \approx 0.0244 \]
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Calculate the confidence interval:
- For a 95% confidence level, we use the z-score of \( 1.96 \).
- Standard error (SE) = \( \text{SD} \approx 0.0244 \)
- The margin of error (ME) = \( z \times \text{SE} = 1.96 \times 0.0244 \approx 0.0479 \)
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Calculate the confidence interval for the proportion:
- Lower limit: \( \hat{p} - \text{ME} = 0.33 - 0.0479 \approx 0.2821 \)
- Upper limit: \( \hat{p} + \text{ME} = 0.33 + 0.0479 \approx 0.3779 \)
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Convert the proportion confidence interval to the population:
- Population size \( N = 29,352 \)
- Lower limit for the number of voters: \( 0.2821 \times 29352 \approx 8286 \)
- Upper limit for the number of voters: \( 0.3779 \times 29352 \approx 11056 \)
Thus, based on our calculations, we estimate that the number of eligible voters who do not support the mayor lies between approximately \( 8,286 \) and \( 11,056 \).
Among the response options provided, the closest reasonable statement is:
Between 8,512 – 10,860 eligible voters do not support the mayor.