To determine which statement is most reasonable, we'll apply the provided information.
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Calculate the sample size and the proportion of voters who disapprove:
- Sample size (n) = 515
- Proportion (p) = 33% = 0.33
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Calculate the standard error (SE): \[ SE = \sqrt{p(1-p)/n} = \sqrt{0.33 \times (1 - 0.33) / 515} = \sqrt{0.33 \times 0.67 / 515} \]
\[ SE \approx \sqrt{0.2211 / 515} \approx \sqrt{0.000428} \approx 0.0207 \]
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Calculate the margin of error (ME) at a 95% confidence level:
- Z-score = 1.96 \[ ME = Z \times SE = 1.96 \times 0.0207 \approx 0.0406 \]
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Calculate the confidence interval for the proportion of voters who do not approve: \[ \text{Lower bound} = p - ME \approx 0.33 - 0.0406 \approx 0.2894 \] \[ \text{Upper bound} = p + ME \approx 0.33 + 0.0406 \approx 0.3706 \]
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Convert these proportions to the total eligible voters (29,352) to find the number of voters:
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Lower bound (number of disapproving voters): \[ 0.2894 \times 29352 \approx 8,507.57 \text{ (approximately 8,508)} \]
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Upper bound (number of disapproving voters): \[ 0.3706 \times 29352 \approx 10,885.99 \text{ (approximately 10,886)} \]
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Conclusion: The confidence interval for the number of voters who do not support the mayor lies between approximately 8,508 and 10,886 eligible voters.
Therefore, the most reasonable statement from the options provided is:
Between 8,512 – 10,860 eligible voters do not support the mayor.