To analyze the data provided, we can first determine the number of voters who do not approve of the mayor based on the sample data.
Step 1: Calculate the sample size and the proportion of disapproval.
From the sample of 515 eligible voters, 33% said they do not approve of the mayor:
\[ \text{Disapproving voters in sample} = 0.33 \times 515 = 169.95 \approx 170 \text{ (rounding to nearest whole number)} \]
Step 2: Calculate the total proportion of voters in the larger population.
We know that the total number of eligible voters in the city is 29,352. To find the estimate of total voters who do not support the mayor:
\[ \text{Estimated total disapproving voters} = 0.33 \times 29,352 \approx 9,669.16 \approx 9,669 \text{ (rounding to nearest whole number)} \]
Step 3: Calculate the standard error for the proportion.
The standard error (SE) of the proportion can be calculated using the formula:
\[ SE = \sqrt{\frac{p(1 - p)}{n}} \]
Where:
- \( p = 0.33 \)
- \( n = 515 \)
\[ SE = \sqrt{\frac{0.33 \times (1 - 0.33)}{515}} = \sqrt{\frac{0.33 \times 0.67}{515}} = \sqrt{\frac{0.2211}{515}} \approx \sqrt{0.0004287} \approx 0.0207 \]
Step 4: Determine the confidence interval.
For a 95% confidence interval, we can use the z-score of 1.96:
\[ \text{Margin of error} = z \times SE = 1.96 \times 0.0207 \approx 0.0406 \]
Now we calculate the confidence interval for the proportion:
\[ \text{Lower limit} = p - \text{Margin of error} = 0.33 - 0.0406 \approx 0.2894 \]
\[ \text{Upper limit} = p + \text{Margin of error} = 0.33 + 0.0406 \approx 0.3706 \]
Step 5: Translate the confidence interval back to numbers of voters.
Calculating the number of voters corresponding to the proportion limits:
For the lower limit: \[ 0.2894 \times 29,352 \approx 8,511.14 \approx 8,511 \text{ (rounding to nearest whole number)} \]
For the upper limit: \[ 0.3706 \times 29,352 \approx 10,861.72 \approx 10,862 \text{ (rounding to nearest whole number)} \]
Thus, the 95% confidence interval for the number of voters who do not support the mayor is:
\[ (8,511, 10,862) \]
Conclusion:
Thus, the most reasonable statement is:
Between 8,512 – 10,860 eligible voters do not support the mayor. (Responding to the rounding, the actual interval is approximately correct given the rounding.)