To interpret the margin of error for a 90% confidence interval based on the information provided, we first need to calculate the margin of error using the formula for the confidence interval of a proportion.
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Sample Proportion (p): Given that 32% of the respondents favor the reelection of the mayor, the sample proportion p = 0.32.
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Sample Size (n): The total number of respondents is 650.
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Standard Error (SE): The formula for the standard error of a proportion is: \[ SE = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.32(1-0.32)}{650}} \]
First, calculate \(0.32(1 - 0.32) = 0.32 \times 0.68 = 0.2176\).
Now, calculate the standard error: \[ SE = \sqrt{\frac{0.2176}{650}} \approx \sqrt{0.000334} \approx 0.0183 \]
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Z-score for 90% Confidence: The Z-score for a 90% confidence level is approximately 1.645.
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Margin of Error (ME): This is calculated as: \[ ME = Z \times SE = 1.645 \times 0.0183 \approx 0.0301 \]
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Confidence Interval: The 90% confidence interval for the proportion who favor reelection can be calculated using: \[ p \pm ME = 0.32 \pm 0.0301 \]
This results in: \[ (0.32 - 0.0301, 0.32 + 0.0301) = (0.2899, 0.3501) \]
Rounding to two decimal places gives us approximately (0.29, 0.35).
Based on this calculation, the correct interpretation of the margin of error for a 90% confidence interval is:
With 90% confidence, the sample proportion of people who favor the reelection of the current mayor falls between (0.29, 0.35).