To construct a 99.8% confidence interval for the proportion of people who volunteered based on your data, we will follow these steps:
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Gather the sample data:
- Sample size (n) = 1309
- Number of successes (x) = 518 (those who said yes)
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Calculate the sample proportion (p-hat): \[ \hat{p} = \frac{x}{n} = \frac{518}{1309} \approx 0.395 \]
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Find the Z-score for a 99.8% confidence level: For a 99.8% confidence interval, the significance level (\(\alpha\)) is: \[ \alpha = 1 - 0.998 = 0.002 \] Since this is a two-tailed test, we will divide by 2: \[ \alpha/2 = 0.001 \] Looking up the Z-score corresponding to 0.001 in a Z-table, we find: \[ Z \approx 3.09 \]
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Calculate the standard error (SE): The standard error for the proportion is calculated as: \[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.395(1 - 0.395)}{1309}} \approx \sqrt{\frac{0.395 \cdot 0.605}{1309}} \approx \sqrt{\frac{0.239775}{1309}} \approx \sqrt{0.000183} \approx 0.0135 \]
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Calculate the margin of error (ME): \[ ME = Z \cdot SE \approx 3.09 \cdot 0.0135 \approx 0.0417 \]
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Construct the confidence interval: The 99.8% confidence interval for the proportion is given by: \[ \hat{p} \pm ME \approx 0.395 \pm 0.0417 \] This results in:
- Lower limit: \( 0.395 - 0.0417 \approx 0.3533 \)
- Upper limit: \( 0.395 + 0.0417 \approx 0.4367 \)
So, the 99.8% confidence interval for the proportion of people who volunteered is approximately:
\[ \boxed{(0.3533, 0.4367)} \]
This means we are 99.8% confident that the true proportion of people who would say "yes" to volunteering lies between 35.33% and 43.67%.
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