To construct a 99% confidence interval for the proportion of residents in favor of the construction of a nuclear power plant, we can follow these steps:
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Identify the sample size and the number of successes:
- Sample size (\(n\)) = 100
- Number of successes (\(x\)) = 35
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Calculate the sample proportion (\(\hat{p}\)): \[ \hat{p} = \frac{x}{n} = \frac{35}{100} = 0.35 \]
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Determine the standard error (SE) of the sample proportion: \[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.35(1 - 0.35)}{100}} = \sqrt{\frac{0.35 \times 0.65}{100}} = \sqrt{\frac{0.2275}{100}} \approx \sqrt{0.002275} \approx 0.0477 \]
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Find the z-score for a 99% confidence level:
- For a 99% confidence level, the z-score (\(z\)) can be found from the standard normal distribution. The critical value for 99% confidence is approximately 2.576.
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Calculate the margin of error (ME): \[ ME = z \times SE = 2.576 \times 0.0477 \approx 0.123 \]
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Construct the confidence interval: \[ \text{Lower limit} = \hat{p} - ME = 0.35 - 0.123 \approx 0.227 \] \[ \text{Upper limit} = \hat{p} + ME = 0.35 + 0.123 \approx 0.473 \]
Therefore, the 99% confidence interval for the proportion of all residents in favor of the nuclear power plant is approximately:
\[ \boxed{(0.227, 0.473)} \]
This means we are 99% confident that the true proportion of all residents in favor of the nuclear power plant falls within this interval.
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