To construct a 95% confidence interval for the proportion of residents in favor of the construction of a nuclear power plant, we can use the formula for the confidence interval for a population proportion.
Step 1: Calculate the sample proportion
The sample proportion \(\hat{p}\) is given by:
\[ \hat{p} = \frac{x}{n} \]
where \(x\) is the number of successes (in favor) and \(n\) is the sample size.
From the problem:
- \(x = 35\)
- \(n = 100\)
So,
\[ \hat{p} = \frac{35}{100} = 0.35 \]
Step 2: Calculate the standard error
The standard error (SE) for the proportion is calculated using the formula:
\[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]
Substitute \(\hat{p} = 0.35\) and \(n = 100\):
\[ SE = \sqrt{\frac{0.35 \times (1 - 0.35)}{100}} = \sqrt{\frac{0.35 \times 0.65}{100}} = \sqrt{\frac{0.2275}{100}} = \sqrt{0.002275} \approx 0.0477 \]
Step 3: Find the Z-score for 95% confidence
For a 95% confidence level, the Z-score (critical value) is approximately 1.96.
Step 4: Calculate the margin of error
The margin of error (ME) is calculated as follows:
\[ ME = Z \times SE \]
So,
\[ ME = 1.96 \times 0.0477 \approx 0.0935 \]
Step 5: Construct the confidence interval
The confidence interval is given by:
\[ \hat{p} \pm ME \]
Calculating the lower and upper bounds:
- Lower bound: \(0.35 - 0.0935 \approx 0.2565\)
- Upper bound: \(0.35 + 0.0935 \approx 0.4435\)
Final Result
Thus, the 95% confidence interval for the proportion of all residents in favor of the nuclear power plant is approximately:
\[ (0.2565, 0.4435) \]
This means we are 95% confident that the true proportion of all residents in favor of the construction of the nuclear power plant lies within the interval \(0.2565\) to \(0.4435\).