To construct a 90% confidence interval for the proportion of residents in favor of the nuclear power plant, we can use the formula for a confidence interval for a proportion. The formula is given by:
\[ \hat{p} \pm z \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]
where:
- \( \hat{p} \) is the sample proportion,
- \( z \) is the z-score corresponding to the desired confidence level,
- \( n \) is the sample size.
Step 1: Calculate the sample proportion (\( \hat{p} \))
From the sample of 100 residents, 35 are in favor, so: \[ \hat{p} = \frac{35}{100} = 0.35 \]
Step 2: Find the z-score for a 90% confidence interval
For a 90% confidence level, the z-score (critical value) can be found using a z-table or standard normal distribution. The z-score that corresponds to 90% confidence is approximately: \[ z \approx 1.645 \]
Step 3: Calculate the standard error (SE)
The standard error of the sample proportion is calculated as follows: \[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.35(1 - 0.35)}{100}} = \sqrt{\frac{0.35 \cdot 0.65}{100}} = \sqrt{\frac{0.2275}{100}} \approx \sqrt{0.002275} \approx 0.0477 \]
Step 4: Construct the confidence interval
Using the formula: \[ \hat{p} \pm z \cdot SE \] we can now calculate the confidence interval: \[ 0.35 \pm 1.645 \cdot 0.0477 \]
Calculating the margin of error: \[ 1.645 \cdot 0.0477 \approx 0.0783 \]
Thus, the confidence interval is: \[ 0.35 - 0.0783 \quad \text{to} \quad 0.35 + 0.0783 \] \[ 0.2717 \quad \text{to} \quad 0.4283 \]
Conclusion
The 90% confidence interval for the proportion of all residents in favor of the construction of the nuclear power plant is approximately: \[ (0.2717, 0.4283) \] This means we are 90% confident that the true proportion of residents in favor of the plant lies between 27.17% and 42.83%.