To determine the 95% confidence interval for the proportion of voters against the fire department bond measures, we follow these steps:
1. Identify the parameters:
- Number of voters surveyed ([tex]\(n\)[/tex]): 1000
- Number of voters against the measure ([tex]\(x\)[/tex]): 180
2. Calculate the sample proportion ([tex]\(\hat{p}\)[/tex]):
[tex]\[
\hat{p} = \frac{x}{n} = \frac{180}{1000} = 0.18
\][/tex]
3. Determine the z-value for a 95% confidence interval:
From the provided table, for a 95% confidence level, the z-value ([tex]\(z^*\)[/tex]) is 1.960.
4. Calculate the standard error (SE):
[tex]\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.18 \times (1 - 0.18)}{1000}} = \sqrt{\frac{0.18 \times 0.82}{1000}}
\][/tex]
[tex]\[
SE = \sqrt{\frac{0.1476}{1000}} = \sqrt{0.0001476} \approx 0.0121
\][/tex]
5. Compute the margin of error (ME):
[tex]\[
ME = z^* \times SE = 1.960 \times 0.0121 = 0.0237
\][/tex]
6. Calculate the confidence interval:
[tex]\[
\text{Lower bound} = \hat{p} - ME = 0.18 - 0.0237 = 0.1563
\][/tex]
[tex]\[
\text{Upper bound} = \hat{p} + ME = 0.18 + 0.0237 = 0.2037
\][/tex]
7. Construct the confidence interval:
[tex]\[
\left(0.1563, 0.2037\right)
\][/tex]
Thus, we are 95% confident that the true proportion of voters who are against the fire department bond measures lies between 15.63% and 20.37%.
