A county official would like to estimate the proportion of homes in their county that were heated by natural gas. They randomly select 1200 homes in their county and find that \( 45 \% \) were heated by natural gas. Construct a \( 95 \% \) confidence interval for the population proportion of of homes in their county that were heated by natural gas. Do not round between steps. Round answers to at least 4 decimal places. Select an answer \( \hat{v}<\square \)

To construct a \( 95\% \) confidence interval for the population proportion of homes heated by natural gas, we first need to calculate the sample proportion (\( \hat{p} \)) and the standard error (SE). 1. Sample proportion \( \hat{p} = 0.45 \) (i.e., \( 45\% \) of \( 1200 \)). 2. Standard error (SE) is calculated using the formula: \[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \] where \( n = 1200 \): \[ SE = \sqrt{\frac{0.45 \times 0.55}{1200}} \approx \sqrt{\frac{0.2475}{1200}} \approx \sqrt{0.00020625} \approx 0.0144 \] 3. For a \( 95\% \) confidence interval, we use a \( z \)-value of approximately \( 1.96 \). 4. Margin of error (ME) is calculated as: \[ ME = z \times SE = 1.96 \times 0.0144 \approx 0.028224 \] 5. The confidence interval is: \[ \hat{p} \pm ME = 0.45 \pm 0.028224 \] Finally, calculating the interval: - Lower limit: \( 0.45 - 0.028224 \approx 0.4218 \) - Upper limit: \( 0.45 + 0.028224 \approx 0.4782 \) Thus, the \( 95\% \) confidence interval for the proportion of homes heated by natural gas is approximately \( (0.4218, 0.4782) \). Select an answer \( \hat{v}<0.4782 \)