In a survey of 1800 US households, 1278 indicated that they owned a pet. 1. Construct the $95 \%$ confidence interval.

Step 1 ：In a survey of 1800 US households, 1278 indicated that they owned a pet. We are asked to construct a 95% confidence interval for the proportion of US households that own a pet.

Step 2 ：The formula for the confidence interval for a proportion is given by: \[\hat{p} \pm z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\] where \(\hat{p}\) is the sample proportion, \(n\) is the sample size, and \(z\) is the z-score corresponding to the desired level of confidence. For a 95% confidence interval, the z-score is approximately 1.96.

Step 3 ：In this case, \(\hat{p}\) is the proportion of households that own a pet, which is 1278 out of 1800, or approximately 0.71. The sample size \(n\) is 1800.

Step 4 ：We can plug these values into the formula to calculate the confidence interval. The standard error (se) is calculated as \(\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\), which is approximately 0.0107.

Step 5 ：The lower and upper bounds of the confidence interval are then calculated as \(\hat{p} - z \times se\) and \(\hat{p} + z \times se\), respectively. This gives us approximately 0.689 and 0.731.

Step 6 ：The final answer is: The 95% confidence interval for the proportion of US households that own a pet is approximately \(\boxed{(0.689, 0.731)}\). This means we are 95% confident that the true proportion of US households that own a pet is between 68.9% and 73.1%.