To construct a 95% confidence interval for the difference in proportions of Seattle and Denver residents who do not believe they could contract an STI, we will follow these steps:

1. Define the Proportions:

   • Let $p_1$ be the proportion of Seattle residents who do not believe they can contract an STI: $p_1 = \frac{750}{1000} = 0.75$
   • Let $p_2$ be the proportion of Denver residents who do not believe they can contract an STI: $p_2 = \frac{893}{1123} \approx 0.794$

2. Calculate the Difference in Proportions: $\hat{p_1} - \hat{p_2 = 0.75 - 0.794 = -0.044}$

3. Calculate the Standard Error (SE) of the Difference: $SE = \sqrt{\frac{p_1(1 - p_1)}{n_1} + \frac{p_2(1 - p_2)}{n_2}}$ Where:

   • $n_1 = 1000$ (Seattle residents)
   • $n_2 = 1123$ (Denver residents)

   Substituting the values: $SE = \sqrt{\frac{0.75(1 - 0.75)}{1000} + \frac{0.794(1 - 0.794)}{1123}}$ $= \sqrt{\frac{0.75 \times 0.25}{1000} + \frac{0.794 \times 0.206}{1123}}$ $= \sqrt{\frac{0.1875}{1000} + \frac{0.163244}{1123}}$ $= \sqrt{0.0001875 + 0.000145227} \approx \sqrt{0.000332727} \approx 0.0182$

4. Calculate the Margin of Error (ME): $ME = z \times SE = 1.96 \times 0.0182 \approx 0.0357$

5. Construct the Confidence Interval: $CI = (\hat{p_1} - \hat{p_2}) \pm ME$ $CI = -0.044 \pm 0.0357$ $= (-0.0797, -0.0083)$

Thus, rounding to four decimal places, the 95% confidence interval for the difference in proportions is approximately:

(-0.0809, -0.0095)

Conclusion:

The correct answer is (-0.0809, -0.0095).

Would you like more details on any part of this calculation, or do you have any questions? Here are some related questions you might consider:

1. How can we interpret the confidence interval in practical terms?
2. What assumptions are made when constructing this confidence interval?
3. How would the interval change with a different confidence level, like 90%?
4. What are the implications of these results for public health messaging?
5. How do sample sizes affect the width of a confidence interval?

Tip: Always ensure your sample sizes are adequate to achieve a reliable confidence interval; larger samples generally provide more accurate estimates.