Question
In a survey funded by the UW school of medicine, 750 of 1000 adult Seattle residents said they did not believe they could come down with a sexually transmitted infection (STI). Construct a 95% confidence interval estimage of the proportion of adult Seattle residents who don't believe they can contract an STI. (Use a
z score of 1.96 for your computations.)
(.728, .772)
(.723, .777)
(.718, .782)
(.713, .878)
(.665, .835)
z score of 1.96 for your computations.)
(.728, .772)
(.723, .777)
(.718, .782)
(.713, .878)
(.665, .835)
Answers
GPT-4o mini
To construct a 95% confidence interval for the proportion of adult Seattle residents who don't believe they can contract an STI, we can follow these steps:
1. **Calculate the sample proportion \(\hat{p}\)**:
\[
\hat{p} = \frac{x}{n} = \frac{750}{1000} = 0.75
\]
2. **Determine the standard error (SE) of the proportion**:
\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.75(1 - 0.75)}{1000}} = \sqrt{\frac{0.75 \cdot 0.25}{1000}} = \sqrt{\frac{0.1875}{1000}} \approx \sqrt{0.0001875} \approx 0.01365
\]
3. **Find the z-score for a 95% confidence level**:
- The z-score given is 1.96.
4. **Calculate the margin of error (ME)**:
\[
ME = z \cdot SE = 1.96 \cdot 0.01365 \approx 0.0268
\]
5. **Construct the confidence interval**:
\[
\text{Lower bound} = \hat{p} - ME = 0.75 - 0.0268 \approx 0.7232
\]
\[
\text{Upper bound} = \hat{p} + ME = 0.75 + 0.0268 \approx 0.7768
\]
Thus, the 95% confidence interval for the proportion of adult Seattle residents who don't believe they can contract an STI is approximately:
\[
(0.7232, 0.7768)
\]
Considering the options provided, the correct confidence interval is:
**(.723, .777)**.
1. **Calculate the sample proportion \(\hat{p}\)**:
\[
\hat{p} = \frac{x}{n} = \frac{750}{1000} = 0.75
\]
2. **Determine the standard error (SE) of the proportion**:
\[
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.75(1 - 0.75)}{1000}} = \sqrt{\frac{0.75 \cdot 0.25}{1000}} = \sqrt{\frac{0.1875}{1000}} \approx \sqrt{0.0001875} \approx 0.01365
\]
3. **Find the z-score for a 95% confidence level**:
- The z-score given is 1.96.
4. **Calculate the margin of error (ME)**:
\[
ME = z \cdot SE = 1.96 \cdot 0.01365 \approx 0.0268
\]
5. **Construct the confidence interval**:
\[
\text{Lower bound} = \hat{p} - ME = 0.75 - 0.0268 \approx 0.7232
\]
\[
\text{Upper bound} = \hat{p} + ME = 0.75 + 0.0268 \approx 0.7768
\]
Thus, the 95% confidence interval for the proportion of adult Seattle residents who don't believe they can contract an STI is approximately:
\[
(0.7232, 0.7768)
\]
Considering the options provided, the correct confidence interval is:
**(.723, .777)**.