MATLAB: An Introduction with Applications
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Author: Amos Gilat
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A random sample of 240 doctors revealed that 108 are satisfied with the current state of US health care.
The conditions for inference are met.
Using z* = 1.645, which expression gives a 90% confidence interval for the true proportion of all US doctors who are satisfied with the current state of US health care?
![**Determining the Critical Value**
A random sample of 240 doctors revealed that 108 are satisfied with the current state of US health care.
The conditions for inference are met.
Using \( z^* = 1.645 \), which expression gives a 90% confidence interval for the true proportion of all US doctors who are satisfied with the current state of US health care?
1. \( 108 \pm 1.645 \sqrt{\frac{108(1-108)}{240}} \)
2. \( 1.645 \pm 108 \sqrt{\frac{0.5(1-0.5)}{240}} \)
3. \( 0.45 \pm 1.645 \sqrt{\frac{0.45(1-0.45)}{240}} \)
4. \( 0.45 \pm 108 \sqrt{\frac{0.5(1-0.5)}{1.645}} \)
**Explanation:**
- The task is to find the correct expression for calculating a 90% confidence interval.
- \( z^* = 1.645 \) is the critical value for a 90% confidence level.
- The sample proportion is calculated as \( \frac{108}{240} = 0.45 \).
- The expression should use this sample proportion in its calculation with the formula:
\[
\text{Sample Proportion} \pm z^* \sqrt{\frac{\text{Sample Proportion}(1-\text{Sample Proportion})}{\text{Sample Size}}}
\]
The correct expression that fits this format is:
- \( 0.45 \pm 1.645 \sqrt{\frac{0.45(1-0.45)}{240}} \)
This choice correctly uses the sample proportion of 0.45 and incorporates the critical value \( z^* \) correctly in the confidence interval formula.](https://content.bartleby.com/qna-images/question/7c6d136e-2074-4849-9b3d-0df6994baeb8/3abeb8b2-8003-42df-ac4a-33e8cdd4795e/c7gekn_processed.png)
Transcribed Image Text:**Determining the Critical Value**
A random sample of 240 doctors revealed that 108 are satisfied with the current state of US health care.
The conditions for inference are met.
Using \( z^* = 1.645 \), which expression gives a 90% confidence interval for the true proportion of all US doctors who are satisfied with the current state of US health care?
1. \( 108 \pm 1.645 \sqrt{\frac{108(1-108)}{240}} \)
2. \( 1.645 \pm 108 \sqrt{\frac{0.5(1-0.5)}{240}} \)
3. \( 0.45 \pm 1.645 \sqrt{\frac{0.45(1-0.45)}{240}} \)
4. \( 0.45 \pm 108 \sqrt{\frac{0.5(1-0.5)}{1.645}} \)
**Explanation:**
- The task is to find the correct expression for calculating a 90% confidence interval.
- \( z^* = 1.645 \) is the critical value for a 90% confidence level.
- The sample proportion is calculated as \( \frac{108}{240} = 0.45 \).
- The expression should use this sample proportion in its calculation with the formula:
\[
\text{Sample Proportion} \pm z^* \sqrt{\frac{\text{Sample Proportion}(1-\text{Sample Proportion})}{\text{Sample Size}}}
\]
The correct expression that fits this format is:
- \( 0.45 \pm 1.645 \sqrt{\frac{0.45(1-0.45)}{240}} \)
This choice correctly uses the sample proportion of 0.45 and incorporates the critical value \( z^* \) correctly in the confidence interval formula.
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