A college statistics professor has office hours from 9:00 A.M. to 10:30 A.M. daily. A sample of waiting times to see the professor (in minutes) is: 10, 12, 20, 15, 17, 10, 30, 28, 35, 28, 19, 27, 25, 22, 33, 37, 14, 21, 20, 23. Assuming o = 7.84, find the 99.74% confidence interval for the population mean. Round to two decimal places.
A college statistics professor has office hours from 9:00 A.M. to 10:30 A.M. daily. A sample of waiting times to see the professor (in minutes) is: 10, 12, 20, 15, 17, 10, 30, 28, 35, 28, 19, 27, 25, 22, 33, 37, 14, 21, 20, 23. Assuming o = 7.84, find the 99.74% confidence interval for the population mean. Round to two decimal places.
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![## Determining the Confidence Interval for Population Mean
### Question:
A college statistics professor has office hours from 9:00 A.M. to 10:30 A.M. daily. A sample of waiting times to see the professor (in minutes) is:
```
10, 12, 20, 15, 17, 10, 30, 28, 35, 28, 19, 27, 25, 22, 33, 37, 14, 21, 20, 23
```
Assuming \( \sigma = 7.84 \), find the 99.74% confidence interval for the population mean. Round to two decimal places.
### Options:
A. 20.55 to 24.05 minutes
B. 19.5 to 35.1 minutes
C. 17.05 to 27.55 minutes
D. 18.8 to 25.8 minutes
### Solution Explanation:
To determine the 99.74% confidence interval for the population mean, we use the formula:
\[ \text{CI} = \bar{x} \pm Z \left(\frac{\sigma}{\sqrt{n}}\right) \]
Where:
- \( \bar{x} \) is the sample mean
- \( Z \) is the Z-value corresponding to the desired confidence level
- \( \sigma \) is the population standard deviation
- \( n \) is the sample size
Steps to solve:
1. **Calculate the Sample Mean (\( \bar{x} \))**:
Calculate the mean of the provided sample data.
2. **Determine the Z-Value**:
Find the Z-value for a 99.74% confidence interval (lookup from Z-tables).
3. **Calculate the Standard Error**:
The standard error is \( \frac{\sigma}{\sqrt{n}} \).
4. **Calculate the Margin of Error**:
Multiply the Z-value by the standard error.
5. **Compute the Confidence Interval**:
Add and subtract the margin of error from the sample mean to find the lower and upper bounds of the confidence interval.
Use the above steps and select the correct confidence interval from the given options:
A. 20.55 to 24.05 minutes
B. 19.5 to 35.1 minutes
C. 17.05 to](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd5c84145-a33d-498c-9246-5dc8646df0e1%2F0eab0901-813f-4b49-8c2b-af2c2e1cea47%2Fiqt81qo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Determining the Confidence Interval for Population Mean
### Question:
A college statistics professor has office hours from 9:00 A.M. to 10:30 A.M. daily. A sample of waiting times to see the professor (in minutes) is:
```
10, 12, 20, 15, 17, 10, 30, 28, 35, 28, 19, 27, 25, 22, 33, 37, 14, 21, 20, 23
```
Assuming \( \sigma = 7.84 \), find the 99.74% confidence interval for the population mean. Round to two decimal places.
### Options:
A. 20.55 to 24.05 minutes
B. 19.5 to 35.1 minutes
C. 17.05 to 27.55 minutes
D. 18.8 to 25.8 minutes
### Solution Explanation:
To determine the 99.74% confidence interval for the population mean, we use the formula:
\[ \text{CI} = \bar{x} \pm Z \left(\frac{\sigma}{\sqrt{n}}\right) \]
Where:
- \( \bar{x} \) is the sample mean
- \( Z \) is the Z-value corresponding to the desired confidence level
- \( \sigma \) is the population standard deviation
- \( n \) is the sample size
Steps to solve:
1. **Calculate the Sample Mean (\( \bar{x} \))**:
Calculate the mean of the provided sample data.
2. **Determine the Z-Value**:
Find the Z-value for a 99.74% confidence interval (lookup from Z-tables).
3. **Calculate the Standard Error**:
The standard error is \( \frac{\sigma}{\sqrt{n}} \).
4. **Calculate the Margin of Error**:
Multiply the Z-value by the standard error.
5. **Compute the Confidence Interval**:
Add and subtract the margin of error from the sample mean to find the lower and upper bounds of the confidence interval.
Use the above steps and select the correct confidence interval from the given options:
A. 20.55 to 24.05 minutes
B. 19.5 to 35.1 minutes
C. 17.05 to
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