c) A sample of 11 nurses working in a large hospital found that they slept an average of 6.1 hours per night with a sample standard deviation of 1.2 hours. Estimate the true mean hours slept per night with 99 % confidence. df.-10 Ce. S= 1.2 4.96uく7.2 3.169
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How do you get the "3.169" on this example problem?
![The text addresses an estimation problem in statistics, specifically calculating a confidence interval for a mean.
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**Problem Statement:**
A sample of 11 nurses working in a large hospital found that they slept an average of 6.1 hours per night with a sample standard deviation of 1.2 hours. Estimate the true mean hours slept per night with 99% confidence.
**Solution:**
1. **Degrees of Freedom (d.f.):** 10
2. **Sample Size (n):** 11
3. **Sample Mean (\(\bar{x}\)):** 6.1
4. **Sample Standard Deviation (s):** 1.2
5. **t-value (\(t_{\alpha/2}\)):** 3.169
**Confidence Interval Calculation:**
The formula for the confidence interval is:
\[
\bar{x} \pm \left(t_{\alpha/2} \times \frac{s}{\sqrt{n}}\right)
\]
**Calculations:**
- Lower Bound:
\[
6.1 - \left(3.169 \times \frac{1.2}{\sqrt{11}}\right)
\]
- Upper Bound:
\[
6.1 + \left(3.169 \times \frac{1.2}{\sqrt{11}}\right)
\]
**Resulting Interval:**
The calculation results in:
\[
4.9 < \mu < 7.2
\]
This means we are 99% confident that the true mean hours slept per night by nurses in this sample lies between 4.9 and 7.2 hours.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc48de0b6-daaf-4f21-b11f-b5dcf0f469cd%2F21cd0083-3537-4271-9191-48bcd25e285d%2Fcjafjqf_processed.png&w=3840&q=75)
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