Question 1188786: (1 point) The heights of Vulcans - an imaginary humanoid in Star Trek- are normally distributed. Suppose that a simple random sample of 11 Vulcans have a standard deviation of 25.8. Find the confidence Interval for the standard deviation of the entire population with 80% confidence.
1. Find the critical values š2šæ=š21āš¼/2 and š2š
=š2š¼/2 that correspond to 80% degree of confidence and the sample size š=11.
š2šæ=
š2š
=
2. Find the upper and lower limits of 80% confidence Interval for the standard deviation of the entire population.
The lower limit of the 80% confidence interval =
The upper limit of the 80% confidence interval =
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to calculate the confidence interval for the standard deviation of Vulcan heights:
**1. Find the critical chi-square values:**
* **Degrees of freedom (df):** df = n - 1 = 11 - 1 = 10
* **Confidence level:** 80%, so Ī± = 1 - 0.80 = 0.20
* **Ī±/2:** 0.20 / 2 = 0.10
* **1 - Ī±/2:** 1 - 0.10 = 0.90
Now, look up the chi-square values in a chi-square table or use a calculator for df = 10:
* ĻĀ²(0.90, 10) = ĻĀ²_L ā 4.865 (Lower critical value)
* ĻĀ²(0.10, 10) = ĻĀ²_R ā 15.987 (Upper critical value)
**2. Calculate the confidence interval limits:**
* **Sample standard deviation (s):** 25.8
* **Sample size (n):** 11
* **Lower Limit:**
sqrt[ (n-1) * sĀ² / ĻĀ²_R ] = sqrt[ (10 * 25.8Ā²) / 15.987 ] ā sqrt(419.92) ā 20.49
* **Upper Limit:**
sqrt[ (n-1) * sĀ² / ĻĀ²_L ] = sqrt[ (10 * 25.8Ā²) / 4.865 ] ā sqrt(1390.94) ā 37.29
**Answers:**
1. ĻĀ²_L ā 4.865
ĻĀ²_R ā 15.987
2. Lower Limit ā 20.49
Upper Limit ā 37.29
Therefore, you are 80% confident that the population standard deviation of Vulcan heights is between approximately 20.49 and 37.29.
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