Question 1188786
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 α = 1 - 0.80 = 0.20
* **α/2:** 0.20 / 2 = 0.10
* **1 - α/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:

* χ²(0.90, 10) = χ²_L ≈ 4.865  (Lower critical value)
* χ²(0.10, 10) = χ²_R ≈ 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) * s² / χ²_R ] = sqrt[ (10 * 25.8²) / 15.987 ] ≈ sqrt(419.92) ≈ 20.49

* **Upper Limit:**
   sqrt[ (n-1) * s² / χ²_L ] = sqrt[ (10 * 25.8²) / 4.865 ] ≈ sqrt(1390.94) ≈ 37.29

**Answers:**

1. χ²_L ≈ 4.865
   χ²_R ≈ 15.987

2. Lower Limit ≈ 20.49
   Upper Limit ≈ 37.29

Therefore, you are 80% confident that the population standard deviation of Vulcan heights is between approximately 20.49 and 37.29.