Question 1198875
**a. Sample Standard Deviation**

1. **Calculate the sample mean (x̄):**
   x̄ = (18 + 16 + 20 + 12 + 11 + 10 + 16 + 17) / 8 = 15

2. **Calculate the squared differences from the mean:**
   (18-15)² = 9
   (16-15)² = 1
   (20-15)² = 25
   (12-15)² = 9
   (11-15)² = 16
   (10-15)² = 25
   (16-15)² = 1
   (17-15)² = 4

3. **Sum the squared differences:**
   9 + 1 + 25 + 9 + 16 + 25 + 1 + 4 = 89

4. **Calculate the sample variance (s²):**
   s² = Σ(x - x̄)² / (n - 1) = 89 / (8 - 1) = 12.714

5. **Calculate the sample standard deviation (s):**
   s = √s² = √12.714 = 3.566

**Therefore, the sample standard deviation (s) is 3.566**

**b. 95% Confidence Interval for the Population Standard Deviation**

* **Find the chi-square values:**
    * Degrees of freedom (df) = n - 1 = 8 - 1 = 7
    * Using a chi-square table or calculator:
        * χ²_lower (for 0.025 area in the right tail) = 2.167
        * χ²_upper (for 0.025 area in the left tail) = 18.475

* **Calculate the confidence interval:**
    * Lower bound: √[(n - 1) * s² / χ²_upper] = √[(7 * 12.714) / 18.475] = 2.172
    * Upper bound: √[(n - 1) * s² / χ²_lower] = √[(7 * 12.714) / 2.167] = 6.390

* **95% Confidence Interval: (2.172, 6.390)**

**c. 99% Confidence Interval for the Population Standard Deviation**

* **Find the chi-square values:**
    * Degrees of freedom (df) = 7
    * Using a chi-square table or calculator:
        * χ²_lower (for 0.005 area in the right tail) = 0.989
        * χ²_upper (for 0.005 area in the left tail) = 20.278

* **Calculate the confidence interval:**
    * Lower bound: √[(7 * 12.714) / 20.278] = 1.974
    * Upper bound: √[(7 * 12.714) / 0.989] = 9.446

* **99% Confidence Interval: (1.974, 9.446)**

**d. As expected, the answer for part c. is wider** than the answer for part b.

**e. The lower bound is lower** and **the upper bound is higher** from s because of the shape of the chi-square distribution.

* The chi-square distribution is skewed to the right. 
* For a higher confidence level (like 99%), we need a wider interval to capture more of the possible values of the population standard deviation. This results in a lower lower bound and a higher upper bound compared to the 95% confidence interval.