6. Battery Lives Find the \( 99 \% \) confidence interval for the variance and standard deviation of the weights of 25 one- gallon containers of motor oil if a sample of 14 containers has a variance of 3.2 . The weights are given in ounces. Assume the variable is normally distributed.

To find the **99% confidence interval** for both the **variance (\( \sigma^2 \))** and the **standard deviation (\( \sigma \))** of the weights of the one-gallon motor oil containers, we'll use the chi-square (\( \chi^2 \)) distribution. Here's a step-by-step guide to solving the problem: ### **Given:** - **Sample size (\( n \))**: 14 - **Sample variance (\( s^2 \))**: 3.2 ounces\(^2\) - **Degrees of freedom (\( df \))**: \( n - 1 = 13 \) - **Confidence level**: 99% (\( \alpha = 0.01 \)) ### **Step 1: Determine the Critical Chi-Square Values** For a 99% confidence interval, the significance level \( \alpha = 0.01 \), which means \( \alpha/2 = 0.005 \) in each tail of the chi-square distribution. Using chi-square tables or a calculator, find: - **\( \chi^2_{\alpha/2, df} = \chi^2_{0.005, 13} \approx 32.801 \)** - **\( \chi^2_{1-\alpha/2, df} = \chi^2_{0.995, 13} \approx 1.771 \)** ### **Step 2: Calculate the Confidence Interval for the Variance (\( \sigma^2 \))** The formula for the confidence interval for variance is: \[ \left( \frac{(n-1)s^2}{\chi^2_{\alpha/2, df}}, \frac{(n-1)s^2}{\chi^2_{1-\alpha/2, df}} \right) \] Plugging in the values: \[ \left( \frac{13 \times 3.2}{32.801}, \frac{13 \times 3.2}{1.771} \right) \approx \left( 1.27, 23.49 \right) \text{ ounces}^2 \] ### **Step 3: Calculate the Confidence Interval for the Standard Deviation (\( \sigma \))** Take the square root of the variance bounds to find the standard deviation bounds: \[ \left( \sqrt{1.27}, \sqrt{23.49} \right) \approx \left( 1.13, 4.85 \right) \text{ ounces} \] ### **Final Result:** - **99% Confidence Interval for the Variance (\( \sigma^2 \))**: **(1.27, 23.49) ounces²** - **99% Confidence Interval for the Standard Deviation (\( \sigma \))**: **(1.13, 4.85) ounces** **Summary:** At the 99% confidence level, the variance lies between 1.27 and 23.49 ounces² and the standard deviation lies between 1.13 and 4.85 ounces. **Mathematical Notation:** \[ \sigma^2 \in (1.27, \ 23.49) \ \text{ounces}^2 \] \[ \sigma \in (1.13, \ 4.85) \ \text{ounces} \]