The standard deviation of the lifetimes of a sample of 200 electric light bulbs was computed 
to be 100 hours.Find (a) 95% ,(b)99% confidence limits for the standard deviationof all such 
electric light bulbs..Step by step answer 

To calculate the confidence limits for the standard deviation of all such electric light bulbs, we will use the Chi-Square distribution. The formula for the confidence interval for the standard deviation \(\sigma\) is:

\[
\left( \sqrt{\frac{(n-1)s^2}{\chi^2_{\alpha/2}}}, \sqrt{\frac{(n-1)s^2}{\chi^2_{1-\alpha/2}}} \right)
\]

where:
- \(n\) is the sample size,
- \(s\) is the sample standard deviation,
- \(\chi^2_{\alpha/2}\) and \(\chi^2_{1-\alpha/2}\) are the critical values from the Chi-Square distribution with \(n-1\) degrees of freedom.

Given:
- \(n = 200\)
- \(s = 100\) hours
- Degrees of freedom = \(n - 1 = 199\)

Let's go step by step.

### (a) 95% Confidence Limits

1. **Determine \(\alpha\)**:
   \[
   \alpha = 1 - 0.95 = 0.05
   \]

2. **Find the Chi-Square critical values**:
   For a 95% confidence level, \(\alpha/2 = 0.025\) and \(1-\alpha/2 = 0.975\).

   We need to find \(\chi^2_{0.025}\) and \(\chi^2_{0.975}\) for 199 degrees of freedom.

   Using Chi-Square tables or a calculator:
   \[
   \chi^2_{0.025, 199} \approx 238.99 \quad \text{and} \quad \chi^2_{0.975, 199} \approx 165.05
   \]

3. **Compute the confidence interval**:
   \[
   \text{Lower limit} = \sqrt{\frac{(199) \times (100^2)}{238.99}} = \sqrt{\frac{1990000}{238.99}} \approx \sqrt{8330.87} \approx 91.26 \text{ hours}
   \]
   \[
   \text{Upper limit} = \sqrt{\frac{(199) \times (100^2)}{165.05}} = \sqrt{\frac{1990000}{165.05}} \approx \sqrt{12055.77} \approx 109.81 \text{ hours}
   \]

   Therefore, the 95% confidence interval for the standard deviation is approximately \(91.26\) hours to \(109.81\) hours.

### (b) 99% Confidence Limits

1. **Determine \(\alpha\)**:
   \[
   \alpha = 1 - 0.99 = 0.01
   \]

2. **Find the Chi-Square critical values**:
   For a 99% confidence level, \(\alpha/2 = 0.005\) and \(1-\alpha/2 = 0.995\).

   Using Chi-Square tables or a calculator:
   \[
   \chi^2_{0.005, 199} \approx 246.90 \quad \text{and} \quad \chi^2_{0.995, 199} \approx 152.97
   \]

3. **Compute the confidence interval**:
   \[
   \text{Lower limit} = \sqrt{\frac{(199) \times (100^2)}{246.90}} = \sqrt{\frac{1990000}{246.90}} \approx \sqrt{8060.43} \approx 89.77 \text{ hours}
   \]
   \[
   \text{Upper limit} = \sqrt{\frac{(199) \times (100^2)}{152.97}} = \sqrt{\frac{1990000}{152.97}} \approx \sqrt{13010.33} \approx 114.07 \text{ hours}
   \]

   Therefore, the 99% confidence interval for the standard deviation is approximately \(89.77\) hours to \(114.07\) hours.

### Summary:

- **95% Confidence Interval**: \(91.26\) hours to \(109.81\) hours.
- **99% Confidence Interval**: \(89.77\) hours to \(114.07\) hours.

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Would you like to dive deeper into any part of this solution, or do you have any other questions?

### Relative Questions:

1. What is the significance of the Chi-Square distribution in calculating confidence intervals for standard deviation?
2. How would the confidence intervals change if the sample size were larger or smaller?
3. Why do we use different critical values for 95% and 99% confidence levels?
4. How does the shape of the Chi-Square distribution affect the confidence interval?
5. Can you explain the relationship between the sample standard deviation and the population standard deviation?

### Tip:
When using Chi-Square tables, always ensure you are looking at the correct degrees of freedom and confidence level to avoid errors in your calculations.

The standard deviation of the lifetimes of a sample of 200 electric light bulbs was computed to be 100 hours. Find (a) 95%, (b) 99% confidence limits for the standard deviation of all such electric light bulbs. Step by step answer.

This page provides a detailed step-by-step solution for calculating the 95% and 99% confidence intervals for the standard deviation of the lifetimes of electric light bulbs, using the Chi-Square distribution. Suitable for college-level statistics students.

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