To construct the confidence interval for the population standard deviation based on the sample standard deviation, we need to use the Chi-square distribution. Given the parameters:

• $n = 20$ (sample size)
• $s = 3.8$ (sample standard deviation)
• $c = 0.8$ (confidence level)

Step 1: Determine Degrees of Freedom

The degrees of freedom $df = n - 1 = 20 - 1 = 19$.

Step 2: Find Chi-Square Critical Values

For a confidence level $c = 0.8$, we want the middle 80% of the chi-square distribution, which leaves 10% in each tail. Therefore:

• The lower critical value $\chi^2_{\alpha/2}$ is found at the 0.10 percentile.
• The upper critical value $\chi^2_{1 - \alpha/2}$ is found at the 0.90 percentile.

Using a chi-square table or calculator for 19 degrees of freedom:

• $\chi^2_{0.10, 19} \approx 10.117$
• $\chi^2_{0.90, 19} \approx 30.144$

Step 3: Calculate the Confidence Interval

The confidence interval for the population standard deviation $\sigma$ is given by: $\sqrt{\frac{(n - 1) s^2}{\chi^2_{1 - \alpha/2}}} < \sigma < \sqrt{\frac{(n - 1) s^2}{\chi^2_{\alpha/2}}}$

Substitute the values: $\sqrt{\frac{(20 - 1) \cdot (3.8)^2}{30.144}} < \sigma < \sqrt{\frac{(20 - 1) \cdot (3.8)^2}{10.117}}$

$\sqrt{\frac{19 \cdot 14.44}{30.144}} < \sigma < \sqrt{\frac{19 \cdot 14.44}{10.117}}$

Calculating each side:

• Lower bound: $\sqrt{\frac{274.36}{30.144}} \approx \sqrt{9.1} \approx 3.0$
• Upper bound: $\sqrt{\frac{274.36}{10.117}} \approx \sqrt{27.12} \approx 5.2$

Answer:

The 80% confidence interval for the population standard deviation $\sigma$ is approximately: $3.0 < \sigma < 5.2$

Would you like further explanation on any part of this process?

Here are some related questions for further exploration:

1. How does changing the confidence level affect the width of the confidence interval?
2. What would the confidence interval be if the sample size increased?
3. Why is the chi-square distribution used for confidence intervals of standard deviation?
4. How do we interpret the confidence interval in a practical context?
5. What assumptions are required for this interval to be valid?

Tip: When using chi-square tables, ensure you choose the correct degrees of freedom and the appropriate percentiles for your confidence level.