To find the degrees of freedom (df), the critical chi-square values $\chi^2_L$ and $\chi^2_R$, and the confidence interval estimate for $\sigma$ (the population standard deviation), follow these steps:

Step 1: Find the degrees of freedom (df)

The degrees of freedom are calculated using the formula: $df = n - 1$ where $n$ is the sample size. Given $n = 25$, $df = 25 - 1 = 24$

Step 2: Find the critical values $\chi^2_L$ and $\chi^2_R$

For a 90% confidence interval, the significance level $\alpha = 1 - 0.90 = 0.10$. Since this is a two-tailed test, each tail will have an area of $\alpha/2 = 0.05$.

From the chi-square table, look up the values for:

• $\chi^2_L$ at $\alpha/2 = 0.05$ with $df = 24$
• $\chi^2_R$ at $1 - \alpha/2 = 0.95$ with $df = 24$

Using the table, you get: $\chi^2_L = 13.848 \quad \text{and} \quad \chi^2_R = 36.415$

Step 3: Confidence Interval for $\sigma$

The confidence interval for the population standard deviation $\sigma$ is given by: $\left( \sqrt{\frac{(n - 1) s^2}{\chi^2_R}}, \sqrt{\frac{(n - 1) s^2}{\chi^2_L}} \right)$ where $n = 25$, $s = 0.28$, $\chi^2_L = 13.848$, and $\chi^2_R = 36.415$.

First, calculate $(n - 1)s^2$: $(n - 1)s^2 = 24 \times (0.28)^2 = 24 \times 0.0784 = 1.8816$

Now, find the confidence interval: $\text{Lower bound} = \sqrt{\frac{1.8816}{36.415}} = \sqrt{0.0517} = 0.2274$ $\text{Upper bound} = \sqrt{\frac{1.8816}{13.848}} = \sqrt{0.1359} = 0.3687$

Final Answer:

• Degrees of freedom (df): 24
• Critical values: $\chi^2_L = 13.848$, $\chi^2_R = 36.415$
• Confidence interval for $\sigma$: $(0.2274, 0.3687)$

Would you like further details or have any questions?

5 Related Questions:

1. How do you interpret the confidence interval for $\sigma$ in this context?
2. What happens to the confidence interval if we increase the sample size?
3. How would the calculation change if the confidence level was 95%?
4. What is the relationship between the sample standard deviation and the population standard deviation?
5. Why do we assume a normal distribution when calculating chi-square values?

Tip:

For higher confidence levels, the critical values $\chi^2_L$ and $\chi^2_R$ will spread out, leading to a wider confidence interval.