To find the critical values of the chi-squared distribution, $\chi^2_L$ (left-tail critical value) and $\chi^2_R$ (right-tail critical value), for a given confidence level $c$ and sample size $n$, we follow these steps:

Step 1: Degrees of Freedom

The degrees of freedom (df) is calculated as:

$\text{df} = n - 1$ Given $n = 27$, we have:

$\text{df} = 27 - 1 = 26$

Step 2: Confidence Level

The confidence level $c = 0.80$ means that the central 80% of the chi-squared distribution is retained. Thus, the area in each tail is:

$\frac{1 - c}{2} = \frac{1 - 0.80}{2} = 0.10$

So, the left tail area is 0.10, and the right tail area is also 0.10.

Step 3: Finding Critical Values

We need to find the critical values $\chi^2_L$ and $\chi^2_R$ such that:

• $\chi^2_L$ corresponds to a cumulative area of 0.10 (left tail),
• $\chi^2_R$ corresponds to a cumulative area of 0.90 (right tail).

Using a chi-squared distribution table or statistical software:

• For $\chi^2_L$, with $df = 26$ and a left-tail area of 0.10, we find $\chi^2_L = 16.151$.
• For $\chi^2_R$, with $df = 26$ and a right-tail area of 0.90, we find $\chi^2_R = 37.652$.

Final Answer:

$\chi^2_L = 16.151 \quad \text{and} \quad \chi^2_R = 37.652$

Would you like further explanation, or do you have any questions?

---

Here are 5 related questions to consider:

1. How do you interpret confidence intervals using the chi-squared distribution?
2. What are some applications of the chi-squared test in statistics?
3. How does changing the confidence level affect $\chi^2_L$ and $\chi^2_R$?
4. What is the relationship between sample size and the degrees of freedom in a chi-squared distribution?
5. How can we use a chi-squared table to approximate critical values?

Tip: The chi-squared distribution becomes more symmetric as the degrees of freedom increase. For small degrees of freedom, the distribution is highly skewed.