To solve these problems, we need to use the Chi-square distribution table (or formula), which provides critical values of $x$ based on the probability $P(X^2 \geq x)$ or $P(X^2 < x)$, with the degree of freedom specified.

The table provided references a Chi-square distribution with 8 degrees of freedom, since it's denoted by $\chi^2_8$.

Given:

1. $P(X^2_8 \geq x) = 0.025$
2. $P(X^2_8 \geq x) = 0.05$
3. $P(X^2_8 < x) = 0.025$
4. $P(X^2_8 < x) = 0.05$

Approach:

1. For cases like $P(X^2_8 \geq x)$, we need to find the corresponding upper-tail critical value in the chi-square table.
2. For $P(X^2_8 < x)$, we need to find the lower-tail critical value.

Let me calculate these values based on the chi-square distribution table. I will compute the values for $x$.Here are the calculated values of $x$ for each condition:

1. $P(X^2_8 \geq x) = 0.025$ → $x = 17.535$
2. $P(X^2_8 \geq x) = 0.05$ → $x = 15.507$
3. $P(X^2_8 < x) = 0.025$ → $x = 2.180$
4. $P(X^2_8 < x) = 0.05$ → $x = 2.733$

These values are rounded to three decimal places as required.

Would you like a detailed explanation of how these values were derived or how to use the chi-square distribution table?

Here are five related questions to deepen your understanding:

1. What is the general formula for the Chi-square probability density function?
2. How does the degrees of freedom affect the shape of the Chi-square distribution?
3. Can you explain the difference between one-tailed and two-tailed tests in the context of chi-square distributions?
4. How would these values change if the degrees of freedom were increased to 10?
5. How can you use Chi-square distribution in hypothesis testing?

Tip: When referencing a chi-square table, always make sure you know if you're dealing with upper-tail or lower-tail probabilities, as they give different critical values.