The image you uploaded contains a table with four conditions where you need to find the values of $x$ corresponding to certain probabilities for a Chi-Square distribution with 20 degrees of freedom, $\chi^2_{20}$.

Here's how we will solve each one:

Given information:

• $\chi^2_{20}$ refers to a chi-square distribution with 20 degrees of freedom.

The values we need to find from the chi-square table are as follows:

1. $P(\chi^2_{20} \geq x) = 0.005$
2. $P(\chi^2_{20} \geq x) = 0.01$
3. $P(\chi^2_{20} \leq x) = 0.005$
4. $P(\chi^2_{20} \leq x) = 0.01$

Solution approach:

1. For the first two (a and b), we will use the chi-square table to find the upper tail critical value corresponding to the given probability (since $P(\chi^2_{20} \geq x)$ is the tail probability).
2. For the last two (c and d), we will look up the lower tail critical value for the given probabilities (since $P(\chi^2_{20} \leq x)$ is for the left tail).

I will calculate the values for you using these criteria.Here are the values of $x$ for the given probabilities, rounded to three decimal places:

1. $P(\chi^2_{20} \geq x) = 0.005$: $x = 39.997$
2. $P(\chi^2_{20} \geq x) = 0.01$: $x = 37.566$
3. $P(\chi^2_{20} \leq x) = 0.005$: $x = 7.434$
4. $P(\chi^2_{20} \leq x) = 0.01$: $x = 8.260$

These values come from the chi-square distribution with 20 degrees of freedom for the corresponding probabilities.

Do you have any questions or need further details?

Here are 5 related questions to explore further:

1. How does the chi-square distribution change with degrees of freedom?
2. What are some applications of the chi-square distribution?
3. How would the values of $x$ change for different probabilities?
4. Can the chi-square distribution be used for hypothesis testing? How?
5. What is the relationship between the chi-square distribution and the normal distribution?

Tip: Always verify tail probabilities carefully when working with chi-square tables, as upper and lower tails are often switched in different problem contexts.