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Find the value *x* for which: **(Round your answers to 3 decimal places. You may find it useful to reference the appropriate table: chi-square table or *F* table)**

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.

Find the value *x* for which P(χ²₈ ≥ x) = 0.025, P(χ²₈ ≥ x) = 0.05, P(χ²₈ < x) = 0.025, and P(χ²₈ < x) = 0.05, using a chi-square distribution table and rounding answers to three decimal places.

Find the critical values of x for the chi-square distribution with 8 degrees of freedom for P(χ²₈ ≥ x) = 0.025, P(χ²₈ ≥ x) = 0.05, P(χ²₈ < x) = 0.025, and P(χ²₈ < x) = 0.05. This solution explains how to use the chi-square table to find the values, rounding to three decimal places.

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