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To solve these questions about the chi-square distribution with 65 degrees of freedom, we need to use the chi-square cumulative distribution function (CDF) and complementary CDF (for the "greater than" probabilities).

Here’s how we can approach each question:

1. **Find \( P(\chi^2 < 85.6) \)**  
   This is the cumulative probability of a chi-square variable with 65 degrees of freedom being less than 85.6. We’ll use the chi-square CDF for this calculation.

2. **Find \( P(\chi^2 > 84.3) \)**  
   This is the probability of a chi-square variable with 65 degrees of freedom being greater than 84.3. For this, we calculate \( P(\chi^2 < 84.3) \) and then use the complementary probability:
   \[
   P(\chi^2 > 84.3) = 1 - P(\chi^2 < 84.3).
   \]

3. **Find \( P(60.7 < \chi^2 < 82.3) \)**  
   This is the probability that the chi-square variable falls between 60.7 and 82.3. To compute this, we use:
   \[
   P(60.7 < \chi^2 < 82.3) = P(\chi^2 < 82.3) - P(\chi^2 < 60.7).
   \]

I’ll calculate each probability value.Here are the calculated probabilities for each question:

1. \( P(\chi^2 < 85.6) = 0.9556 \)
2. \( P(\chi^2 > 84.3) = 0.0541 \)
3. \( P(60.7 < \chi^2 < 82.3) = 0.5557 \)

Would you like any further details on these calculations, or do you have any questions?

Here are some related questions for practice:

1. How would you calculate \( P(\chi^2 < 90) \) with 65 degrees of freedom?
2. What is the impact of increasing the degrees of freedom on the shape of the chi-square distribution?
3. How do we interpret the probability \( P(\chi^2 > x) \) in hypothesis testing?
4. How would you find \( P(50 < \chi^2 < 100) \) for a chi-square distribution with a different number of degrees of freedom?
5. Why is the chi-square distribution skewed, especially with lower degrees of freedom?

**Tip:** When calculating probabilities for "greater than" cases, it’s often easier to compute the CDF for the "less than" scenario and then subtract it from 1.

Consider a chi-square distribution with 65 degrees of freedom. Find P(χ² < 85.6), P(χ² > 84.3), and P(60.7 < χ² < 82.3).

Explore chi-square probability calculations with 65 degrees of freedom, including finding P(χ² < 85.6), P(χ² > 84.3), and P(60.7 < χ² < 82.3). This guide covers the chi-square CDF and complementary probability concepts.

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