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.