Let's break down the two parts of your question:

Part (a)

You are asked to find the probability: $P(15 \leq \chi^2 \leq 29)$ where $\chi^2$ follows a chi-square distribution with 21 degrees of freedom.

1. To solve this, we need to find the cumulative probability at $\chi^2 = 29$ and subtract the cumulative probability at $\chi^2 = 15$.
2. Use a chi-square cumulative distribution table, a statistical calculator, or the ALEKS calculator:
   • First, compute $P(\chi^2 \leq 29)$ for 21 degrees of freedom.
   • Then, compute $P(\chi^2 \leq 15)$ for 21 degrees of freedom.
   • Finally, subtract the two values to get $P(15 \leq \chi^2 \leq 29)$.

Part (b)

You are asked to find $k$ such that: $P(\chi^2 > k) = 0.025$ for 21 degrees of freedom.

1. This involves finding the value of $k$ such that the upper tail probability is 0.025.
2. This is the chi-square critical value for the 0.025 upper tail and 21 degrees of freedom.
3. Use a chi-square distribution table or the ALEKS calculator to find the value of $k$. This is typically done using the inverse chi-square function.

Would you like help with the specific steps using an example of the calculations or additional details?

Here are 5 related questions for more practice:

1. What is the probability $P(\chi^2 \leq 10)$ for a chi-square distribution with 21 degrees of freedom?
2. How does changing the degrees of freedom affect the shape of the chi-square distribution?
3. How do you find a chi-square critical value for a given confidence level?
4. What is the difference between the cumulative distribution function and the probability density function for chi-square?
5. How do you compute a confidence interval using the chi-square distribution?

Tip: In hypothesis testing, chi-square distribution is often used for tests involving variance or goodness-of-fit. Understanding the tail probabilities is crucial for setting up critical regions for your tests.