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Midterm Exam 2 Review (Nonadaptive)
Question 7 of 16 (1 point)|Question Attempt: 2 of Unlimited

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Question 7
Suppose that χ2 follows a chi-square distribution with 21 degrees of freedom.
Use the ALEKS calculator to answer the following.
(a) Compute P≤15≤χ229. Round your answer to at least three decimal places.

=P≤15≤χ229
(b) Find k such that =P>χ2k0.025. Round your answer to at least two decimal places.

=k

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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.

Suppose that χ2 follows a chi-square distribution with 21 degrees of freedom. (a) Compute P(15 ≤ χ2 ≤ 29). (b) Find k such that P(χ2 > k) = 0.025.

Explore how to calculate the probability for a chi-square distribution with 21 degrees of freedom between two values and find the critical value for a 0.025 upper tail probability. Learn to use the chi-square cumulative distribution and inverse functions for accurate results.

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