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

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Question 7
Suppose that χ2 follows a chi-square distribution with 28 degrees of freedom.
Use the ALEKS calculator to answer the following.
(a) Compute P≤χ242. Round your answer to at least three decimal places.

=P≤χ242
(b) Find k such that =P>χ2k0.05. Round your answer to at least two decimal places.

=k

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Let's break this problem down into its two parts:

### (a) Compute \( P(\chi^2 \leq 42) \) for 28 degrees of freedom.

You are asked to compute the probability that a chi-square random variable with 28 degrees of freedom is less than or equal to 42. This is represented as:

\[
P(\chi^2 \leq 42)
\]

Using a chi-square distribution table or a statistical calculator, you would look up this probability or compute it using a chi-square cumulative distribution function (CDF). 

For \( \chi^2 \) distribution with 28 degrees of freedom, and a chi-square value of 42:

\[
P(\chi^2 \leq 42) \approx 0.914
\]

So, the answer rounded to three decimal places is approximately:

\[
P(\chi^2 \leq 42) = 0.914
\]

### (b) Find \( k \) such that \( P(\chi^2 > k) = 0.05 \) for 28 degrees of freedom.

Here, you are looking for a chi-square value \( k \) such that the upper tail probability (to the right of \( k \)) is 0.05. This can be found by looking up the critical value corresponding to a significance level of 0.05 in a chi-square distribution table or using a statistical calculator for 28 degrees of freedom.

The critical value \( k \) for \( P(\chi^2 > k) = 0.05 \) and 28 degrees of freedom is approximately:

\[
k \approx 41.34
\]

Thus, the value of \( k \) rounded to two decimal places is:

\[
k = 41.34
\]

Would you like further details or have any questions about this?

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Here are 5 related questions for further practice:
1. What is the probability that a chi-square random variable with 15 degrees of freedom is less than 25?
2. Compute \( P(\chi^2 > 50) \) for 30 degrees of freedom.
3. Find \( k \) such that \( P(\chi^2 > k) = 0.01 \) for 20 degrees of freedom.
4. What is the mean and variance of a chi-square distribution with 10 degrees of freedom?
5. How does increasing the degrees of freedom affect the shape of the chi-square distribution?

**Tip:** The chi-square distribution becomes more symmetric as the degrees of freedom increase.

Suppose that χ2 follows a chi-square distribution with 28 degrees of freedom. (a) Compute P(≤ χ2 42). (b) Find k such that P(> χ2 k) = 0.05.

This detailed solution explains how to compute P(χ^2 ≤ 42) and find the critical value k for P(χ^2 > k) = 0.05 with 28 degrees of freedom. Learn about chi-square distribution, probability, and statistical significance for college-level statistics.

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