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

Dennis

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 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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Solution

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

(a) Compute P(χ242)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(χ242)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 χ2\chi^2 distribution with 28 degrees of freedom, and a chi-square value of 42:

P(χ242)0.914P(\chi^2 \leq 42) \approx 0.914

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

P(χ242)=0.914P(\chi^2 \leq 42) = 0.914

(b) Find kk such that P(χ2>k)=0.05P(\chi^2 > k) = 0.05 for 28 degrees of freedom.

Here, you are looking for a chi-square value kk such that the upper tail probability (to the right of kk) 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 kk for P(χ2>k)=0.05P(\chi^2 > k) = 0.05 and 28 degrees of freedom is approximately:

k41.34k \approx 41.34

Thus, the value of kk rounded to two decimal places is:

k=41.34k = 41.34

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


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(χ2>50)P(\chi^2 > 50) for 30 degrees of freedom.
  3. Find kk such that P(χ2>k)=0.01P(\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.

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Math Problem Analysis

Mathematical Concepts

Chi-Square Distribution
Probability
Cumulative Distribution Function

Formulas

P(χ^2 ≤ x)
P(χ^2 > k) = α

Theorems

Chi-Square Distribution Properties
Critical Value for Chi-Square

Suitable Grade Level

College-level Statistics