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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 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:
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 distribution with 28 degrees of freedom, and a chi-square value of 42:
So, the answer rounded to three decimal places is approximately:
(b) Find such that for 28 degrees of freedom.
Here, you are looking for a chi-square value such that the upper tail probability (to the right of ) 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 for and 28 degrees of freedom is approximately:
Thus, the value of rounded to two decimal places is:
Would you like further details or have any questions about this?
Here are 5 related questions for further practice:
- What is the probability that a chi-square random variable with 15 degrees of freedom is less than 25?
- Compute for 30 degrees of freedom.
- Find such that for 20 degrees of freedom.
- What is the mean and variance of a chi-square distribution with 10 degrees of freedom?
- 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
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