Question

A) For a chi-squared distribution with v = 4 degrees of freedom, find χ²ₐ such that P(X² > χ²ₐ) = 0.99.

b) For a chi-squared distribution with v = 19 degrees of freedom, find χ²ₐ such that P(X² > χ²ₐ) = 0.025.

c) For a chi-squared distribution with v = 25 degrees of freedom, find χ²ₐ such that P(37.652 < X² < χ²ₐ) = 0.045.

Answer 1

To find χ²ₐ for different degrees of freedom and probabilities, we can refer to the chi-squared distribution table. For a), v = 4 and P(X² > χ²ₐ) = 0.99, the critical value is approximately 13.28. For b), v = 19 and P(X² > χ²ₐ) = 0.025, the critical value is approximately 34.169. For c), v = 25 and P(37.652 < X² < χ²ₐ) = 0.045, we need to find the cumulative probability and subtract it from 0.045 to find the remaining probability, then refer to the table.

Step-by-step explanation:

In order to find χ²ₐ, we need to find the critical value from the chi-squared distribution table. We can find this value by looking up the appropriate degrees of freedom (v) and the desired probability (P(X² > χ²ₐ)).

a) For v = 4 and P(X² > χ²ₐ) = 0.99, the critical value χ²ₐ is approximately 13.28.

b) For v = 19 and P(X² > χ²ₐ) = 0.025, the critical value χ²ₐ is approximately 34.169.

c) For v = 25 and P(37.652 < X² < χ²ₐ) = 0.045, we need to find the cumulative probability for 37.652 and subtract it from 0.045 to find the remaining probability. Then, we can refer to the chi-squared distribution table to find the corresponding critical value.