Question 1178445: Let Y have a chi-square distribution with 7 degrees of freedom. Find the following probabilities.
(a) P (Y > y0) = 0.025
(b) P (a < Y < b) = 0.90
(c) P (Y > 1.239).
thank you
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Let's solve these probability problems using the chi-square distribution with 7 degrees of freedom.
**(a) P(Y > y0) = 0.025**
* We need to find the value y0 such that the area to the right of it under the chi-square curve is 0.025.
* This is the 97.5th percentile of the chi-square distribution.
* Using a chi-square table or calculator, we find y0.
**(b) P(a < Y < b) = 0.90**
* We need to find the values a and b such that the area between them under the chi-square curve is 0.90.
* Since we want the middle 90%, we need to exclude the lower 5% and the upper 5%.
* Thus, 'a' is the 5th percentile, and 'b' is the 95th percentile of the chi-square distribution.
**(c) P(Y > 1.239)**
* We need to find the area to the right of 1.239 under the chi-square curve.
* This is equivalent to 1 - P(Y ≤ 1.239), where P(Y ≤ 1.239) is the cumulative distribution function (CDF) at 1.239.
**Calculations**
Using a chi-square calculator or statistical software:
**(a) P(Y > y0) = 0.025**
* y0 = 16.0128 (approximately)
**(b) P(a < Y < b) = 0.90**
* a = 2.1673 (approximately)
* b = 14.0671 (approximately)
**(c) P(Y > 1.239)**
* P(Y ≤ 1.239) = 0.0100 (approximately)
* P(Y > 1.239) = 1 - 0.0100 = 0.9900 (approximately)
**Results**
* **(a) y0 = 16.0128**
* **(b) a = 2.1673, b = 14.0671**
* **(c) P(Y > 1.239) = 0.9900**
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