Question 1178442: Let Y have a chi-square distribution with 15 degrees of freedom. Find the following probabilities.
(a) P (Y ≤ y0) = 0.025
(b) P (a
(c) P (Y ≥ 22.307).
thank you
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's find the probabilities for a chi-square distribution with 15 degrees of freedom.
**(a) P(Y ≤ y0) = 0.025**
* We are looking for the value y0 such that 2.5% of the chi-square distribution is to the left of it.
* This is the 2.5th percentile of the chi-square distribution with 15 degrees of freedom.
**(b) P(a < Y < b) = 0.90**
* We want to find the values a and b such that 90% of the chi-square distribution lies between them.
* This means we want to exclude the lower 5% and the upper 5%.
* Therefore, 'a' is the 5th percentile, and 'b' is the 95th percentile.
**(c) P(Y ≥ 22.307)**
* We want to find the probability that Y is greater than or equal to 22.307.
* This is the area to the right of 22.307 under the chi-square curve.
**Calculations**
Using the `scipy.stats` library in Python:
```python
import scipy.stats as stats
df = 15 # Degrees of freedom
# (a)
y0 = stats.chi2.ppf(0.025, df)
print(f"(a) y0 = {y0:.4f}")
# (b)
a = stats.chi2.ppf(0.05, df)
b = stats.chi2.ppf(0.95, df)
print(f"(b) a = {a:.4f}, b = {b:.4f}")
# (c)
prob_c = 1 - stats.chi2.cdf(22.307, df)
print(f"(c) P(Y >= 22.307) = {prob_c:.4f}")
```
**Results**
* **(a) y0 = 6.2621**
* **(b) a = 7.2609, b = 24.9958**
* **(c) P(Y ≥ 22.307) = 0.1000**
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