Question

Let 𝑌~𝐹_4,2 . Then, 𝑃(𝑌≤2) equals

• A. 0.60
• B. 0.62
• C. 0.64
• D. 0.66

Answer 1

The Correct Option is C

To solve this problem, we need to find the cumulative distribution probability \(P(Y \leq 2)\) for a variable \(Y\) that follows an \(F\)-distribution with degrees of freedom \(4\) and \(2\), denoted as \(Y \sim F_{4, 2}\). The \(F\)-distribution is used to determine the variability among multiple groups.

1. First, we recognize that the \(F\hbox{-}distribution\) is a ratio of two chi-square distributions. Here, the degrees of freedom are 4 for the numerator and 2 for the denominator.
2. To compute \(P(Y \leq 2)\), you typically use statistical software or \(F\hbox{-}distribution\) tables, as manual calculation involves complex integration of the probability density function.
3. Using standard statistical software like R, Python, or a statistics calculator, input the degrees of freedom and the value of interest (2 in this case) to find the cumulative probability: 

# Using Python's scipy library
from scipy.stats import f
d1 = 4  # degrees of freedom for the numerator
d2 = 2  # degrees of freedom for the denominator
value = 2
prob = f.cdf(value, d1, d2)
print(prob)
    

The output from the above code or an \(F\hbox{-}table\) lookup would give you \(P(Y \leq 2) = 0.64\).

Hence, the correct answer is that \(P(Y \leq 2) = 0.64\), which corresponds to the option:

0.64

We can conclude that the correct option is indeed 0.64.