Question

Let $U \sim F_{5,8}$ and $V \sim F_{8,5}$. If $P[U > 3.69] = 0.05$, then the value of $c$ such that $P[V > c] = 0.95$ equals ................ (round off to two decimal places).
 

Hint:

Remember: If $U \sim F_{v_1, v_2}$, then its reciprocal $1/U \sim F_{v_2, v_1}$. Use this relationship to convert right-tail to left-tail probabilities.

Answer 1

Correct Answer: 0.25

Step 1: Relationship between $F$-distributions.
If $U \sim F_{v_1, v_2}$, then $\frac{1}{U} \sim F_{v_2, v_1}$. Hence, since $V \sim F_{8,5}$ and $U \sim F_{5,8}$, \[ V = \frac{1}{U} \text{ in distribution sense.} \]

Step 2: Find critical value correspondence.
\[ P(U > 3.69) = 0.05 \implies P\left(\frac{1}{U} < \frac{1}{3.69}\right) = 0.05. \] For $V \sim F_{8,5}$, \[ P(V < 1/3.69) = 0.05 $\Rightarrow$ P(V > 1/3.69) = 0.95. \]

Step 3: Compute numerical value.
\[ c = \frac{1}{3.69} = 0.271. \]

Step 4: Round off.
\[ \boxed{c = 0.27.} \]