This set of Probability and Statistics Multiple Choice Questions & Answers (MCQs) focuses on “F-Distribution”.
1. The mean of the f – distribution is equal to ___________
a) v2 / (v2 – 2) for v2 > 2
b) v2 / (v2 – 2)2 for v2 > 2
c) v2 / (v2 – 2)3 for v2 > 2
d) v2 / (v2 – 2)-1 for v2 > 2
Explanation: The mean of the distribution is equal to v2 / (v2 – 2) for v2 > 2. v2 denotes the degree of freedom of F-Distribution.
2. Variance is equal to [(v1 + v2 – 2)] / [v1 * (v2 – 2)2 * (v2 – 4)] for v2 > 4 for a f-Distribution.
Explanation: Variance is equal to [2 * \(v_2^2\) * (v1 + v2 – 2)] / [ v1 * (v2 – 2)2 * (v2 – 4)] for v2 > 4 for a f-Distribution where v1 and v2 denote the degrees of freedom of f-Distribution.
3. Which of the following distributions is Continuous?
a) Binomial Distribution
b) Hyper-geometric Distribution
d) Poisson Distribution
Explanation: Binomial, Poisson and Hyper geometric distributions are Discrete Distributions. Only F- Distribution is Continuous Distribution in the given Distributions.
4. Which of the following distributions is used to compare two variances?
a) T – Distribution
b) F – Distribution
c) Normal Distribution
d) Poisson Distribution
Explanation: F – Distribution is used when we require for comparing two variances. It uses a f-Test to compare two values of variances.
5. F-Distribution cannot take negative values.
Explanation: The value of the F-distribution is always positive, or zero. The variances are the square of the deviations and hence cannot assume negative values. Its value lies between 0 and ∞.
6. Find Variance for an F-Distribution with v1=5 and v2=9.
Explanation: For a f – Distribution:
Var(X) = [2 * \(v_2^2\) * (v1 + v2 – 2)] / [v1 * (v2 – 2)2 * (v2 – 4)] for v2 > 4 where v1 and v2 denote the degrees of freedom of f-Distribution.
Hence Var(X) = [2 * 92 * (14 – 2)] / [5 * (9 – 2)2 * (9- 4)] = 1.587.
7. The table shows the standard Deviation and Sample Standard Deviation for both men and women. Find the f statistic considering the Men population in numerator.
|Population||Population Standard Deviation||Sample Standard Deviation|
Explanation: The f -statistic is calculated using the following equation:
f = [s12/σ12]/[s22/σ22] where σ1 is the standard deviation of population 1
s1 is the standard deviation of the sample drawn from population 1
σ2 is the standard deviation of population 2
s1 is the standard deviation of the sample drawn from population 2.
f = (352/302)/(452/502)
f = (1225/900)/(2025/2500)
f = 1.361/0.81 = 1.68.
8. Calculate the value of f-statistic having a cumulative probability of 0.95.
Explanation: The relation between f – statistic and cumulative probability is given as
If f – statistic = f α then, cumulative probability = (1 – α)
Hence, for cumulative probability 0.95
f – statistic = (1 – 0.95) = 0.05.
9. There is only 1 parameter in F-Distribution.
Explanation: There are 2 parameters in F-Distribution v1 and v2. They are called degrees of freedom of F-Distribution.
10. Find the Expectation for a F- Distribution variable with v1 = 7 and v2 = 8.
Explanation: The Expectation for F-Distribution is given as
E(X) = v2 / (v2 – 2) for v2 > 2
Hence, E(X) = 8 / (8-2)
E(X) = 4/3.
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