F Distribution and Ratio of Variances

Question 1

What is the F-distribution used for in hypothesis testing?

Answer 1

To test whether two population variances are equal using a ratio of independent chi-square variables.

Question 2

How is the F-statistic defined using chi-square variables X1 ~ χ²(n1) and X2 ~ χ²(n2)?

Answer 2

F = (X1 / n1) / (X2 / n2), which follows an F-distribution with (n1, n2) degrees of freedom.

Question 3

What are the degrees of freedom in the F-distribution F(n1, n2)?

Answer 3

n1 is the numerator degrees of freedom and n2 is the denominator degrees of freedom.

Question 4

What is the mean of the F-distribution when n2 > 2?

Answer 4

E[F] = n2 / (n2 - 2)

Question 5

What condition must be met for the variance of the F-distribution to exist?

Answer 5

n2 must be greater than 4.

Question 6

What is the null hypothesis for testing equality of two variances?

Answer 6

H0: σ1² = σ2²

Question 7

What is the test statistic used to compare two sample variances S1² and S2²?

Answer 7

F = S1² / S2² or F = S2² / S1² (larger variance in numerator)

Question 8

What is the distribution of the F-statistic under the null hypothesis?

Answer 8

F ~ F(n1 - 1, n2 - 1)

Question 9

What is the convention when forming the F-statistic for a two-sample test?

Answer 9

Place the larger sample variance in the numerator to ensure F ≥ 1.

Question 10

In R, how do you compute the 95th percentile critical value of the F-distribution with df1=5 and df2=1?

Answer 10

qf(0.95, 5, 1)

Question 11

When do you reject the null hypothesis in a two-tailed F-test?

Answer 11

If F > F_α/2,df1,df2 using the critical value from the F-distribution.

Question 12

In the example with s1²=15.3 and s2²=19.7, what was the computed F-statistic?

Answer 12

F = 19.7 / 15.3 ≈ 1.288

Question 13

What was the conclusion in the example when F = 1.288 and the critical value was 2.900?

Answer 13

Fail to reject H0; insufficient evidence to say the variances are different.