Student/Statistics/TwoSampleFTest/overview - Maple Help

Student[Statistics][TwoSampleFTest] Overview

overview of the Two Sample F-Test

Description

 • Two Sample F Test is used to test if the ratio of two population variances is equal to the test value. For this test, the two samples do not have to be the same size, but they must be independent. This test is particularly sensitive if the distributions are normal; approximately normal is typically not good enough.
 • Requirements for using Two Sample F Test:
 1 The two studied populations are assumed to be normally distributed.
 2 The samples drawn from the populations are independent.
 • The formula is:

$F=\frac{\mathrm{Variance}\left(X\right)}{\mathrm{\beta }\mathrm{Variance}\left(Y\right)}$

 where $\mathrm{Variance}\left(X\right)$ is the sample variance of the sample drawn from the first population, $\mathrm{Variance}\left(Y\right)$ is the sample variance of the sample drawn from the second population, and $\mathrm{\beta }$ is the test value of the ratio.
 $F$ follows an F-distribution with degrees of freedom equal to $N-1,M-1$, where $N$ is the sample size of $X$, and $M$ is the sample size of $Y$.

Example

Suppose there are two samples drawn from two normally distributed populations as shown in the following table:

 Sample Size Sample Variance Sample One 121 66 Sample Two 61 60

Now use the formula in the previous section to determine if the two populations have the same variance, and if the confidence level is equal to 0.95. $\mathrm{\beta }$ is equal to one in this case.

 1 Determine the null hypothesis:
 Null Hypothesis: The two studied populations have the same variance.
 2 Substitute the information into the formula:
 $f=\frac{66}{1\cdot 60}=1.1$
 ${d}.f=\left(121-1,61-1\right)=\left(120,60\right)$
 3 Compute the p-value:
 $p-\mathrm{value}=2*\mathrm{Probability}\left(F>1.1\right)=0.69078$ (two-tailed)
 $F˜\mathrm{FRatio}\left(120,60\right)$
 4 Draw the conclusion:
 This statistical test does not provide enough evidence to conclude that the null hypothesis is false, so we fail to reject the null hypothesis.