Two-Sample F-Test for Population Variances

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Introduction

 

The aim of this test is to compare two population variances, image002t and image004t, based on two independent random samples, image006t and image008t, of size image010t and image012t, respectively.

 

 

Method

 

The following method assumes that the two independent random samples, image006t and image008t, of size image010t and image012t, respectively, come from two normally distributed populations, image022t and image024t.

 

F-Test

 

If the samples under consideration are normally distributed in their respective populations, then the sample statistic,

 

image026t,

 

has an F-distribution with image028t degrees of freedom for the numerator, and image030t degrees of freedom for the denominator, where image032t and image034t are the unbiased sample variances. If the population variances, image002t and image004t, are assumed equal, then the sample statistic,

 

image038t,

 

has also an F-distribution with image028t and image030t degrees of freedom.

 

 

Hypotheses

 

The null hypothesis takes the form image014t, with the following alternative hypotheses:

 

§image016t for two-tailed tests,
§image018t for left-tailed tests, and
§image020t for right-tailed tests.

 

Refer to Alternative Hypothesis for additional information of this subject.

 

Output Format

 

Two-Sample:

<The two-Sample Number>

 

Sample:

1

2

Observations:

<Observations in Sample>

<Observations in Sample>

Sample Mean:

<The Sample Mean>

<The Sample Mean>

Sample Variance:

<The Sample Variance>

<The Sample Variance>

Degrees of Freedom:

<Degrees of Freedom>

<Degrees of Freedom>

Significance Level:

<The Significance Level>

 

F:

<F>

 

Left-Tailed P(F):

<The Left-Tailed P-Value of F>

 

Left-Tailed F Critical:

<The Left-Tailed Critical Point>

 

Left-Tailed Test:

"[Do not ]Reject Ho"

 

Right-Tailed P(F):

<The Right-Tailed P-Value of F>

 

Right-Tailed F Critical:

<The Right-Tailed Critical Point>

 

Right-Tailed Test:

"[Do not ]Reject Ho"

 

Two-Tailed (Left) 2P(F):

<Two-Tailed Left P-Value of F, doubled>

 

Two-Tailed (Right) 2P(F):

<Two-Tailed Right P-Value of F, doubled>

 

Two-Tailed (Left) F Critical:

<Two-Tailed Left Critical Point>

 

Two-Tailed (Right) F Critical:

<Two-Tailed Right Critical Point>

 

Two-Tailed Test:

"[Do not ]Reject Ho"