Chi-Square Test for Population Variances

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Introduction

 

The aim of this test is to compare a population variance, image002r, to a value, image004r, referred here to as the hypothetical variance of the test, based on a random sample, image006r, of size image008r, from the population in question.

 

 

Method

 

If the population from which the sample is drawn is normally distributed, then the test statistic

 

image010r        (1)

 

is chi-square distributed with image012r degrees of freedom.

 

To test the sample variance, image014r, against the hypothetical variance of the test, image004r, we calculate the test statistic

 

image016r,

 

and find the likelihood that image002r is equal to image004r.

 

Refer to Statistical Inference, Decision Theory and Hypothesis Testing for additional information on this subject.

 

 

Hypotheses

 

The null hypothesis takes the form image017r, with the following alternative hypotheses:

 

§image019r for two-tailed tests,
§image021r for left-tailed tests, and
§image023r for right-tailed tests.

 

Refer to Alternative Hypothesis for additional information of this subject.

 

Output Format

 

One-Sample Chi-Square Test for Variances

 

Sample:

<The Sample Number>

Observations:

<Observations in the Sample>

Hypothetical Variance:

<The Hypothetical Variance>

Significance Level:

<The Significance Level>

Mean:

<The Sample Mean>

Sample Variance:

<The Sample Variance>

Chi-Square (X2):

<X2>

Degrees of Freedom:

<Number of Degrees of Freedom>

Left-Tailed P(X2):

<The Left-Tailed P-Value of X2>

Left-Tailed X2 Critical:

<The Left-Tailed Critical Point>

Left-Tailed Test:

"[Do not ]Reject Ho"

Right-Tailed P(X2):

<The Right-Tailed P-Value of X2>

Right-Tailed X2 Critical:

<The Right-Tailed Critical Point>

Right-Tailed Test:

"[Do not ]Reject Ho"

Two-Tailed (Left) 2P(X2):

<The Two-Tailed Left P-Value of X2, doubled>

Two-Tailed (Right) 2P(X2):

<The Two-Tailed Right P-Value of X2, doubled>

Two-Tailed (Left) X2 Critical:

<The Two-Tailed Left Critical Point>

Two-Tailed (Right) X2 Critical:

<The Two-Tailed Right Critical Point>

Two-Tailed Test:

"[Do not ]Reject Ho"