tests for variances

SADC Course in Statistics

Tests for Variances

(Session 11)

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Learning Objectives

By the end of this session, you will be able to:

• identify situations where testing for a population variance, or comparing variances, may be applicable

• conduct a chi-square test for a single population variance or an F-test for comparing two population variances

• interpret results of tests concerning population variances


Session ContentsIn this session you will• be shown why in some applications the

study of population means alone is not adequate

• be introduced to the chi-square and F tests for testing one population variance and comparing two population variances respectively

• see examples of the applicability of these tests


Is the packing machine working properly?• Suppose people have lodged complaints

about the weight of the 12.5 Kg mealie-meal bags.

• A consultant took a sample of mealie-meal bags and did not find any problem with the average weight. That is, she could not reject the null hypothesis that the population mean weight = 12.5 Kg

What could be the problem?


• Although the mean is OK in the above example, there could be a problem with the variance

• Packaging plants are designed to operate within certain specified precision

• Ideally it would be desirable to have the machine pack exactly 12.5 Kg in every bag but this is practically impossible. So a certain pre-specified variation is tolerated

Why study variance?


• After years of operation it is always important to check whether the machine variation is still at the initially set level of precision (say )

• This implies testing the hypothesis

against the alternative

20

20 : H

20

20

21 : H

Testing for a single variance

2


• A similar problem could occur if a factory manager is considering whether to buy packaging Machine A or Machine B.

• During test runs, Machine A produced sample variance while Machine B produced sample variance .

Question:

• Are these variances significantly different?

2As

2Bs

Comparing variances


• Suppose the population variances for weights of mealie-meal bags packaged from machines A and B are respectively

and .

• We can answer the question concerning whether the variances are different by testing the null hypothesis

against the alternative .

We will return to this later in the session.

220 : BAH

2A

221 : BAH

2B

Test for comparing variances


Other applicationsOther applications where testing for variancemay be important includes the following:• Foreign exchange stability is important in

any economy. Too much variation of a currency is not good.

• Price stability of other commodities is also important.

Question:Can you name other possible areas of application where testing that the variation remains stable at a pre-set value is important?


The chi-square test

• This test applies when we want to test for a single variance.

• The null hypothesis is of the form

• Need to test this against the alternative

• The test is based on the comparison between and using the ratio

20

20 : H

20

21 : H

2s 20

20

2)1(

sn


• Calculate the chi-square test statistic

Under H0, this is known to have a chi-square distribution with n-1 degrees of freedom.

• Compare this with chi-square tables, or use statistics software to get the p-value.

• Here, p-value = where is a chi-square random variable.

22

20

1( n )sX

2 21nP( X ) 2

1n

Conducting the test


Form of Chi-square distribution

Value of calculated test-statistic

Shaded area represent the p-value

21n


Back to Example

• Suppose the mealie-meal packaging machine is designed to operate with precision of .

• Suppose that data from a sample of 12 mealie-meal bags gave .

• Does the data indicate a significant increase in the variation?

2 20 0 0016. Kg

2 20 0025s . Kg


Test computations and results

• The calculated chi-square value

• The p-value (based on a chi-square with 11 d.f.) is

indicating no significant increase in the variance.

22

20

11 0 00251 17 2

0 0016

s * .X ( n ) .

.

143.0)2.17( 2 P


The F-test• The F-test is used for comparing two

variances, say and .

• The hypothesis being tested is

with either a one-sided alternative ; ;

or a two-sided alternative

220 : BAH

221 : BAH

221 : BAH

2A 2

B

2 21 A BH :


The F-test• The null hypothesis is rejected, for large

values of the F-statistic below, in the case of a one-sided test

2

1 1 2

1

1A B

A

An ,n

B

B

sn

F .sn

• For a 2-sided test, need to pay attention to both sides of the F-distribution (see below).


Example of an F-distribution

99.234.0

1% region (0.5% x2)

24 19,F

0.49 2.11

To use just the upper tail value, ensure F-ratio is calculated so it is >1, then use upper tail of the 2½% F-tabled value when testing at 5% significance.


Numerical Example• Suppose 20 items produced on test trial of

Machine A gave

while 27 items produced by Machine B gave

• Does the data provide evidence that the working precision of the two machines are significantly different?

0016.02 Bs

0035.02 As


Computations

• The value of the F-statistic is

• The p-value is 0.01 (from statistics software).

This indicate a significant difference in variance.

2

1 1 2

0 00351 19 2 990 0016

261A B

A

An ,n

B

B

s .nF .

.sn


Tests for comparing several variances• Levene’s test – is robust in the face of

departures from normality. It is used automatically in some software before conducting other tests which are based on the assumption of equal variance

• Bartlett’s test – based on a chi-square statistic. The test is dependent on meeting the assumption of normality. It is therefore weaker than Levene’s test.


References

• Gallagher, J. (2006) The F-test for comparing two normal variances: correct and incorrect calculation of the two-sided p-value. Teaching Statistics, 28, 58-60.

(this gives an example to show that some statistics software packages can give incorrect p-values for F-values close to 1.)


Some practical work follows…

tests for variances

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