tests for variances
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Tests for Variances. (Session 11). 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 - PowerPoint PPT PresentationTRANSCRIPT
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
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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
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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?
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• 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?
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• 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
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• 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
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• 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
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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?
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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
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• 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.
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20
1( n )sX
2 21nP( X ) 2
1n
Conducting the test
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Form of Chi-square distribution
Value of calculated test-statistic
Shaded area represent the p-value
21n
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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
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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
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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 :
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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).
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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.
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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
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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
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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.
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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.)
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Some practical work follows…