1 1 slide © 2008 thomson south-western. all rights reserved chapter 11 inferences about population...

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© 2008 Thomson South-Western. All Rights Reserved© 2008 Thomson South-Western. All Rights Reserved

Chapter 11Chapter 11 Inferences About Population Variances Inferences About Population Variances

Inference about a Population VarianceInference about a Population Variance Inferences about Two Population VariancesInferences about Two Population Variances

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Inferences About a Population VarianceInferences About a Population Variance

Chi-Square DistributionChi-Square Distribution Interval EstimationInterval Estimation Hypothesis TestingHypothesis Testing

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Chi-Square DistributionChi-Square Distribution

We can use the chi-square distribution to developWe can use the chi-square distribution to develop interval estimates and conduct hypothesis testsinterval estimates and conduct hypothesis tests about a population variance.about a population variance.

The sampling distribution of (The sampling distribution of (nn - 1) - 1)ss22//22 has a has a chi-chi- square distribution whenever a simple random square distribution whenever a simple random samplesample of size of size nn is selected from a normal population. is selected from a normal population.

The chi-square distribution is based on samplingThe chi-square distribution is based on sampling from a normal population.from a normal population.

The The chi-square distributionchi-square distribution is the sum of is the sum of squaredsquared standardized normal random variables standardized normal random variables such assuch as

((zz11))22+(+(zz22))22+(+(zz33))22 and so on. and so on.

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Examples of Sampling Distribution of (Examples of Sampling Distribution of (nn - - 1)1)ss22//22

00

With 2 degreesWith 2 degrees of freedomof freedomWith 2 degreesWith 2 degrees of freedomof freedom

2

2

( 1)n s



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2 2 2.975 .025 2 2 2.975 .025

Chi-Square DistributionChi-Square Distribution

For example, there is a .95 probability of For example, there is a .95 probability of obtaining a obtaining a 22 (chi-square) value such that (chi-square) value such that

We will use the notation to denote the We will use the notation to denote the value for the chi-square distribution that value for the chi-square distribution that provides an area of provides an area of to the right of the stated to the right of the stated value. value.

2 2

2 2

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95% of thepossible 2 values 95% of thepossible 2 values

22

00

.025.025

2.025 2.025

.025.025

2.975 2.975

Interval Estimation of Interval Estimation of 22

22 2.975 .0252

( 1)n s

2

2 2.975 .0252

( 1)n s

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( ) ( )

/ ( / )

n s n s

1 12

22

22

1 22

( ) ( )

/ ( / )

n s n s

1 12

22

22

1 22

2 2 2(1 / 2) / 2 2 2 2(1 / 2) / 2

22 2(1 / 2) / 22

( 1)n s

2

2 2(1 / 2) / 22

( 1)n s

Substituting (Substituting (nn – 1) – 1)ss22//22 for the for the 22 we get we get

Performing algebraic manipulation we getPerforming algebraic manipulation we get

There is a (1 – There is a (1 – ) probability of obtaining a ) probability of obtaining a 22 valuevalue

such thatsuch that

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Interval Estimate of a Population VarianceInterval Estimate of a Population Variance


( ) ( )

/ ( / )

n s n s

1 12

22

22

1 22

( ) ( )

/ ( / )

n s n s

1 12

22

22

1 22

where the where the values are based on a chi-squarevalues are based on a chi-square

distribution with distribution with nn - 1 degrees of freedom and - 1 degrees of freedom and

where 1 - where 1 - is the confidence coefficient. is the confidence coefficient.

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Interval Estimation of Interval Estimation of

Interval Estimate of a Population Standard DeviationInterval Estimate of a Population Standard Deviation

Taking the square root of the upper and lowerTaking the square root of the upper and lower

limits of the variance interval provides the confidencelimits of the variance interval provides the confidence

interval for the population standard deviation.interval for the population standard deviation.

2 2

2 2/ 2 (1 / 2)

( 1) ( 1)n s n s

2 2

2 2/ 2 (1 / 2)

( 1) ( 1)n s n s

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Left-Tailed TestLeft-Tailed Test

Hypothesis TestingHypothesis TestingAbout a Population VarianceAbout a Population Variance

22

02

1 ( )n s

22

02

1 ( )n s

where is the hypothesized valuewhere is the hypothesized valuefor the population variancefor the population variance

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•Test StatisticTest Statistic

•HypothesesHypotheses2 2

0 0: H 2 20 0: H

2 20: aH 2 20: aH

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Left-Tailed Test (continued)Left-Tailed Test (continued)


Reject Reject HH00 if if pp-value -value << pp-Value approach:-Value approach:

Critical value approach:Critical value approach:•Rejection RuleRejection Rule

Reject Reject HH00 if if 2 2(1 ) 2 2(1 )

where is based on a chi-squarewhere is based on a chi-squaredistribution with distribution with nn - 1 d.f. - 1 d.f.

2(1 ) 2(1 )

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Right-Tailed TestRight-Tailed Test


H02

02: H0

202:

Ha : 202Ha : 202

22

02

1 ( )n s

22

02

1 ( )n s


20 20


•HypothesesHypotheses

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Right-Tailed Test (continued)Right-Tailed Test (continued)


Reject Reject HH00 if if 2 2 2 2

Reject Reject HH00 if if pp-value -value <<

22where is based on a chi-squarewhere is based on a chi-square

distribution with distribution with nn - 1 d.f. - 1 d.f.

pp-Value approach:-Value approach:


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Two-Tailed TestTwo-Tailed Test


22

02

1 ( )n s

22

02

1 ( )n s


20 20



Ha : 202Ha : 202

H02

02: H0

202:

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Two-Tailed Test (continued)Two-Tailed Test (continued)





2 2 2 2(1 / 2) / 2 or 2 2 2 2(1 / 2) / 2 or Reject Reject HH00 if if

where are based on awhere are based on achi-square distribution with chi-square distribution with nn - 1 d.f. - 1 d.f.

2 2(1 / 2) / 2 and 2 2(1 / 2) / 2 and

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One-Tailed TestOne-Tailed Test



Hypothesis Testing About theHypothesis Testing About theVariances of Two PopulationsVariances of Two Populations

Denote the population providing theDenote the population providing thelarger sample variance as population 1.larger sample variance as population 1.

2 20 1 2: H 2 20 1 2: H

2 21 2: aH 2 21 2: aH

21

22

sFs

21

22

sFs

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One-Tailed Test (continued)One-Tailed Test (continued)


where the value of where the value of FF is based on anis based on an

F F distribution with distribution with nn11 - 1 (numerator) - 1 (numerator)

and and nn2 2 - 1 (denominator) d.f.- 1 (denominator) d.f.




Reject Reject HH00 if if FF >> FF

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Two-Tailed TestTwo-Tailed Test




H0 12

22: H0 1

222:

Ha : 12

22Ha : 1

222

Denote the population providing theDenote the population providing thelarger sample variance as population 1.larger sample variance as population 1.

21

22

sFs

21

22

sFs

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Two-Tailed Test (continued)Two-Tailed Test (continued)

Reject Reject HH00 if if pp-value -value << pp-Value approach:-Value approach:



Reject Reject HH00 if if FF >> FF/2/2

where the value of where the value of FF/2 /2 is based on anis based on an

F F distribution with distribution with nn11 - 1 (numerator) - 1 (numerator)

and and nn2 2 - 1 (denominator) d.f.- 1 (denominator) d.f.

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