1 1 Slide Chapter 11 Inferences About Population Variances n Inference about a Population Variance n...

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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

2 2 Slide Slide

Inferences About a Population VarianceInferences About a Population Variance

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

3 3 Slide Slide

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.

4 4 Slide Slide

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



5 5 Slide Slide

6 6 Slide Slide

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

7 7 Slide Slide

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

8 8 Slide Slide


( ) ( )

/ ( / )

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

9 9 Slide Slide

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.

10 10 Slide Slide

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

11 11 Slide Slide

Buyer’s Digest rates thermostatsBuyer’s Digest rates thermostats

manufactured for home temperaturemanufactured for home temperature

control. In a recent test, 10 thermostatscontrol. In a recent test, 10 thermostats

manufactured by ThermoRite weremanufactured by ThermoRite were

selected and placed in a test room thatselected and placed in a test room that

was maintained at a temperature of 68was maintained at a temperature of 68ooF.F.

The temperature readings of the ten thermostats The temperature readings of the ten thermostats areare

shown on the next slide. shown on the next slide.


Example: Buyer’s Digest (A)Example: Buyer’s Digest (A)

12 12 Slide Slide


We will use the 10 readings below toWe will use the 10 readings below todevelop a 95% confidence intervaldevelop a 95% confidence intervalestimate of the population variance.estimate of the population variance.

Example: Buyer’s Digest (A)Example: Buyer’s Digest (A)

Temperature Temperature 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.267.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2

ThermostatThermostat 1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10

13 13 Slide Slide

Degrees

of Freedom .99 .975 .95 .90 .10 .05 .025 .01

5 0.554 0.831 1.145 1.610 9.236 11.070 12.832 15.0866 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.8127 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.4758 1.647 2.180 2.733 3.490 13.362 15.507 17.535 20.0909 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666

10 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209

Area in Upper TailDegrees

of Freedom .99 .975 .95 .90 .10 .05 .025 .01

5 0.554 0.831 1.145 1.610 9.236 11.070 12.832 15.0866 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.8127 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.4758 1.647 2.180 2.733 3.490 13.362 15.507 17.535 20.0909 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666

10 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209

Area in Upper Tail


Selected Values from the Chi-Square Distribution TableSelected Values from the Chi-Square Distribution Table

Our Our value value

Our Our value value

2.9752.975

For For nn - 1 = 10 - 1 = 9 d.f. and - 1 = 10 - 1 = 9 d.f. and = .05= .05

14 14 Slide Slide


22

00

.025.025

2

2.0252

( 1)2.700

n s

2

2.0252

( 1)2.700

n s

Area inArea inUpper TailUpper Tail

= .975= .975

2.7002.700


15 15 Slide Slide

Degrees

of Freedom .99 .975 .95 .90 .10 .05 .025 .01

5 0.554 0.831 1.145 1.610 9.236 11.070 12.832 15.0866 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.8127 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.4758 1.647 2.180 2.733 3.490 13.362 15.507 17.535 20.0909 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666

10 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209


of Freedom .99 .975 .95 .90 .10 .05 .025 .01

5 0.554 0.831 1.145 1.610 9.236 11.070 12.832 15.0866 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.8127 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.4758 1.647 2.180 2.733 3.490 13.362 15.507 17.535 20.0909 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666

10 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209

Area in Upper Tail


Selected Values from the Chi-Square Distribution TableSelected Values from the Chi-Square Distribution Table


Our Our value value

Our Our value value 2

.025 2

.025

16 16 Slide Slide

22

00

.025.025

2.7002.700


nn - 1 = 10 - 1 = 9 degrees of freedom and - 1 = 10 - 1 = 9 degrees of freedom and = .05= .05

2

2

( 1)2.700 19.023

n s

2

2

( 1)2.700 19.023

n s

19.02319.023

Area in UpperArea in UpperTail = .025Tail = .025Area in UpperArea in UpperTail = .025Tail = .025

17 17 Slide Slide

Sample variance Sample variance ss22 provides a point estimate of provides a point estimate of 22..

sx xni2

2

16 39

70

( ) .

.sx xni2

2

16 39

70

( ) .

.

( )..

( )..

10 1 7019 02

10 1 702 70

2 ( ).

.( ).

.10 1 70

19 0210 1 70

2 702


.33 .33 << 2 2 << 2.33 2.33

A 95% confidence interval for the population A 95% confidence interval for the population variance is given by:variance is given by:

18 18 Slide Slide

Hypothesis TestingHypothesis TestingAbout a Population VarianceAbout a Population Variance

Lower-tail test:

H0: σ2 σ02

H1: σ2 < σ02

Upper-tail test:

H0: σ2 ≤ σ02

H1: σ2 > σ02

Two-tail test:

H0: σ2 = σ02

H1: σ2 ≠ σ02

/2 /2

Reject H0 ifReject H0 if Reject H0 if

or

2, 1n χ

2,1 1n χ

2,1 1n 2/χ

2, 1n 2/χ

2,1 1n

21n χχ

2, 1n

21n χχ

2, 1n

21n 2/ χχ

2,1 1n

21n 2/ χχ

22

02

1

( )n s

22

02

1

( )n s

19 19 Slide Slide

Recall that Buyer’s Digest is ratingRecall that Buyer’s Digest is rating

ThermoRite thermostats. Buyer’s DigestThermoRite thermostats. Buyer’s Digest

gives an “acceptable” rating to a thermo-gives an “acceptable” rating to a thermo-

stat with a temperature variance of 0.5stat with a temperature variance of 0.5

or less.or less.


Example: Buyer’s Digest (B)Example: Buyer’s Digest (B)

We will conduct a hypothesis test (withWe will conduct a hypothesis test (with

= .10) to determine whether the ThermoRite= .10) to determine whether the ThermoRite

thermostat’s temperature variance is “acceptable”.thermostat’s temperature variance is “acceptable”.

20 20 Slide Slide


Using the 10 readings, we willUsing the 10 readings, we will

conduct a hypothesis test (with conduct a hypothesis test (with = .10) = .10)

to determine whether the ThermoRiteto determine whether the ThermoRite

thermostat’s temperature variance isthermostat’s temperature variance is

““acceptable”.acceptable”.

Example: Buyer’s Digest (B)Example: Buyer’s Digest (B)



21 21 Slide Slide

HypothesesHypotheses2

0 : 0.5H 20 : 0.5H

2: 0.5aH 2: 0.5aH


Reject Reject HH00 if if 22 >> 14.684 14.684

Rejection RuleRejection Rule

22 22 Slide Slide

Degrees

of Freedom .99 .975 .95 .90 .10 .05 .025 .01

5 0.554 0.831 1.145 1.610 9.236 11.070 12.832 15.0866 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.8127 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.4758 1.647 2.180 2.733 3.490 13.362 15.507 17.535 20.0909 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666

10 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209


of Freedom .99 .975 .95 .90 .10 .05 .025 .01

5 0.554 0.831 1.145 1.610 9.236 11.070 12.832 15.0866 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.8127 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.4758 1.647 2.180 2.733 3.490 13.362 15.507 17.535 20.0909 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666

10 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209

Area in Upper TailSelected Values from the Chi-Square Distribution TableSelected Values from the Chi-Square Distribution Table



Our Our value value Our Our value value 2

.10 2

.10

23 23 Slide Slide

22

00 14.68414.684

Area in UpperArea in UpperTail = .10Tail = .10Area in UpperArea in UpperTail = .10Tail = .10


Rejection RegionRejection Region

2 22

2

( 1) 9.5

n s s

2 2

22

( 1) 9.5

n s s

Reject Reject HH00Reject Reject HH00

24 24 Slide Slide

Test StatisticTest Statistic

2 9(.7)12.6

.5 2 9(.7)

12.6.5


Because Because 22 = 12.6 is less than 14.684, we cannot = 12.6 is less than 14.684, we cannot

reject reject HH00. The sample variance . The sample variance ss22 = .7 is insufficient = .7 is insufficient

evidence to conclude that the temperature varianceevidence to conclude that the temperature variance

for ThermoRite thermostats is unacceptable.for ThermoRite thermostats is unacceptable.

ConclusionConclusion

The sample variance The sample variance s s 22 = 0.7 = 0.7

25 25 Slide Slide

Using the Using the pp-Value-Value

• The sample variance of The sample variance of s s 22 = .7 is = .7 is insufficient evidence to conclude that theinsufficient evidence to conclude that the temperature variance is unacceptable (>.5).temperature variance is unacceptable (>.5).

• Because the Because the pp –value > –value > = .10, we = .10, we cannot reject the null hypothesis.cannot reject the null hypothesis.

• The rejection region for the ThermoRite The rejection region for the ThermoRite thermostat example is in the upper tail; thus, thethermostat example is in the upper tail; thus, the appropriate appropriate pp-value is less than .90 (-value is less than .90 (22 = 4.168) = 4.168) and greater than .10 (and greater than .10 (22 = 14.684). = 14.684).


A precise A precise pp-value-valuecan be found usingcan be found usingMinitab or Excel.Minitab or Excel.

A precise A precise pp-value-valuecan be found usingcan be found usingMinitab or Excel.Minitab or Excel.

26 26 Slide Slide

27 27 Slide Slide

28 28 Slide Slide

Hypothesis Tests for Two VariancesHypothesis Tests for Two Variances

Tests for TwoPopulation Variances

F test statistic

H0: σx2 = σy

2

H1: σx2 ≠ σy

2Two-tail test

Lower-tail test

Upper-tail test

H0: σx2 σy

2

H1: σx2 < σy

2

H0: σx2 ≤ σy

2

H1: σx2 > σy

2

Goal: Test hypotheses about two population variances

The two populations are assumed to be independent and normally distributed

29 29 Slide Slide

FF 分布分布

FF 分布定义为分布定义为 :: 设设 XX 、、 YY 为两个独立的随机变量为两个独立的随机变量，， XX 服从自由度为服从自由度为 mm 的卡方分布，的卡方分布， YY 服从自由服从自由度为度为 nn 的卡方分布，这的卡方分布，这 2 2 个独立的卡方分布被个独立的卡方分布被各自的自由度除以后的比率这一统计量的分布即各自的自由度除以后的比率这一统计量的分布即：：

F=F= （（ x/mx/m ）） /(y/n)/(y/n)

服从自由度为（服从自由度为（ m,n)m,n) 的的 F-F- 分布，上式分布，上式 FF 服从服从第一自由度为第一自由度为 mm ，第二自由度为，第二自由度为 nn 的的 FF 分布分布

30 30 Slide Slide

FF 分布分布

FF 分布的性质分布的性质 11 、它是一种非对称分布；、它是一种非对称分布； 22 、它有两个自由度，即、它有两个自由度，即 n1 -1n1 -1 和和 n2-1n2-1 ，相应的，相应的

分布记为分布记为 FF （（ n1 –1n1 –1 ，， n2-1n2-1 ），）， n1 –1n1 –1 通常通常称为分子自由度，称为分子自由度， n2-1n2-1 通常称为分母自由度；通常称为分母自由度；

33 、、 FF 分布是一个以自由度分布是一个以自由度 n1 –1n1 –1 和和 n2-1n2-1 为参数为参数的分布族，不同的自由度决定了的分布族，不同的自由度决定了 F F 分布的形状。分布的形状。

44 、、 FF 分布的倒数性质：分布的倒数性质： FFα,df1,df2α,df1,df2=1/F=1/F1-1-α,df2,df1α,df2,df1

31 31 Slide Slide

FF 分布分布

32 32 Slide Slide

Hypothesis Tests for Two VariancesHypothesis Tests for Two Variances


F test statistic

2y

2y

2x

2x

/σs

/σsF

The random variable

Has an F distribution with (nx – 1) numerator degrees of freedom and (ny – 1) denominator degrees of freedom

Denote an F value with 1 numerator and 2

denominator degrees of freedom by

(continued)

33 33 Slide Slide



F test statistic 2y

2x

s

sF

The critical value for a hypothesis test about two population variances is

where F has (nx – 1) numerator degrees of freedom and (ny – 1) denominator degrees of freedom

34 34 Slide Slide

One-Tailed TestOne-Tailed Test

•Test StatisticTest Statistic

•HypothesesHypotheses

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

35 35 Slide Slide

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

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

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.

pp-Value approach:-Value approach:

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


Reject Reject HH00 if if FF >> FF

36 36 Slide Slide

Two-Tailed TestTwo-Tailed Test

•Test StatisticTest Statistic

•HypothesesHypotheses


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

37 37 Slide Slide

虽然我们可以通过以上公式求左侧虽然我们可以通过以上公式求左侧 FF 值，但通常在进行假设检验值，但通常在进行假设检验计算时，只需知道右侧计算时，只需知道右侧 FF 值。进行假设检验时，可以随意值。进行假设检验时，可以随意规定哪个总体是总体规定哪个总体是总体 11 或总体或总体 22 ，我们用总体，我们用总体 11 来代表方差较大来代表方差较大的总体时，只有在右侧才有可能出现的总体时，只有在右侧才有可能出现 HH00 被拒绝的情况。虽然左被拒绝的情况。虽然左侧临界值仍存在，但我们不需要知道它值，因为使用方差较大的侧临界值仍存在，但我们不需要知道它值，因为使用方差较大的总体作为总体总体作为总体 11 ，， ss11

22/s/s2222 往往出现在右侧。往往出现在右侧。

38 38 Slide Slide

Two-Tailed Test (continued)Two-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 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.

39 39 Slide Slide

Decision Rules: Two VariancesDecision Rules: Two Variances

rejection region for a two-tail test is:

F 0

Reject H0Do not reject H0

F 0 /2

Reject H0Do not reject H0

H0: σx2 = σy

2

H1: σx2 ≠ σy

2H0: σx2 ≤ σy

2

H1: σx2 > σy

2

Use sx2 to denote the larger variance.

α1,n1,n yxF

2/α1,n1,n0 yxFF if H Reject

2/α1,n1,n yxF

where sx2 is the larger of the

two sample variances

α1,n1,n0 yxFF if H Reject

40 40 Slide Slide

Buyer’s Digest has conducted theBuyer’s Digest has conducted the

same test, as was described earlier, onsame test, as was described earlier, on

another 10 thermostats, this timeanother 10 thermostats, this time

manufactured by TempKing. Themanufactured by TempKing. The

temperature readings of the tentemperature readings of the ten

thermostats are listed on the next slide. thermostats are listed on the next slide.


Example: Buyer’s Digest (C)Example: Buyer’s Digest (C)

We will conduct a hypothesis test with We will conduct a hypothesis test with = .10 = .10 to seeto seeif the variances are equal for ThermoRite’s if the variances are equal for ThermoRite’s thermostatsthermostatsand TempKing’s thermostats.and TempKing’s thermostats.

41 41 Slide Slide


Example: Buyer’s Digest (C)Example: Buyer’s Digest (C)

ThermoRite SampleThermoRite Sample

TempKing SampleTempKing Sample





42 42 Slide Slide

HypothesesHypotheses

H0 12

22: H0 1

222:

Ha : 12

22Ha : 1

222


Reject Reject HH00 if if FF >> 3.18 3.18

The The FF distribution table (on next slide) shows distribution table (on next slide) shows that withthat withwith with = .10, 9 d.f. (numerator), and 9 d.f. = .10, 9 d.f. (numerator), and 9 d.f. (denominator),(denominator),

FF.05.05 = 3.18. = 3.18.

(Their variances are not equal)(Their variances are not equal)

(TempKing and ThermoRite thermostats(TempKing and ThermoRite thermostatshave the same temperature variance)have the same temperature variance)

Rejection RuleRejection Rule

43 43 Slide Slide

Denominator Area in

Degrees Upper

of Freedom Tail 7 8 9 10 158 .10 2.62 2.59 2.56 2.54 2.46

.05 3.50 3.44 3.39 3.35 3.22.025 4.53 4.43 4.36 4.30 4.10.01 6.18 6.03 5.91 5.81 5.52

9 .10 2.51 2.47 2.44 2.42 2.34.05 3.29 3.23 3.18 3.14 3.01

.025 4.20 4.10 4.03 3.96 3.77.01 5.61 5.47 5.35 5.26 4.96

Numerator Degrees of Freedom

Denominator Area in

Degrees Upper

of Freedom Tail 7 8 9 10 158 .10 2.62 2.59 2.56 2.54 2.46

.05 3.50 3.44 3.39 3.35 3.22.025 4.53 4.43 4.36 4.30 4.10.01 6.18 6.03 5.91 5.81 5.52

9 .10 2.51 2.47 2.44 2.42 2.34.05 3.29 3.23 3.18 3.14 3.01

.025 4.20 4.10 4.03 3.96 3.77.01 5.61 5.47 5.35 5.26 4.96

Numerator Degrees of Freedom

Selected Values from the Selected Values from the FF Distribution Table Distribution Table


44 44 Slide Slide



We We cannotcannot reject reject HH00. . FF = 2.53 < = 2.53 < FF.05.05 = 3.18. = 3.18.There is insufficient evidence to conclude thatThere is insufficient evidence to conclude thatthe population variances differ for the twothe population variances differ for the twothermostat brands.thermostat brands.

ConclusionConclusion

21

22

sFs

21

22

sFs

= 1.768/.700 = 2.53= 1.768/.700 = 2.53

TempKing’s sample variance is 1.768TempKing’s sample variance is 1.768

ThermoRite’s sample variance is .700ThermoRite’s sample variance is .700

45 45 Slide Slide

Determining and Using the Determining and Using the pp-Value-Value


• Because Because = .10, we have = .10, we have pp-value > -value > and therefore and therefore we cannot reject the null hypothesis.we cannot reject the null hypothesis.

• But this is a two-tailed test; after doubling the But this is a two-tailed test; after doubling the upper-uppertail area, the tail area, the pp-value is between .20 and .10. -value is between .20 and .10. (A precise(A precise pp-value can be found using Minitab or Excel.)-value can be found using Minitab or Excel.)

• Because Because FF = 2.53 is between 2.44 and 3.18, the area = 2.53 is between 2.44 and 3.18, the area in the upper tail of the distribution is between .10in the upper tail of the distribution is between .10 and .05.and .05.

Area in Upper TailArea in Upper Tail .10 .05 .025 .01 .10 .05 .025 .01

FF Value (df Value (df11 = 9, df = 9, df22 = 9) 2.44 3.18 4.03 5.35 = 9) 2.44 3.18 4.03 5.35

46 46 Slide Slide

47 47 Slide Slide

48 48 Slide Slide

End of Chapter 11End of Chapter 11

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