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Basic Business Statistics

11th Edition

Chapter 11 Analysis of Variance

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.

Chap 11-1

Learning Objectives

In this chapter, you learn:

The basic concepts of experimental design How to use one-way analysis of variance to test for differences among the means of several populations (also referred to as "groups" in this chapter) When to use a randomized block design How to use two-way analysis of variance and interpret the interaction effect How to perform multiple comparisons in a one-way analysis of variance, a two-way analysis of variance, and a randomized block design

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Chap 11-2

Chapter Overview

Analysis of Variance (ANOVA) One-Way ANOVA

F-test TukeyKramer Multiple Comparisons Levene Test For Homogeneity of Variance

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Randomized Block Design

Tukey Multiple Comparisons

Two-Way ANOVA

Interaction Effects Tukey Multiple Comparisons

Chap 11-3

General ANOVA Setting

Investigator controls one or more factors of interest Each factor contains two or more levels Levels can be numerical or categorical Different levels produce different groups Think of each group as a sample from a different population Observe effects on the dependent variable Are the groups the same? Experimental design: the plan used to collect the data

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Chap 11-4

Completely Randomized Design

Experimental units (subjects) are assigned randomly to groups

Subjects are assumed homogeneous

Only one factor or independent variable

With two or more levels

Analyzed by one-factor analysis of variance (ANOVA)

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Chap 11-5

One-Way Analysis of Variance

Evaluate the difference among the means of three or more groups

Examples: Accident rates for 1st, 2nd, and 3rd shift Expected mileage for five brands of tires

Assumptions Populations are normally distributed Populations have equal variances Samples are randomly and independently drawn

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Chap 11-6

Hypotheses of One-Way ANOVA

H0 : 1 = 2 = 3 = = c

All population means are equal i.e., no factor effect (no variation in means among groups)

H1 : Not all of the population means are the same

At least one population mean is different i.e., there is a factor effect Does not mean that all population means are different (some pairs may be the same)

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 11-7

One-Way ANOVA

H0 : 1 = 2 = 3 = = c

H1 : Not all j are the same

The Null Hypothesis is True All Means are the same: (No Factor Effect)

1 = 2 = 3

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 11-8

One-Way ANOVA

H0 : 1 = 2 = 3 = = c

H1 : Not all j are the same

The Null Hypothesis is NOT true At least one of the means is different (Factor Effect is present)

(continued)

or

1 = 2 ≠ 3

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

1 ≠ 2 ≠ 3

Chap 11-9

Partitioning the Variation

Total variation can be split into two parts:

SST = SSA + SSW

SST = Total Sum of Squares (Total variation) SSA = Sum of Squares Among Groups (Among-group variation) SSW = Sum of Squares Within Groups (Within-group variation)

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Chap 11-10

Partitioning the Variation

(continued)

SST = SSA + SSW

Total Variation = the aggregate variation of the individual data values across the various factor levels (SST) Among-Group Variation = variation among the factor sample means (SSA) Within-Group Variation = variation that exists among the data values within a particular factor level (SSW)

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Chap 11-11

Partition of Total Variation

Total Variation (SST)

=

Variation Due to Factor (SSA)

+

Variation Due to Random Error (SSW)

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Chap 11-12

Total Sum of Squares

SST = SSA + SSW

SST = ∑∑ ( Xij X)

Where:

j=1 i=1

c

nj

2

SST = Total sum of squares c = number of groups or levels nj = number of observations in group j Xij = ith observation from group j X = grand mean (mean of all data values)

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 11-13

Total Variation

(continued)

SST = ( X 11 X ) + ( X 12 X ) + + ( X cn X )

2 2

c

2

Response, X

X

Group 1

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Group 2

Group 3

Chap 11-14

Among-Group Variation

SST = SSA + SSW

SSA = ∑ n j ( X j X)2

Where:

j= j =1

c

SSA = Sum of squares among groups c = number of groups nj = sample size from group j Xj = sample mean from group j X = grand mean (mean of all data values)

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 11-15

Among-Group Variation

(continued)

SSA = ∑ n j ( X j X)2

j =1

c

Variation Due to Differences Among Groups

SSA MSA = c 1

Mean Square Among = SSA/degrees of freedom

i

j

Chap 11-16

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Among-Group Variation

(continued)

SSA = n 1 (X1 X) + n 2 (X 2 X) + + n c (X c X)

2 2

2

Response, X

X3

X1

Group 1

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

X2

Group 3

X

Group 2

Chap 11-17

Within-Group Variation

SST = SSA + SSW

nj

SSW = ∑

j =1

c

∑

i =1

( Xij X j )

2

Where:

SSW = Sum of squares within groups c = number of groups nj = sample size from group j Xj = sample mean from group j Xij = ith observation in group j

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 11-18

Within-Group Variation

(continued)

SSW = ∑

j =1

c

∑

i =1

nj

( Xij X j )

2

Summing the variation within each group and then adding over all groups

SSW MSW = nc

Mean Square Within = SSW/degrees of freedom

j

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 11-19

Within-Group Variation

(continued)

SSW = (X11 X1 ) + (X12 X 2 ) + + (X cn c X c )

2 2

2

Response, X

X3

X1

Group 1

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

X2

Group 3

Chap 11-20

Group 2

Obtaining the Mean Squares

The Mean Squares are obtained by dividing the various sum of squares by their associated degrees of freedom

SSA MSA = c 1 SSW MSW = nc SST MST = n 1

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Mean Square Among (d.f. = c-1)

Mean Square Within (d.f. = n-c)

Mean Square Total (d.f. = n-1)

Chap 11-21

One-Way ANOVA Table

Source of Variation Among Groups Within Groups Total

Degrees of Freedom Sum Of Squares Mean Square (Variance) F

c-1 n-c n–1

SSA SSW SST

SSA MSA = c-1 SSW MSW = n-c

FSTAT = MSA MSW

c = number of groups n = sum of the sample sizes from all groups df = degrees of freedom

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 11-22

One-Way ANOVA F Test Statistic

H0: 1= 2 = … = c H1: At least two population means are different

Test statistic

MSA FSTAT = MSW

MSA is mean squares among groups MSW is mean squares within groups

Degrees of freedom

df1 = c – 1 df2 = n – c (c = number of groups) (n = sum of sample sizes from all populations)

Chap 11-23

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Interpreting One-Way ANOVA F Statistic

The F statistic is the ratio of the among estimate of variance and the within estimate of variance

The ratio must always be positive df1 = c -1 will typically be small df2 = n - c will typically be large

Decision Rule: Reject H0 if FSTAT > Fα, otherwise do not reject H0

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

α

0

Do not reject H0

Reject H0

Fα

Chap 11-24

One-Way ANOVA F Test Example

You want to see if three different golf clubs yield different distances. You randomly select five measurements from trials on an automated driving machine for each club. At the 0.05 significance level, is there a difference in mean distance? Club 1 254 263 241 237 251 Club 2 234 218 235 227 216 Club 3 200 222 197 206 204

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Chap 11-25

One-Way ANOVA Example: Scatter Plot

Club 1 254 263 241 237 251 Club 2 234 218 235 227 216 Club 3 200 222 197 206 204

Distance 270 260 250 240 230 220 210

X1

X2

X

X3

x1 = 249.2 x 2 = 226.0 x 3 = 205.8 x = 227.0

200 190 1 2 Club

3

Chap 11-26

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

One-Way ANOVA Example Computations

Club 1 254 263 241 237 251 Club 2 234 218 235 227 216 Club 3 200 222 197 206 204

X1 = 249.2 X2 = 226.0 X3 = 205.8 X = 227.0 n1 = 5 n2 = 5 n3 = 5 n = 15 c=3

SSA = 5 (249.2 – 227)2 + 5 (226 – 227)2 + 5 (205.8 – 227)2 = 4716.4 SSW = (254 – 249.2)2 + (263 – 249.2)2 +…+ (204 – 205.8)2 = 1119.6 MSA = 4716.4 / (3-1) = 2358.2 MSW = 1119.6 / (15-3) = 93.3

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

F STAT

2358.2 = 93.3

= 25.275

Chap 11-27

One-Way ANOVA Example Solution

H0: 1 = 2 = 3 H1: j not all equal α = 0.05 df1= 2 df2 = 12

Critical Value: Fα = 3.89 α = .05

Test Statistic:

MSA 2358.2 FSTAT = = = 25.275 MSW 93.3

Decision: Reject H0 at α = 0.05

0

Do not reject H0

Conclusion: There is evidence that at least one j differs Reject H Fα = 3.89 FSTAT = 25.275 from the rest

0

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Chap 11-28

One-Way ANOVA Excel Output

SUMMARY Groups Club 1 Club 2 Club 3 ANOVA Source of Variation Between Groups Within Groups Total SS 4716.4 1119.6 5836.0 df 2 12 14

Chap 11-29

Count 5 5 5

Sum 1246 1130 1029

Average 249.2 226 205.8

Variance 108.2 77.5 94.2

MS 2358.2 93.3

F 25.275

P-value 4.99E-05

F crit 3.89

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

One-Way ANOVA Minitab Output

One-way ANOVA: Distance versus Club Source DF SS MS Club 2 4716.4 2358.2 Error 12 1119.6 93.3 Total 14 5836.0 F P 25.28 0.000

S = 9.659 R-Sq = 80.82% R-Sq(adj) = 77.62%

Individual 95% CIs For Mean Based on Pooled StDev Level 1 2 3 N Mean StDev -------+---------+---------+---------+-5 249.20 10.40 (-----*-----) 5 226.00 8.80 (-----*-----) 5 205.80 9.71 (-----*-----) -------+---------+---------+---------+-208 224 240 256

Pooled StDev = 9.66

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Chap 11-30

The Tukey-Kramer Procedure

Tells which population means are significantly different

e.g.: 1 = 2 ≠ 3 Done after rejection of equal means in ANOVA

Allows paired comparisons

Compare absolute mean differences with critical range

1= 2

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

3

x

Chap 11-31

Tukey-Kramer Critical Range

MSW 1 1 + 2 n j n j'

Critical Range = Q α

where: Qα = Upper Tail Critical Value from Studentized Range Distribution with c and n - c degrees of freedom (see appendix E.10 table) MSW = Mean Square Within nj and nj' = Sample sizes from groups j and j'

Chap 11-32

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

The Tukey-Kramer Procedure: Example

Club 1 254 263 241 237 251 Club 2 234 218 235 227 216 Club 3 200 222 197 206 204

1. Compute absolute mean differences:

x1 x 2 = 249.2 226.0 = 23.2 x1 x 3 = 249.2 205.8 = 43.4 x 2 x 3 = 226.0 205.8 = 20.2

2. Find the Qα value from the table in appendix E.10 with c = 3 and (n – c) = (15 – 3) = 12 degrees of freedom:

Q α = 3.77

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 11-33

The Tukey-Kramer Procedure: Example

(continued)

3. Compute Critical Range:

MSW 1 1 = 3.77 93.3 1 + 1 = 16.285 Critical Range = Q α + n j n j' 2 2 5 5

4. Compare: 5. All of the absolute mean differences are greater than critical range. Therefore there is a significant difference between each pair of means at 5% level of significance.

Thus, with 95% confidence we can conclude that the mean distance for club 1 is greater than club 2 and 3, and club 2 is greater than club 3.

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

x1 x 2 = 23.2 x1 x 3 = 43.4 x 2 x 3 = 20.2

Chap 11-34

ANOVA Assumptions

Randomness and Independence

Select random samples from the c groups (or randomly assign the levels)

Normality

The sample values for each group are from a normal population

Homogeneity of Variance

All populations sampled from have the same variance Can be tested with Levene's Test

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 11-35

ANOVA Assumptions Levene's Test

Tests the assumption that the variances of each population are equal. First, define the null and alternative hypotheses:

H0: σ21 = σ22 = …=σ2c H1: Not all σ2j are equal

Second, compute the absolute value of the difference between each value and the median of each group. Third, perform a one-way ANOVA on these absolute differences.

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Chap 11-36

Levene Homogeneity Of Variance Test Example

H0: σ21 = σ22 = σ23 H1: Not all σ2j are equal

Calculate Medians Club 1 237 241 251 254 263 Club 2 216 218 227 234 235 Club 3 197 200 204 Median 206 222 Calculate Absolute Differences Club 1 14 10 0 3 12 Club 2 11 9 0 7 8 Club 3 7 4 0 2 18

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Chap 11-37

Levene Homogeneity Of Variance Test Example (continued)

Anova: Single Factor SUMMARY Groups Club 1 Club 2 Club 3 Count 5 5 5 Sum Average Variance 39 35 31 7.8 7 6.2 36.2 17.5 50.2 Pvalue

Source of Variation Between Groups Within Groups

SS 6.4 415.6

df 2 12

MS

F

F crit

3.2 0.092 34.6

0.912 3.885

Since the p-value is greater than 0.05 we fail to reject H0 & conclude the variances are equal.

Total

422

14

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Chap 11-38

The Randomized Block Design

Like One-Way ANOVA, we test for equal population means (for different factor levels, for example)... ...but we want to control for possible variation from a second factor (with two or more levels) Levels of the secondary factor are called blocks

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Chap 11-39

Partitioning the Variation

Total variation can now be split into three parts:

SST = SSA + SSBL + SSE

SST = Total variation SSA = Among-Group variation SSBL = Among-Block variation SSE = Random variation

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Chap 11-40

Sum of Squares for Blocks

SST = SSA + SSBL + SSE

SSBL = c ∑ ( Xi. X)

i =1

r

2

Where:

c = number of groups r = number of blocks Xi. = mean of all values in block i X = grand mean (mean of all data values)

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 11-41

Partitioning the Variation

Total variation can now be split into three parts:

SST = SSA + SSBL + SSE

SST and SSA are computed as they were in One-Way ANOVA SSE = SST – (SSA + SSBL)

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Chap 11-42

Mean Squares

SSBL MSBL = Mean square blocking = r 1

MSA = Mean square among groups =

SSA c 1

SSE MSE = Mean square error = (r 1)(c 1)

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 11-43

Randomized Block ANOVA Table

Source of Variation

Among Blocks Among Groups Error

SS SSBL SSA SSE SST

df r-1 c-1 (r–1)(c-1) rc - 1

MS MSBL MSA MSE

F MSBL MSE MSA MSE

Total

c = number of populations r = number of blocks

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

rc = total number of observations df = degrees of freedom

Chap 11-44

Testing For Factor Effect

H 0 : .1 = .2 = .3 = = . c

H1 : Not all population means are equal

MSA FSTAT = MSE

Main Factor test: df1 = c – 1 df2 = (r – 1)(c – 1)

Reject H0 if FSTAT > Fα

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Chap 11-45

Test For Block Effect

H 0 : 1. = 2. = 3. = ... = r. H1 : Not all block means are equal

MSBL FSTAT = MSE

Blocking test:

df1 = r – 1 df2 = (r – 1)(c – 1)

Reject H0 if FSTAT > Fα

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Chap 11-46

The Tukey Procedure

To test which population means are significantly different

e.g.: 1 = 2 ≠ 3 Done after rejection of equal means in randomized block ANOVA design

Allows pair-wise comparisons

Compare absolute mean differences with critical range

1= 2

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

3

x

Chap 11-47

The Tukey Procedure

(continued)

Critical Range = Q α

MSE r

Compare:

Is x.j x.j' > Critical Range ?

If the absolute mean difference is greater than the critical range then there is a significant difference between that pair of means at the chosen level of significance.

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

x.1 x.2 x.1 x.3 x.2 x.3 etc...

Chap 11-48

Factorial Design: Two-Way ANOVA

Examines the effect of

Two factors of interest on the dependent variable

e.g., Percent carbonation and line speed on soft drink bottling process

Interaction between the different levels of these two factors

e.g., Does the effect of one particular carbonation level depend on which level the line speed is set?

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Chap 11-49

Two-Way ANOVA

(continued)

Assumptions Populations are normally distributed Populations have equal variances Independent random samples are drawn

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Chap 11-50

Two-Way ANOVA Sources of Variation

Two Factors of interest: A and B r = number of levels of factor A c = number of levels of factor B n' = number of replications for each cell n = total number of observations in all cells n = (r)(c)(n') Xijk = value of the kth observation of level i of factor A and level j of factor B

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 11-51

Two-Way ANOVA Sources of Variation

SST = SSA + SSB + SSAB + SSE

SSA

Factor A Variation

(continued) Degrees of Freedom: r–1

SST Total Variation

SSB

Factor B Variation

c–1

SSAB

n-1 Variation due to interaction between A and B (r – 1)(c – 1)

SSE

Random variation (Error)

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

rc(n' – 1)

Chap 11-52

Two-Way ANOVA Equations

Total Variation:

SST = ∑∑∑ ( Xijk X)

i =1 j=1 k =1

r

r

c

n′

2

Factor A Variation:

′∑ ( Xi.. X)2 SSA = cn

i=1

Factor B Variation:

SSB = rn′∑ ( X. j. X)2

j=1

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 11-53

c

Two-Way ANOVA Equations

(continued)

Interaction Variation:

SSAB = n′∑∑ ( Xij. Xi.. X.j. + X)2

i=1 j=1

r

c

Sum of Squares Error:

SSE = ∑∑∑ ( Xijk Xij. )

i=1 j=1 k =1

r

c

n′

2

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Chap 11-54

Two-Way ANOVA Equations

where:

∑∑∑ X

X=

n′ ijk

r

c

n′

(continued)

ijk

i=1 j =1 k =1

∑∑ X

Xi.. =

j=1 k =1

c

rcn′

= Grand Mean

cn′

r

= Mean of ith level of factor A (i = 1, 2, ..., r)

X. j. =

n′

∑∑ X

i=1 k =1

n′

ijk

rn′

= Mean of jth level of factor B (j = 1, 2, ..., c)

r = number of levels of factor A c = number of levels of factor B n' = number of replications in each cell

Chap 11-55

Xijk Xij. = ∑ = Mean of cell ij k =1 n′

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Mean Square Calculations

SSA MSA = Mean square factor A = r 1 SSB MSB = Mean square factor B = c 1

SSAB MSAB = Mean square interaction = (r 1)(c 1)

SSE MSE = Mean square error = rc(n'1)

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 11-56

Two-Way ANOVA: The F Test Statistics

H0: 1..= 2.. = 3..= = r.. H1: Not all i.. are equal F Test for Factor A Effect

F STAT

=

MSA MSE

Reject H0 if FSTAT > Fα

H0: .1. = .2. = .3.= = .c. H1: Not all .j. are equal

F Test for Factor B Effect

F STAT

MSB = MSE

Reject H0 if FSTAT > Fα

H0: the interaction of A and B is equal to zero zero

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

F Test for Interaction Effect

H1: interaction of A and B is not F STAT

=

MSAB MSE

Reject H0 if FSTAT > Fα

Chap 11-57

Two-Way ANOVA Summary Table

Source of Variation Factor A Factor B AB (Interaction) Error Total Sum of Squares SSA SSB Degrees of Freedom r–1 c–1 Mean Squares MSA

= SSA /(r – 1)

F MSA MSE MSB MSE MSAB MSE

MSB

= SSB /(c – 1)

SSAB

(r – 1)(c – 1)

MSAB

= SSAB / (r – 1)(c – 1)

SSE SST

rc(n' – 1) n–1

MSE =

SSE/rc(n' – 1)

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Chap 11-58

Features of Two-Way ANOVA F Test

Degrees of freedom always add up

n-1 = rc(n'-1) + (r-1) + (c-1) + (r-1)(c-1) Total = error + factor A + factor B + interaction

The denominators of the F Test are always the same but the numerators are different The sums of squares always add up

SST = SSE + SSA + SSB + SSAB Total = error + factor A + factor B + interaction

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 11-59

Examples: Interaction vs. No Interaction

No interaction: line segments are parallel Interaction is present: some line segments not parallel

Mean Response

Factor B Level 1 Mean Response Factor B Level 1 Factor B Level 2 Factor B Level 3 Factor B Level 3 Factor B Level 2

Factor A Levels

Factor A Levels

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Chap 11-60

Multiple Comparisons: The Tukey Procedure

Unless there is a significant interaction, you can determine the levels that are significantly different using the Tukey procedure Consider all absolute mean differences and compare to the calculated critical range

Example: Absolute differences for factor A, assuming three levels:

X1.. X 2.. X1.. X3.. X 2.. X3..

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Chap 11-61

Multiple Comparisons: The Tukey Procedure

Critical Range for Factor A:

MSE Critical Range = Qα c n'

(where Qα is from Table E.10 with r and rc(n'–1) d.f.)

Critical Range for Factor B:

MSE Critical Range = Qα r n'

(where Qα is from Table E.10 with c and rc(n'–1) d.f.)

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc..

Chap 11-62

Chapter Summary

Described one-way analysis of variance

The logic of ANOVA ANOVA assumptions F test for difference in c means The Tukey-Kramer procedure for multiple comparisons The Levene test for homogeneity of variance

Considered the Randomized Block Design

Factor and Block Effects Multiple Comparisons: Tukey Procedure

Described two-way analysis of variance

Examined effects of multiple factors Examined interaction between factors

Basic Business Statistics, 11e 2009 Prentice-Hall, Inc.. Chap 11-63

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