Chapter 11 - ANOVA

 

Business Statistics: A Decision-Making Appr Approach oach 6th $%ition

Chapter 11 Anal&sis o' (ariance

Business Statistics: A Decision-Making Approach, 6e © 2005 PrenticeHall, nc!

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"hapter )oals After completing this chapter, you should be able to:  

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Recognize situations in which to use analysis of variance Understand different analysis of variance designs Perform a single-factor single-factor hypothesis test and interpret results Conduct and interpret post-analysis of variance pairwise comparisons procedures Set up and perform randomized blocks analysis  Analyze two-factor two-factor analysis analysis of variance variance test with replicatio replications ns results

Business Statistics: A Decision-Making Approach, 6e © 2005 PrenticeHall, nc!

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"hapter +erie  Analysis  Analysi s of a ariance !A"# !A"#A$

#ne-+ay  A"#A  A"# A

Randomized Complete ,lock A"#A A"#A

&wo-factor  A"#A  A"# A with replication

%-test %-test &ukey'ramer test Business Statistics: A Decision-Making Approach, 6e © 2005 PrenticeHall, nc!

%isher(s )east Significant *ifference test "hap ##-*

 

)eneral A/+(A )eneral A/+(A Setting Setting 

nvestigator controls one or more independent variables 

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#bserve effects on dependent variable 

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Called factors factors !or  !or treatment variables$ .ach factor contains two or more levels !or levels !or categories/classifications$ Response to levels of independent variable

.0perimental design1 the plan used to test hypothesis

Business Statistics: A Decision-Making Approach, 6e © 2005 PrenticeHall, nc!

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ne- a& na &s s o (ariance

 

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.valuate the difference among the means of three or more populations .0amples1  .0amples1  Accident rates for 2st3 4nd3 and 5rd shift   .0pected mileage for five brands of tires

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 Assumptions 

Populations are normally distributed

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Populations have e6ual variances

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Samples are randomly and independently drawn

Business Statistics: A Decision-Making Approach, 6e © 2005 PrenticeHall, nc!

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o1p ete & an o1 3e Design

 

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.0perimental units !sub7ects$ are assigned randomly to treatments

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#nly one factor or independent variable 

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 Analyzed by 

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+ith two or more m ore treatment levels #ne-factor analysis of variance !one-way A"#A$

Called a ,alanced *esign *esign if  if all factor levels have e6ual sample size

Business Statistics: A Decision-Making Approach, 6e © 2005 PrenticeHall, nc!

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Hall, nc!

&pot eses o ne- a& A/+(A

 

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  1: ; 9 2 

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= :4  = :5 =  = :k

 All population means are e6ual i8e83 no treatment effect !no variation in means among groups$

  1 "ot all of  the population means are the same ;  A 

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 At least one population mean is different i8e83 there is a treatment effect *oes not mean that all population means are different !some pairs may be the same$

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-

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Hall, nc!  

+ne-actor A/+(A ;9 1 :2

= :4  =  :5 =  = :k

; A 1 "ot all :i are the same

 All SS+ &otal ariation = ariation = the aggregate dispersion of the individual data values across the various factor levels !SS&$ ariation = dispersion among the factor ,etween-Sample ariation = sample means !SS,$ +ithin-Sample ariation = ariation = dispersion that e0ists among the data values within a particular factor level !SS+$

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-

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Hall, nc!  

Partition o' 8otal (ariation #otal Variatio Variation n ((SS# SS#

=    

Variation Due to Factor (SSB/SSG

Variation Due to !andom Sampling (SS"

>

Commonly referred to as1 Sum of S6uares ,etween Sum of S6uares Among Sum of S6uares .0plained  Among ?roups ar ariation iation

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-

   

Commonly referred to as1 Sum of S6uares +ithin Sum of S6uares .rror  Sum of S6uares Une0plained +ithin ?roups ariation

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 8otal otal Su1 o' S9uares S9uares  8 SS& = SS, > SS+ ni

k

2

SS& = +here1

i7 − 0 $ ! 0  7=1

∑∑ i =1

SS& = &otal sum of s6uares k = number of populations !levels !levels or treatments$ ni = sample size from population i 0i7 = 7th measurement from population i 0 = grand mean !mean of all data values$ Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-

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 8otal otal (ariation (ariation  8 (continued)

SS&

= ! 0 − 0 $ + ! 0 − 0 $ + 888 + ! 0 kn   − 0 $ 2

11

2

12

2

k

! esp on se, '

 X  

G ro u p $

G ro u p %

Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-

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Su1 o' S9uar S9uares es Beteen SS& = SS, > SS+ k 2

+here1

SS, =

i i n ! 0   i=1

 ∑

− 0$

SS, = Sum of s6uares between k = number of populations ni = sample size from population i 0i = sample mean from populatio population ni 0 = grand mean !mean of all data values$ Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-

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Beteen-)roup (ariation k

 ∑

SS, =

ni ! 0  i

− 0$

2

i=1

ariation *ue to *ifferences Among ?roups

east Signi?cant Signi?cant Di@erence >SD 8est 4 )S* = t α/4