Copyright 2011 Pearson Education, Inc. publishing as Prentice
Hall 12-1 Business Statistics: A Decision-Making Approach 8 th
Edition Chapter 12 Analysis of Variance
Slide 2
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Hall 12-2 Chapter Goals After completing this chapter, you should
be able to: Recognize situations in which to use analysis of
variance Understand different analysis of variance designs Perform
a single-factor hypothesis test and interpret results (manually and
with the aid of Excel) Conduct and interpret post-analysis of
variance pairwise comparisons procedures Set up and perform
randomized blocks analysis Analyze two-factor analysis of variance
test with replications results with Excel or Minitab
Slide 3
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Hall 12-3 Chapter Overview Analysis of Variance (ANOVA) F-test
Tukey- Kramer test Fishers Least Significant Difference test
One-Way ANOVA Randomized Complete Block ANOVA Two-factor ANOVA with
replication
Slide 4
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Hall 12-4 One-Factor 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 Club 2 Club 3 254 234 200 263 218 222 241 235
197 237 227 206 251 216 204
Slide 5
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Hall 12-5 One-Factor ANOVA Example: Scatter Diagram 270 260 250 240
230 220 210 200 190 Distance Club 1 Club 2 Club 3 254 234 200 263
218 222 241 235 197 237 227 206 251 216 204 Club 1 2 3
Slide 6
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Hall 12-6 One-Factor ANOVA Example Computations Club 1 Club 2 Club
3 254 234 200 263 218 222 241 235 197 237 227 206 251 216 204 x 1 =
249.2 x 2 = 226.0 x 3 = 205.8 x = 227.0 n 1 = 5 n 2 = 5 n 3 = 5 n T
= 15 k = 3 SSB = 5 [ (249.2 227) 2 + (226 227) 2 + (205.8 227) 2 ]
= 4716.4 SSW = (254 249.2) 2 + (263 249.2) 2 ++ (204 205.8) 2 =
1119.6 MSB = 4716.4 / (3-1) = 2358.2 MSW = 1119.6 / (15-3) =
93.3
Slide 7
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Hall 12-7 F = 25.275 One-Factor ANOVA Example Solution H 0 : 1 = 2
= 3 H A : i not all equal = 0.05 df 1 = 2 df 2 = 12 Test Statistic:
Decision: Conclusion: Reject H 0 at = 0.05 There is evidence that
at least one i differs from the rest 0 = 0.05 F 0.05 = 3.885 Reject
H 0 Do not reject H 0 Critical Value: F = 3.885
Slide 8
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Hall 12-8 SUMMARY GroupsCountSumAverageVariance Club
151246249.2108.2 Club 25113022677.5 Club 351029205.894.2 ANOVA
Source of Variation SSdfMSFP-valueF crit Between Groups
4716.422358.225.2754.99E-053.885 Within Groups 1119.61293.3
Total5836.014 ANOVA -- Single Factor: Excel Output EXCEL: tools |
data analysis | ANOVA: single factor
Slide 9
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Hall 12-9 Logic of Analysis of Variance Investigator controls one
or more independent variables Called factors (or treatment
variables) Each factor contains two or more levels (or
categories/classifications) Observe effects on dependent variable
Response to levels of independent variable Experimental design: the
plan used to test hypothesis
Slide 10
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Hall 12-10 Completely Randomized Design Experimental units
(subjects) are assigned randomly to treatments Only one factor or
independent variable With two or more treatment levels Analyzed by
One-factor analysis of variance (one-way ANOVA) Called a Balanced
Design if all factor levels have equal sample size
Slide 11
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Hall 12-11 One-Way Analysis of Variance Evaluate the difference
among the means of three or more populations Examples: Accident
rates for 1 st, 2 nd, and 3 rd shift Expected mileage for five
brands of tires Assumptions Populations are normally distributed
Populations have equal variances Samples are randomly and
independently drawn Datas measurement level is interval or
ratio
Slide 12
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Hall 12-12 Hypotheses of One-Way ANOVA All population means are
equal i.e., no treatment effect (no variation in means among
groups) At least one population mean is different i.e., there is a
treatment effect Does not mean that all population means are
different (some pairs may be the same)
Slide 13
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Hall 12-13 One-Factor ANOVA All Means are the same: The Null
Hypothesis is True (No Treatment Effect)
Slide 14
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Hall 12-14 One-Factor ANOVA At least one mean is different: The
Null Hypothesis is NOT true (Treatment Effect is present) or
(continued)
Slide 15
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Hall 12-15 Partitioning the Variation Total variation can be split
into two parts: SST = Total Sum of Squares (total variation) SSB =
Sum of Squares Between (variation between samples) SSW = Sum of
Squares Within (within each factor level) SST = SSB + SSW
Slide 16
Copyright 2011 Pearson Education, Inc. publishing as Prentice
Hall 12-16 Partitioning the Variation Total Variation (SST) = the
aggregate dispersion of the individual data values across the
various factor levels Within-Sample Variation (SSW) = dispersion
that exists among the data values within a particular factor level
Between-Sample Variation (SSB) = dispersion among the factor sample
means SST = SSB + SSW (continued)
Slide 17
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Hall 12-17 Partition of Total Variation Variation Due to Factor
(SSB) Variation Due to Random Sampling (SSW) Total Variation (SST)
Commonly referred to as: Sum of Squares Within Sum of Squares Error
Sum of Squares Unexplained Within Groups Variation Commonly
referred to as: Sum of Squares Between Sum of Squares Among Sum of
Squares Explained Among Groups Variation = +
Slide 18
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Hall 12-18 Total Sum of Squares Where: SST = Total sum of squares k
= number of populations (levels or treatments) n i = sample size
from population i x ij = j th measurement from population i x =
grand mean (mean of all data values) SST = SSB + SSW
Slide 19
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Hall 12-19 Total Variation (continued)
Slide 20
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Hall 12-20 Sum of Squares Between Where: SSB = Sum of squares
between k = number of populations n i = sample size from population
i x i = sample mean from population i x = grand mean (mean of all
data values) SST = SSB + SSW
Slide 21
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Hall 12-21 Between-Group Variation Variation Due to Differences
Among Groups Mean Square Between = SSB/degrees of freedom
Slide 22
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Hall 12-22 Between-Group Variation (continued)
Slide 23
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Hall 12-23 Sum of Squares Within Where: SSW = Sum of squares within
k = number of populations n i = sample size from population i x i =
sample mean from population i x ij = j th measurement from
population i SST = SSB + SSW
Slide 24
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Hall 12-24 Within-Group Variation Summing the variation within each
group and then adding over all groups Mean Square Within =
SSW/degrees of freedom
Slide 25
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Hall 12-25 Within-Group Variation (continued)
Slide 26
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Hall 12-26 Hartleys F-test statistic One-Way ANOVA Table Source of
Variation dfSSMS Between Samples SSBMSB = Within Samples n T -
kSSWMSW = Totaln T - 1 SST = SSB+SSW k - 1 MSB MSW F ratio k =
number of populations n T = sum of the sample sizes from all
populations df = degrees of freedom SSB k - 1 SSW n T - k F =
Slide 27
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Hall 12-27 One-Factor ANOVA F Test Statistic Test statistic MSB is
mean squares between variances MSW is mean squares within variances
Degrees of freedom df 1 = k 1 (k = number of populations) df 2 = n
T k (n T = sum of sample sizes from all populations) H 0 : 1 = 2 =
= k H A : At least two population means are different
Slide 28
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Hall 12-28 Interpreting One-Factor ANOVA F Statistic The F
statistic is the ratio of the between estimate of variance and the
within estimate of variance The ratio must always be positive df 1
= k -1 will typically be small df 2 = n T - k will typically be
large The ratio should be close to 1 if H 0 : 1 = 2 = = k is true
The ratio will be larger than 1 if H 0 : 1 = 2 = = k is false
Slide 29
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Hall 12-29 ANOVA Steps 1.Specify parameter of interest 2.Formulate
hypotheses 3.Specify the significance level, 4.Select independent,
random samples Compute sample means and grand mean 5.Determine the
decision rule 6.Verify the normality and equal variance assumptions
have been satisfied 7.Create ANOVA table 8.Reach a decision and
draw a conclusion
Slide 30
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Hall 12-30 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 pair-wise comparisons Compare absolute
mean differences with critical range x 1 = 2 3
Slide 31
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Hall 12-31 Tukey-Kramer Critical Range where: q = Value from
standardized range table with k and n T - k degrees of freedom for
the desired level of MSW = Mean Square Within n i and n j = Sample
sizes from populations (levels) i and j
Slide 32
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Hall 12-32 The Tukey-Kramer Procedure: Example 1. Compute absolute
mean differences: Club 1 Club 2 Club 3 254 234 200 263 218 222 241
235 197 237 227 206 251 216 204 2. Find the q value from the table
in appendix J with k and n T - k degrees of freedom for the desired
level of
Slide 33
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Hall 12-33 The Tukey-Kramer Procedure: Example 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. 3. Compute Critical Range: 4.
Compare:
Slide 34
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Hall 12-34 Tukey-Kramer in PHStat
Slide 35
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Hall 12-35 Randomized Complete Block ANOVA 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) Used when more than one
factor may influence the value of the dependent variable, but only
one is of key interest Levels of the secondary factor are called
blocks
Slide 36
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Hall 12-36 Assumptions Populations are normally distributed
Populations have equal variances The observations within samples
are independent The date measurement must be interval or ratio
Application examples Testing 5 routes to a destination through 3
different cab companies to see if differences exist Determining the
best training program (out of 4 choices) for various departments
within a company Randomized Complete Block ANOVA (continued)
Slide 37
Copyright 2011 Pearson Education, Inc. publishing as Prentice
Hall 12-37 Partitioning the Variation Total variation can now be
split into three parts: SST = Total sum of squares SSB = Sum of
squares between factor levels SSBL = Sum of squares between blocks
SSW = Sum of squares within levels SST = SSB + SSBL + SSW
Slide 38
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Hall 12-38 Sum of Squares for Blocking Where: k = number of levels
for this factor b = number of blocks x j = sample mean from the j
th block x = grand mean (mean of all data values) SST = SSB + SSBL
+ SSW
Slide 39
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Hall 12-39 Partitioning the Variation Total variation can now be
split into three parts: SST and SSB are computed as they were in
One-Way ANOVA SST = SSB + SSBL + SSW SSW = SST (SSB + SSBL)
Slide 40
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Hall 12-40 Mean Squares
Slide 41
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Hall 12-41 Randomized Block ANOVA Table Source of Variation dfSSMS
Between Samples SSBMSB Within Samples (k1)(b-1)SSWMSW Totaln T -
1SST k - 1 MSBL MSW F ratio k = number of populationsn T = sum of
the sample sizes from all populations b = number of blocksdf =
degrees of freedom Between Blocks SSBLb - 1MSBL MSB MSW
Slide 42
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Hall 12-42 Blocking Test Blocking test: df 1 = b - 1 df 2 = (k 1)(b
1) MSBL MSW F = Reject H 0 if F > F
Slide 43
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Hall 12-43 Main Factor test: df 1 = k - 1 df 2 = (k 1)(b 1) MSB MSW
F = Reject H 0 if F > F Main Factor Test
Slide 44
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Hall 12-44 Fishers Least Significant Difference Test 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 x = 123
Slide 45
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Hall 12-45 Fishers Least Significant Difference (LSD) Test where: t
/2 = Upper-tailed value from Students t-distribution for /2 and (k
- 1)(b - 1) degrees of freedom MSW = Mean square within from ANOVA
table b = number of blocks k = number of levels of the main factor
NOTE: This is a similar process as Tukey-Kramer
Slide 46
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Hall 12-46 Fishers Least Significant Difference (LSD) Test
(continued) If the absolute mean difference is greater than LSD
then there is a significant difference between that pair of means
at the chosen level of significance Compare:
Slide 47
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Hall 12-47 Two-Factor ANOVA With Replication Examines the effect of
Two or more 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 percentage of carbonation depend
on which level the line speed is set?
Slide 48
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Hall 12-48 Two-Factor ANOVA Assumptions Populations are normally
distributed Populations have equal variances Independent random
samples are drawn Data must be interval or ratio level
(continued)
Slide 49
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Hall 12-49 Two-Way ANOVA Sources of Variation Two Factors of
interest: A and B a = number of levels of factor A b = number of
levels of factor B n T = total number of observations in all
cells
Slide 50
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Hall 12-50 Two-Way ANOVA Sources of Variation SST Total Variation
SS A Variation due to factor A SS B Variation due to factor B SS AB
Variation due to interaction between A and B SSE Inherent variation
(Error) Degrees of Freedom: a 1 b 1 (a 1)(b 1) n T ab n T - 1 SST =
SS A + SS B + SS AB + SSE (continued)
Slide 51
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Hall 12-51 Two Factor ANOVA Equations Total Sum of Squares: Sum of
Squares Factor A: Sum of Squares Factor B:
Slide 52
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Hall 12-52 Two Factor ANOVA Equations Sum of Squares Interaction
Between A and B: Sum of Squares Error: (continued)
Slide 53
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Hall 12-53 Two Factor ANOVA Equations where: a = number of levels
of factor A b = number of levels of factor B n = number of
replications in each cell (continued)
Slide 54
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Hall 12-54 Mean Square Calculations
Slide 55
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Hall 12-55 Two-Way ANOVA: The F Test Statistic F Test for Factor B
Main Effect F Test for Interaction Effect H 0 : A1 = A2 = A3 = H A
: Not all Ai are equal H 0 : factors A and B do not interact to
affect the mean response H A : factors A and B do interact F Test
for Factor A Main Effect H 0 : B1 = B2 = B3 = H A : Not all Bi are
equal Reject H 0 if F > F
Slide 56
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Hall 12-56 Two-Way ANOVA Summary Table Source of Variation Sum of
Squares Degrees of Freedom Mean Squares F Statistic Factor ASS A a
1 MS A = SS A /(a 1) MS A MSE Factor BSS B b 1 MS B = SS B /(b 1)
MS B MSE AB (Interaction) SS AB (a 1)(b 1) MS AB = SS AB / [(a 1)(b
1)] MS AB MSE ErrorSSEn T ab MSE = SSE/(n T ab) TotalSSTn T 1
Slide 57
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Hall 12-57 Features of Two-Way ANOVA F Test Degrees of freedom
always add up n T - 1 = (n T - ab) + (a - 1) + (b - 1) + (a - 1)(b
- 1) Total = error + factor A + factor B + interaction The
denominator of the F Test is always the same but the numerator is
different The sums of squares always add up SST = SSE + SS A + SS B
+ SS AB Total = error + factor A + factor B + interaction
Slide 58
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Hall 12-58 Interaction vs. No Interaction No interaction: 12 Factor
B Level 1 Factor B Level 3 Factor B Level 2 Factor A Levels 1 2
Factor B Level 1 Factor B Level 3 Factor B Level 2 Factor A Levels
Mean Response Interaction is present:
Slide 59
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Hall 12-59 When conducting tests for a Two-Factor ANOVA: Test for
interaction If present, conduct a one-way ANOVA to test the levels
of one of the other factors using only one level of the other
factor If NO interaction, test Factor A and Factor B Interaction
vs. No Interaction (continued)
Slide 60
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Hall 12-60 Chapter Summary Described one-way analysis of variance
The logic of ANOVA ANOVA assumptions F test for difference in k
means The Tukey-Kramer procedure for multiple comparisons Described
randomized complete block designs F test Fishers least significant
difference test for multiple comparisons Described two-way analysis
of variance Examined effects of multiple factors and
interaction
Slide 61
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Hall 12-61 All rights reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted, in any
form or by any means, electronic, mechanical, photocopying,
recording, or otherwise, without the prior written permission of
the publisher. Printed in the United States of America.