Business statistics, 7e, by groebner ch12

Business Statistics: A Decision-Making Approach 7th Edition Chapter 12 Analysis of Variance Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc Chap 12-1 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  Conduct and interpret post-analysis of variance pairwise comparisons procedures Set up and perform randomized blocks analysis Business Statistics: A Decision Analyze two-factor analysis of variance test with replications Making Approach, 7e © 2008 results Prentice-Hall, Inc Chap 12-2  Chapter Overview Analysis of Variance (ANOVA) One-Way ANOVA Randomized Complete Block ANOVA Two-factor ANOVA with replication F-test TukeyKramer test A DecisionBusiness Statistics: Making Approach, 7e © 2008 Prentice-Hall, Inc F-test Fisher’s Least Significant Difference test Chap 12-3 General ANOVA Setting  Investigator controls one or more independent variables    Observe effects on dependent variable   Called factors (or treatment variables) Each factor contains two or more levels (or categories/classifications) Response to levels of independent variable Experimental design: the plan used to test hypothesis Business Statistics: A DecisionMaking Approach, 7e © 2008 Prentice-Hall, Inc Chap 12-4 One-Way Analysis of Variance  Evaluate the difference among the means of three or more populations 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 Business Statistics: drawn A Decision Making Approach, 7e © 2008 Prentice-Hall, Inc Chap 12-5 Completely Randomized Design   Experimental units (subjects) are assigned randomly to treatments Only one factor or independent variable   Analyzed by   With two or more treatment levels One-factor analysis of variance (one-way ANOVA) Called a Balanced Design if all factor levels have equal sample size Business Statistics: A DecisionMaking Approach, 7e © 2008 Prentice-Hall, Inc Chap 12-6 Hypotheses of One-Way ANOVA   H0 : μ1 μ2 μ3  μk  All population means are equal  i.e., no treatment effect (no variation in means among groups) HA : Not all of the population means are the same  At least one population mean is different  i.e., there is a treatment effect Does not mean that all population means are different (some A pairs may be the same) Business Statistics: DecisionMaking Approach, 7e © 2008 Prentice-Hall, Inc Chap 12-7  One-Factor ANOVA H0 : μ1 μ2 μ3  μk HA : Not all μi are the same All Means are the same: The Null Hypothesis is True (No Treatment Effect) Business Statistics: A Decisionμ1 μ μ3 Making Approach, 7e2 ©2008 Prentice-Hall, Inc Chap 12-8 One-Factor ANOVA (continued) H0 : μ1 μ2 μ3  μk HA : Not all μi are the same At least one mean is different: The Null Hypothesis is NOT true (Treatment Effect is present) or Business Statistics: A DecisionMaking Approach, μ1 μ2 7eμ3© 2008 Prentice-Hall, Inc μ1 μ2 μ3 Chap 12-9 Partitioning the Variation  Total variation can be split into two parts: SST = SSB + SSW SST = Total Sum of Squares SSB = Sum of Squares Between SSW = Sum of Squares Within Business Statistics: A DecisionMaking Approach, 7e © 2008 Prentice-Hall, Inc Chap 12-10 Fisher’s Least Significant Difference (LSD) Test LSD t /2 MSW b where: t/2 = Upper-tailed value from Student’s t-distribution for /2 and (k - 1)(b - 1) degrees of freedom MSW = Mean square within from ANOVA table b = number of blocks Business Statistics: A Decisionk = number of levels of the main factor Making Approach, 7e © 2008 Prentice-Hall, Inc Chap 12-43 Fisher’s Least Significant Difference (LSD) Test (continued) LSD t /2 MSW b Compare: Is x i  x j  LSD ? If the absolute mean difference is greater than LSD then there is a significant difference between that pair of means at the chosen of significance Business Statistics:level A DecisionMaking Approach, 7e © 2008 Prentice-Hall, Inc x1  x x1  x x2  x3 etc Chap 12-44 Two-Factor ANOVA  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 levelAthe line speed is set? Business Statistics: Decision Making Approach, 7e © 2008 Prentice-Hall, Inc Chap 12-45 Two-Factor ANOVA (continued)  Assumptions  Populations are normally distributed  Populations have equal variances  Independent random samples are drawn Business Statistics: A DecisionMaking Approach, 7e © 2008 Prentice-Hall, Inc Chap 12-46 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 nT = total number of observations in all cells Business Statistics: A DecisionMaking Approach, 7e © 2008 Prentice-Hall, Inc Chap 12-47 Two-Way ANOVA Sources of Variation SST = SSA + SSB + SSAB + SSE SSA Variation due to factor A SST Total Variation SSB Variation due to factor B SSAB nT - Variation due to interaction between A and B Business Statistics: A DecisionSSE Making Approach, 7e © 2008 Inherent variation (Error) Prentice-Hall, Inc (continued) Degrees of Freedom: a–1 b–1 (a – 1)(b – 1) nT – ab Chap 12-48 Two Factor ANOVA Equations Total Sum of Squares: a n b SST    ( x ijk  x ) i1 j1 k 1 Sum of Squares Factor A: a  SS A bn  ( x i  x ) i1 Sum of Squares Factor B: b  SS  a n ( x  x ) Business Statistics: A Decision j B Making Approach, 7e © 2008 Prentice-Hall, Inc j1 Chap 12-49 Two Factor ANOVA Equations (continued) Sum of Squares Interaction Between A and B: SS a AB b n  ( x ij  x i  x j  x )2 i1 j1 Sum of Squares Error: a b n SSE   ( x ijk  x ij ) i1 j 1 k 1 Business Statistics: A DecisionMaking Approach, 7e © 2008 Prentice-Hall, Inc Chap 12-50 Two Factor ANOVA Equations a x i 1 j1 k 1 abn n x xi  n x where: b b j1 k 1 bn (continued) ijk Grand Mean ijk Mean of each level of factor A a n x x j  i1 k 1 an ijk Mean of each level of factor B x ijk Business Statistics: A Decisionx ij  Mean of each cell Making Approach, 7e © 2008 k 1 n Prentice-Hall, Inc n a = number of levels of factor A b = number of levels of factor B n’ = number of replications in each cell Chap 12-51 Mean Square Calculations SS A MS A Mean square factor A  a SSB MSB Mean square factor B  b MS AB SS AB Mean square interaction  (a  1)(b  1) SSE Business Statistics: DecisionMSEA Mean square error  nT  ab Making Approach, 7e © 2008 Prentice-Hall, Inc Chap 12-52 Two-Way ANOVA: The F Test Statistic H0: μA1 = μA2 = μA3 = • • • HA: Not all μAi are equal H0: μB1 = μB2 = μB3 = • • • HA: Not all μBi are equal H0: factors A and B not interact to affect the mean response Business Statistics: A DecisionHA: factors A and B7edo©interact Making Approach, 2008 Prentice-Hall, Inc F Test for Factor A Main Effect MS A F MSE Reject H0 if F > F F Test for Factor B Main Effect MSB F MSE Reject H0 if F > F F Test for Interaction Effect MS AB F MSE Reject H0 if F > F Chap 12-53 Two-Way ANOVA Summary Table Source of Variation Sum of Squares Degrees of Freedom Factor A SSA Factor B SSB AB (Interaction) SSAB (a – 1)(b – 1) Error SSE nT – ab a–1 b–1 Business Statistics: A DecisionTotal Making Approach,SST 7e © 2008nT – Prentice-Hall, Inc Mean Squares F Statistic MSA MSA MSE = SSA /(a – 1) MSB MSB MSE = SSB /(b – 1) MSAB = SSAB / [(a – 1)(b – 1)] MSAB MSE MSE = SSE/(nT – ab) Chap 12-54 Features of Two-Way ANOVA F Test  Degrees of freedom always add up  nT - = (nT - 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 + SSA + SSB + SSAB BusinessStatistics: A DecisionTotal = error + factor A + factor B + interaction Making Approach, 7e © 2008 Prentice-Hall, Inc Chap 12-55  Examples: Interaction vs No Interaction No interaction: Interaction is present: Factor B Level Factor B Level Factor B Level BusinessFactor Statistics: A Levels A Decision1 Approach, 7e © 2008 Making Prentice-Hall, Inc Mean Response Mean Response   Factor B Level Factor B Level Factor B Level Factor A Levels Chap 12-56 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 Fisher’s least significant difference test for multiple comparisons  Described two-way analysis of variance Business Statistics: A DecisionMakingApproach, 7e effects © 2008of multiple factors and interaction Examined Prentice-Hall, Inc Chap 12-57 ... variable   Analyzed by   With two or more treatment levels One-factor analysis of variance (one-way ANOVA) Called a Balanced Design if all factor levels have equal sample size Business Statistics:... Randomized Complete Block ANOVA Two-factor ANOVA with replication F-test TukeyKramer test A DecisionBusiness Statistics: Making Approach, 7e © 2008 Prentice-Hall, Inc F-test Fisher’s Least Significant... Response to levels of independent variable Experimental design: the plan used to test hypothesis Business Statistics: A DecisionMaking Approach, 7e © 2008 Prentice-Hall, Inc Chap 12-4 One-Way Analysis