Business statistics a decision making approach 6th edition ch11ppln

Business Statistics: A Decision-Making Approach 6th Edition Chapter 11 Analysis of Variance Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 11-1 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  Analyze two-factor analysis of variance test with replications results Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 11-2 Chapter Overview Analysis of Variance (ANOVA) One-Way ANOVA Randomized Complete Block ANOVA Two-factor ANOVA with replication F-test TukeyKramer test Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc F-test Fisher’s Least Significant Difference test Chap 11-3 General ANOVA Setting   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 Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 11-4 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 drawn Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 11-5 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 Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 11-6 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 pairs may be the same) Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 11-7 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) μ1 = μ2 = μ3 Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 11-8 One-Factor ANOVA H0 : μ1 = μ2 = μ3 =  = μk (continue d) HA : Not all μi are the same At least one mean is different: The Null Hypothesis is NOT true (Treatment Effect is present) or μ1 = μ2 ≠ μ3 Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc μ1 ≠ μ2 ≠ μ3 Chap 11-9 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 Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 11-10 Fisher’s Least Significant Difference (LSD) Test LSD = t α/2 MSW b where: tα/2 = Upper-tailed value from Student’s t-distribution for α/2 and (k -1)(n - 1) degrees of freedom MSW = Mean square within from ANOVA table b = number of blocks k = number of levels of the main factor Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 11-43 Fisher’s Least Significant Difference (LSD) Test (continued) LSD = t α/2 Is x i − x j > LSD ? MSW b Compare: 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 Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc x1 − x x1 − x x2 − x3 etc Chap 11-44 Two-Way 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 level the line speed is set? Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 11-45 Two-Way ANOVA (continued)  Assumptions  Populations are normally distributed  Populations have equal variances  Independent random samples are drawn Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 11-46 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 N = total number of observations in all cells Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 11-47 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 N-1 Variation due to interaction between A and B SSE Inherent variation (Error) Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc (continued) Degrees of Freedom: a–1 b–1 (a – 1)(b – 1) N – ab Chap 11-48 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 SSB = an′∑ ( x j − x )2 j=1 Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 11-49 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 Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 11-50 Two Factor ANOVA Equations a where: b xi = x= j=1 k =1 n′ ∑∑∑ x i =1 j=1 k =1 abn′ n′ ∑∑ x b (continued) ijk = Grand Mean ijk bn′ = Mean of each level of factor A a xj = n′ ∑∑ x i=1 k =1 an′ ijk = Mean of each level of factor B x ijk x ij = ∑ = Mean of each cell k =1 n′ n′ Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc a = number of levels of factor A b = number of levels of factor B n’ = number of replications in each cell Chap 11-51 Mean Square Calculations SS A MS A = Mean square factor A = a −1 SSB MSB = Mean square factor B = b −1 MS AB SS AB = Mean square interaction = (a − 1)(b − 1) SSE MSE = Mean square error = N − ab Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 11-52 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 HA: factors A and B interact Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc F Test for Factor A Main Effect MS A F= MSE Reject H0 if F > Fα F Test for Factor B Main Effect MSB F= MSE Reject H0 if F > Fα F Test for Interaction Effect MS AB F= MSE Reject H0 if F > Fα Chap 11-53 Two-Way ANOVA Summary Table Source of Variation Sum of Squares Degrees of Freedom Factor A SSA a–1 Factor B SSB AB (Interaction) SSAB (a – 1)(b – 1) Error SSE N – ab Total SST N–1 b–1 Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Mean Squares F Statistic MSA MSA MSE = SSA /(a – 1) MSB = SSB /(b – 1) MSAB = SSAB / [(a – 1)(b – 1)] MSB MSE MSAB MSE MSE = SSE/(N – ab) Chap 11-54 Features of Two-Way ANOVA F Test  Degrees of freedom always add up  N-1 = (N-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  Total = error + factor A + factor B + interaction Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 11-55 Examples: Interaction vs No Interaction  Interaction is present: No interaction: Factor B Level Factor B Level Factor A Levels Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Mean Response Factor B Level Mean Response  Factor B Level Factor B Level Factor B Level Factor A Levels Chap 11-56 Chapter Summary  Described one-way analysis of variance      Described randomized complete block designs    The logic of ANOVA ANOVA assumptions F test for difference in k means The Tukey-Kramer procedure for multiple comparisons F test Fisher’s least significant difference test for multiple comparisons Described two-way analysis of variance  Examined effects of multiple factors and interaction Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc Chap 11-57 ... analysis of variance test with replications results Business Statistics: A Decision- Making Approach, 6e © 2010 PrenticeHall, Inc Chap 11-2 Chapter Overview Analysis of Variance (ANOVA) One-Way... Does not mean that all population means are different (some pairs may be the same) Business Statistics: A Decision- Making Approach, 6e © 2010 PrenticeHall, Inc Chap 11-7 One-Factor ANOVA H0 : μ1... Within-Sample Variation = dispersion that exists among the data values within a particular factor level (SSW) Business Statistics: A Decision- Making Approach, 6e © 2010 PrenticeHall, Inc Chap 11-11