Slide 1Learning ObjectivesSlide 3TerminologyAsideMini-case: Package Design SelectionNew Product Marketing ProblemThe Sales Data by DesignTerminologyOne-Way Analysis of Variance (in a nutshell)Required ConditionsVariability in X-bar When H0 Is TrueVariability in X-bar When H0 Is FalseMore on Within vs. Between VarianceTest Statistic = Ratio of VariancesMaking the DecisionThe F Sampling DistributionSlide 18Analysis of the Mini-case: Sales by Package DesignMini-Case: New Product MarketingANOVA : The Research QuestionThe Combined (Total) Sales DataThe Sales Data by DesignPartitioning the Total VariationOne-way ANOVA Partitions SSTotalThe One-way ANOVA Summary TableCompare the MSG and the MSEThe Decision and ConclusionThe P-valueCause-and-Effect?Analysis of the Mini-case: Sales by Package DesignWithin Groups (Error) VariationCalculation of MSEBetween Groups VariationCalculation of MSGRe-capRe-capTo DoOne-Way ANalysis Of Variance(ANOVA)Part ILecture #7“What is the variation trying to tell us about a process, about the people in the process?” ---- W. Edwards Deming (1900 – 1993)Learning Objectives1. Understand how the analysis of variance procedure can be used to determine if the means from two or more populations are equal.2. Know how to calculate the sums of squares and mean squares in the one-way ANOVA.3. Know how to determine the degrees of freedom for the different sources of variation in the One-way ANOVA.4. Know the assumptions that form the basis of the ANOVA procedure.5. Understand the use of the F-Distribution in performing the ANOVA.Terminology•ANOVA•Family Significance Level•Response Variable•Dependent Variable•Factor•Independent Variable•Treatment•Experimental Unit•CRD•Total Variation•Between Variation•Within Variation•Pooled Variance Estimate•F-ratio•Post Hoc Analysis•Sum of Squares•Mean Squares•Degrees of FreedomAside if only random variance is present•MINI-CASE:PACKAGE DESIGN SELECTIONNew Product Marketing ProblemA marketing manager is interested in the effect of different types of packaging on the sales of a particular new item. Three different types of packaging have been suggested for the item. Will average sales be the same for all three package types?The manager selects 15 stores from the population of stores that would stock the item. The 3 package types are then randomly assigned to the stores (5 stores per package type). The sales of the item at each of the stores are carefully recorded for a period of one month.i jDesign 1 Design 2 Design 31 52 28 152 48 35 143 43 34 234 50 32 21ni = 543 34 14The Sales Data by Design_______________2.471x6.322x4.173x7.1621s8.722s3.1823sTerminology9•Response Variable: ________________•Factor: ____________•Treatments: ____________•Experimental Units: _______________Comparing multiple (2 or more) population meansH0: µ1 = µ2 = ···= µIHa: Not all µi are equalONE-WAY ANALYSIS OF VARIANCE(in a nutshell)Required Conditions1. Independent ____________.2. The treatment populations are _________ distributed.Use Q-Q Normal plots or other suitable tests 3. Homogeneity of ____________.Bartlett’s Test, Levine’s TestRule of Thumb: The largest sample standard deviation should be no more than twice as large as the smallest sample ____________________.Variability in X-bar When H0 Is TrueIf All of the ’s should be “similar.” Differences in the ’s are due to sampling error only. �2=� �´�2 321ixix1x2x3xVariability in X-bar When H0 Is FalseIf “Not all µi equal each other,” e. g., µ2 < µ1 < µ3 At least one of the ’s differs significantly from anotherVariability in the ’s now includes the effect of the group differences in addition to random (sampling) error.ixix1231x2x3xH0 is True H0 is FalseMore on Within vs. Between VarianceCombinedGroup 3Group 2Group 16055504540CombinedGroup 3Group 2Group 1807060504030´� ´� ´�1 ´�1 ´�2 ´�2 ´�3 ´�3 MSW: Estimate σ2 with pooled variance: . MSB: Estimate σ2 using variance of = .Test Statistic = Ratio of VariancesWe measure the variability based on differences in the , and compare it to a measure of random variability ----The Test Statistic: The F-ratios'ixMSWMSBFobsvariancegroupwithinresiduals'xinsdifferencefrommeasuredvarianceMaking the DecisionIf the variance associated with the (Between) grouping effect is significantly large relative to the (Within) random variance, we have statistical evidence that the group population means are not all equal, i.e.,If Fobs > F*, Reject H0. We would conclude that the population means are not all the same.The F Sampling Distributionα= 0.05ν1 = 2ν2 = 123.8853ANALYSIS OF THE MINI-CASE:SALES BY PACKAGE DESIGNMini-Case: New Product MarketingA marketing manager is interested in the effect of different types of packaging on the sales of a particular new item. Three different types of packaging have been suggested for the item. Will average sales be the same for all three package types?The manager selects 15 stores from the population of stores that would stock the item. The 3 package types are then randomly assigned to the stores (5 stores per package type). The sales of the item at each of the stores are carefully recorded for a period of one month.ANOVA : The Research QuestionDo sales of the product differ on average as a result of the package design? H0: µ1 = µ2 = µ3(Mean sales of the product are the same for all three package designs)Ha: ___________ (Mean sales are not the same for all three package designs)i jDesign 1 Design 2 Design 3152 28 15248 35 14343 34 23450 32 21ni = 543 34 14The Combined (Total) Sales Data Grand Mean: Total Variance:nx = xijn=jIii 11nT =iIi14.321587163236 =Totali jijdfTotalSquaresofSumnxxs 1)(2283.170146.2391=i jDesign 1 Design 2 Design 3152 28 15248 35 14343 34 23450 32 21ni = 543 34 14s1 = 4.1 s2 = 2.8 s3 = 4.3The Sales Data by DesignTreatment Average:Treatment Variance:Balanced Design: ______________________________________2.471x6.322x4.173x7.1621s8.722s3.1823sijijinxx1)(2ijiijinxxsPartitioning the Total Variation19.632.4-52total xxij4.847.2-52withiniijxx14.832.4-47.2between xxiOne-way ANOVA Partitions SSTotalExplained SSTotal SSUnexplained SS•SS Treatments•SS Between•SS Groups•SS Error•SS