# OSU BUSMGT 2320 - s_7_One-way ANOVA Part I Autumn 2014 (38 pages)

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## s_7_One-way ANOVA Part I Autumn 2014

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## s_7_One-way ANOVA Part I Autumn 2014

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Pages:
38
School:
Ohio State University
Course:
Busmgt 2320 - Decision Sciences: Statistical Techniques
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Lecture 7 One Way ANalysis Of Variance ANOVA Part I What is the variation trying to tell us about a process about the people in the process W Edwards Deming 1900 1993 Learning Objectives 1 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 Freedom Aside if only random variance is present MINI CASE PACKAGE DESIGN SELECTION New Product Marketing Problem A 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 The Sales Data by Design i j 1 2 3 4 ni 5 Design 1 Design 2 Design 3 52 48 43 50 43 28 35 34 32 34 15 14 23 21 14 x1 47 2 s12 16 7 x2 32 6 s 22 7 8 x3 17 4 s32 18 3 Terminology Response Variable Factor Treatments Experimental Units 9 ONE WAY ANALYSIS OF VARIANCE in a nutshell Comparing multiple 2 or more population means H0 1 2 I Ha Not all i are equal Required Conditions 1 Independent 2 The treatment populations are distributed Use Q Q Normal plots or other suitable tests 3 Homogeneity of Bartlett s Test Levine s Test Rule 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 True If 1 2 3 xi All of the s should be similar Differences in the s xi are due to sampling error only x 2 x1 x3 2 2 Variability in X bar When H0 Is False If Not all i equal each other e g 2 1 3 At least one of the xi s differs significantly from another Variability in the xi s now includes the effect of the group differences in addition to random sampling error x2 2 1 x1 3 x3 More on Within vs Between Variance H0 is True Group 1 H0 is False 1 Group 1 Group 2 2 Group 2 Group 3 3 Group 3 Combined Combined 40 45 50 55 60 1 2 3 30 40 50 MSB Estimate 2 using variance of MSW Estimate 2 with pooled variance 60 70 80 Test Statistic Ratio of Variances We measure the variability based on differences in the xi s and compare it to a measure of random variability The Test Statistic The F ratio Fobs variance measured from difference s in x s MSB residual within group variance MSW Making the Decision If 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 Distribution 0 05 1 2 2 12 3 8853 ANALYSIS OF THE MINI CASE SALES BY PACKAGE DESIGN Mini Case New Product Marketing A 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 Question Do sales of the product differ on average as a result of the package design H0 1 2 3 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 The Combined Total Sales Data i j 1 2 3 4 ni 5 I Grand Mean x Design 1 Design 2 Design 3 52 48 43 50 43 28 35 34 32 34 15 14 23 21 14 ni I xij i 1 j 1 i 1 n x ij Total Variance 2 s i j n 1 n x 2 Ti 236 163 87 32 4 15 Sum of Squares Total 2391 6 170 83 14 dfTotal The Sales Data by Design i j 1 2 3 4 ni 5 Design 1 Design 2 Design 3 52 48 43 50 43 28 35 34 32 34 15 14 23 21 14 x1 47 2 s12 16 7 s1 4 1 x2 32 6 x3 17 4 2 s 7 8 3 18 3 s2 2 8 s3 4 3 s 22 Treatment Average x ij xi j ni Treatment Variance x ij si2 xi j ni 1 Balanced Design Partitioning the Total Variation xij xi within 52 47 2 4 8 xi x between 47 2 32 4 14 8 xij x total 52 32 4 19 6 One way ANOVA Partitions SSTotal Total SS X ij X i SS Error SS Within Explained SS i j SS Treatments SS Between SS Groups ni X i X 2 2 Unexplained SS i j X ij X i 2 The One way ANOVA Summary Table OneWay ANOVA Table Between Variation Within Variation Total Variation Sum of Degrees of Mean Squares Freedom Squares 2220 400 171 200 2391 600 2 12 14 1110 200 14 267 The Total SS is partitioned into two additive pieces F Ratio p Value 77 818 0 0001 The Total df are partitioned into two additive pieces Compare the MSG and the MSE MSG MSB 1110 2 77 82 MSE MSE 14 267 The MSB is 77 82 times bigger than the MSW If H0 1 2 3 is true MSG and MSE both estimate the grand variance 2 MSG should be close to 1 MSE Large values of MSG MSEindicate that H 0 is false The Decision and Conclusion Since Fobs FC decide to Reject H0 Conclude that there is statistically significant evidence at the 5 level that mean sales are not the same with all package types Conduct a post hoc …

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