2 Analysis of Variance : the simultaneous comparizon of several population means F-distribution : used to test whether two samples are from population having equal variances and it is also applied when we want to compare several population means simultaneously.
3 Comparing two population variance 2Ho : σ = σH1 : σ = σTest statistic for comparing two variances :F = ~ F- distribution22122122s12s2
4 Contoh :Sean Lammers, president of the Lammers Limos Service Company, membandingkan variasi waktu tempuh (menit) dari Cityhall di Toledo, Ohio ke Metro Airport di Detroit melalui 2 rute ( via US -25 dan Interstate-75), dengan taraf α = 10 %US-25 routeInterstate-75 route5267564570546459606151635765
5 US-Route-25Interstate-75Rata-rata = 28.29Std deviasi =Rata-rata = 59.00Std deviasi =US-Route25 is more variation then Interstate-75 route,This is somewhat consistent with his knowledge of two route; US-25 contain more stoplights, I-75 is limited-access interstate highway, but I-75 is several miles longer.
6 Nilai F = 4.23,Nilai tabel F distribution pada Appendix G pada buku Douglas A. Lind,dkk,Statistical Techniques in Business of Economics,International Edition,yaitu : nilai F dengan α/2 = 5 %, dan df numerator 6 dan df denominator 7 adalah sebesar 3.87.Hasil ini menunjukkan bahwa Ho di tolakBerikan kesimpulan anda.
7 The Anova TestTotal Variation : the sum of the squared differences between each observation and the overall mean (grand mean); SS-totalTreatment Variation : the sum of the squared differences between each treatment mean and the overall mean (grand mean); SSTRandom Variation : the sum of the squared differences between each observation and its treatment mean; SSE
9 ANOVA Table Source of Variation Sum of Squares Degrees of Freedom Mean SquareFTreatmentsErrorTotalSSTSSESS- Tolalk-1n-kn-1SST/(k-1) = MSTSSE/(n-k) = MSEMST/ MSE
10 Example :Prof. James had students in his marketing class rate his performance as Excellent, Good, Fair or Poor. A graduate student collected the rate and from records office, Prof. James was matched with his or her course grade. The sample information is reported below. Is there a difference in the mean score of the students in each of the four rating categories ? Use the 0.01 significance level. (note : The rating is the treatment variable)Formulasikan hipotesis, buat tabel anova, tentukan nilai F tabeldan buat keputusan !
12 Ho : μ1= μ2 = μ3 = μ4H1 : The mean scores are not all equal
13 Berikan kesimpulan anda ! ANOVA TableSource of VariationSum of SquaresDegrees of FreedomMean SquareFTreatmentsErrorTotal890.68594.41SSESS- Tolal31821296.8933.028.99Nilai F tabel untuk α = 0.01, df numerator = 3 dan df denominator =18, adalah 5.09Berikan kesimpulan anda !
14 Two-way AnovaWe have the second treatment variable, that is Blocking variableBlocking variable : A second treatment variable that when included in the ANOVA analysis will have the effect of reducing the SSE term
15 SSB = k Σ(Xb-XG)2k is the number of treatmentb is the number of blocksXb is the sample mean of block bXG is the overall or grand meanSSE = SS total – SST - SSB
16 ANOVA Table Source of Variation Sum of Squares Degrees of Freedom Mean SquareFTreatmentsBlocksErrorTotalSSTSSBSSESS- Tolalk-1b-1(k-1)(b-1)n-1SST/(k-1) = MSTSSB/(b-1) =MSBSSE/(k-1)(b-1) = MSEMST/ MSEMSB/MSE
17 ExampleThe Chapin Manufacturing Company operates 24 hours a day, five days a week. The workers rotate shifts each week. Management is interested in whether there is a difference in the number of units produced when the employees work on various shifts. A sample of five workers is selected and their output recorded on each shift. At the 0.05 significance level, can we conclude there is a difference in the mean production rate by shift or by employee ?
19 ANOVA Table For Block : For treatment : Ho :μ1 = μ2 = μ3 = μ4 = μ5 H1 : Not all means equalReject if F > 4.46For Block :Ho :μ1 = μ2 = μ3 = μ4 = μ5H1 : Not all means equalReject if F > 3.84ANOVA TableSource of VariationSum of SquaresDegrees of FreedomMean SquareFTreatment (rotate shift)Blocks (employee)ErrorTotal62.5333.7343.47139.73248148.43255.43385.751.55There is a difference in shifts but not by employees.