2Analysis 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.
3Comparing two population variance 2Ho : σ = σH1 : σ = σTest statistic for comparing two variances :F = ~ F- distribution22122122s12s2
4Contoh :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
5US-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.
6Nilai 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.
7The 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
9ANOVA Table Source of Variation Sum of Squares Degrees of Freedom Mean SquareFTreatmentsErrorTotalSSTSSESS- Tolalk-1n-kn-1SST/(k-1) = MSTSSE/(n-k) = MSEMST/ MSE
10Example :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 !
12Ho : μ1= μ2 = μ3 = μ4H1 : The mean scores are not all equal
13Berikan 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 !
14Two-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
15SSB = 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
16ANOVA 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
17ExampleThe 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 ?
19ANOVA 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.