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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.

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Comparing two population variance Ho : σ = σ H1 : σ = σ Test statistic for comparing two variances : F = ~ F- distribution s 2 1 s 2 2

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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 route

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US-Route-25Interstate-75 Rata-rata = Std deviasi = Rata-rata = Std 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.

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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 Hasil ini menunjukkan bahwa Ho di tolak Berikan kesimpulan anda.

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The Anova Test Total Variation : the sum of the squared differences between each observation and the overall mean (grand mean); SS-total Treatment Variation : the sum of the squared differences between each treatment mean and the overall mean (grand mean); SST Random Variation : the sum of the squared differences between each observation and its treatment mean; SSE

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SS-total = Σ(X-X G ) 2 SSE = Σ(X-X C ) 2 SST = Σ(X c -X G ) 2 SST = SS-total - SSE

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ANOVA Table Source of VariationSum of Squares Degrees of Freedom Mean SquareF Treatments Error Total SST SSE SS- Tolal k-1 n-k n-1 SST/(k-1) = MST SSE/(n-k) = MSE MST/ MSE

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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 tabel dan buat keputusan !

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Course Grades ExcellentGoodFairPoorTotal Column Total n Mean

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Ho : μ 1 = μ 2 = μ 3 = μ 4 H1 : The mean scores are not all equal

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ANOVA Table Source of VariationSum of Squares Degrees of Freedom Mean SquareF Treatments Error Total SSE SS- Tolal Nilai F tabel untuk α = 0.01, df numerator = 3 dan df denominator =18, adalah 5.09 Berikan kesimpulan anda !

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Two-way Anova We have the second treatment variable, that is Blocking variable Blocking variable : A second treatment variable that when included in the ANOVA analysis will have the effect of reducing the SSE term

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SSB = k Σ(X b -X G ) 2 k is the number of treatment b is the number of blocks X b is the sample mean of block b X G is the overall or grand mean SSE = SS total – SST - SSB

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ANOVA Table Source of Variation Sum of Squares Degrees of Freedom Mean SquareF Treatments Blocks Error Total SST SSB SSE SS- Tolal k-1 b-1 (k-1)(b-1) n-1 SST/(k-1) = MST SSB/(b-1) =MSB SSE/(k-1)(b-1) = MSE MST/ MSE MSB/MSE

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Example The 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 ?

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Employee Units Produced DayAfternoonNight Skaff Lum Clark Treece Morgan282627

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ANOVA Table Source of VariationSum of Squares Degrees of Freedom Mean SquareF Treatment (rotate shift) Blocks (employee) Error Total For treatment : Ho :μ 1 = μ 1 = μ 3 H1 : Not all means equal Reject if F > 4.46 For Block : Ho :μ 1 = μ 2 = μ 3 = μ 4 = μ 5 H1 : Not all means equal Reject if F > 3.84 There is a difference in shifts but not by employees.

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