Two Way ANOVA – Problem

In this ppt i have solved a problem on Two Way ANOVA.
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1. Sundar B. N. Assistant Professor Two Way ANOVA – Problem #1 ANOVA

2. Problem The following table gives the number of refrigerators sold by 4 salesmen in three months May, June and July Month Salesmen A B C D May 50 40 48 39 June 46 48 50 45 July 39 44 40 39 Is there a significant difference in the sales made by the four salesmen? Is there a significant difference in the sales made during different months?

3. Two Way ANOVA Model Since in each call there is one observation. So we will use the model yij = µ + α i + β j + eij where, yij is the yield corresponding to ith salesmen and jth months, µ is the general mean, α i is the effect due to ith salesmen, βj is the effect due to jth months and eij ~ ii d N (0, σ2) Month Salesmen A B C D May 50 40 48 39 June 46 48 50 45 July 39 44 40 39

4. Steps for Calculating Two Way ANOVA Set up Hₒ and H₁ Grand Total = Sum of all the data Find the N = The number of observations Correction Factor = G²/N Raw Sum of Square (RSS) Total Sum of Square (TSS) = RSS-CF Sum of Squares due to Factor A (SSA) Sum of Squares due to Factor B (SSB)

5. Steps for Calculating Two Way ANOVA 9) Sum of Squares due to Errors (SSE) = TSS − SSA-SSB 10) Compute MSSA =SSA/ df MSSB = SSB/ df MSSE = SSE/ df 11) Find Fᴀ=MSSA/MSSE Fᴃ=MSSB/MSSE 12) Decision

6. Step I: Set up Hₒ and H₁ Test of hypothesis that there is a significant difference in the sales made by the four salesmen H o ᴀ: α₁= α₂ = α₃ = α₄ OR H o ᴀ: αᵢ=0 for all i =1,2,3,4 H ₁ ᴀ: α₁≠ α₂ ≠ α₃ ≠ α₄ OR H ₁ ᴀ: αᵢ≠0 at least one i Similarly, there is a significant difference in the sales made during different months H oʙ : β₁= β₂ = β₃ OR H o ʙ : βj =0 for all i =1,2,3 H ₁ β : β₁≠ β₂ ≠ β₃ OR H ₁ β : βj≠0 at least one j

7. Step II, III and IV Step II: Grand Total G Step IV: Correction Factor(CF) = G²/N = 528²/12 = 2,78,784/12 CF= 23,232 Month Salesmen Total A B C D May 50 40 48 39 177 June 46 48 50 45 189 July 39 44 40 39 162 Total 135 132 138 123 528 Step III: Find the N = 12 = 135+132+138+123 = 528 = 177+189+162 = 528

8. Raw Sum of Squares Step V: Raw Sum of Squares (RSS) = Month Salesmen Total A² B² C² D² Total A B C D May 50 40 48 39 177 2500 1600 2304 1521 7925 June 46 48 50 45 189 2116 2304 2500 2025 8945 July 39 44 40 39 162 1521 1936 1600 1521 6578 Total 135 132 138 123 528 6137 5840 6404 5067 23448 Step VI:Total Sum of Square (TSS) = RSS-CF = 23448-23232 = 216 = 6137+5840+6404+5067 = 23448 = 7925+8945+6578 = 23448

9. Step VII: Sum of Squares due to Sales made during Months (SSM) Month Salesmen Total A² B² C² D² Total A B C D May 50 40 48 39 177 2500 1600 2304 1521 7925 June 46 48 50 45 189 2116 2304 2500 2025 8945 July 39 44 40 39 162 1521 1936 1600 1521 6578 Total 135 132 138 123 528 6137 5840 6404 5067 23448

10. Step VIII: Sum of Squares due to Salesmen (SSS) Month Salesmen Total A² B² C² D² Total A B C D May 50 40 48 39 177 2500 1600 2304 1521 7925 June 46 48 50 45 189 2116 2304 2500 2025 8945 July 39 44 40 39 162 1521 1936 1600 1521 6578 Total 135 132 138 123 528 6137 5840 6404 5067 23448

11. Steps IX: Sum of Squares due to Error(SSE) Step VII: Sum of Squares due to Errors (SSE) TSS = 216 SSM = 95.5 SSS = 42 = TSS-SSM-SSS = 216 – 95.5 - 42 SSE= 78.5

12. Two Way ANOVA Table for One-way Classified Data Source of Variation Degrees of Freedom ( df ) Sum of Squares (SS) Mean Sum of Squares (MSS) Variance Ratio F Between the months (V1) 3 −1=2 SSM = 95.5 MSSM= SSM/p-1 =95.5/3-1 = 47.75 F=MSSM/MSSE =47.75/13.083 =3.65 Between the Salesmen (V1) 4-1=3 SSS = 42 MSSS= SSS/q-1 =42/4-1 = 14 F=MSSS/MSSE =14/13.083 =1.07 Error (V2) (3-1)(4-1)=6 SSE = 78.5 MSSE = SSE/(p-1)(q-1) =78.5/6=13.083 Total 3x4-1=11 TSS = 216 TSS = 216 SSM = 95.5 SSS = 42 SSE = 78.5 Month Salesmen A B C D May 50 40 48 39 June 46 48 50 45 July 39 44 40 39

13. Decision Between the months Calculated F = 3.65 Tabulated F at 5% level of significance with (2, 6) degree of freedom is 5.14. Since Calculated F (CV) < F(TV), so we may accept H ₒ and conclude that sales made during months May, June and July do not differ significantly. Between the Salesmen Calculated F = 1.07 Tabulated F at 5% level of significance with (3, 6) degree of freedom is 4.76. Since Calculated F (CV) < F(TV), so we may accept H ₒ and conclude that sales made by the A, B, C, and D salesmen do not differ significantly.

14. Reference Prasad, J. (2017). “Unit-7 Introduction to Analysis of Variance. IGNOU

15. Thank You