2. Example-2
Solve using Two-way ANOVA method
| Observation | A | B | C | D | E | F | | 1 | 1200 | 1000 | 980 | 900 | 750 | 800 | | 2 | 1000 | 1100 | 700 | 800 | 500 | 700 | | 3 | 890 | 650 | 1100 | 900 | 400 | 350 |
Solution:
Given problem
| Observation | `A` | `B` | `C` | `D` | `E` | `F` | | `1` | 1200 | 1000 | 980 | 900 | 750 | 800 | | `2` | 1000 | 1100 | 700 | 800 | 500 | 700 | | `3` | 890 | 650 | 1100 | 900 | 400 | 350 |
| `A` | `B` | `C` | `D` | `E` | `F` | Row total `(x_(r))` | | `1` | 1200 | 1000 | 980 | 900 | 750 | 800 | `5630` | | `2` | 1000 | 1100 | 700 | 800 | 500 | 700 | `4800` | | `3` | 890 | 650 | 1100 | 900 | 400 | 350 | `4290` | | Col total `(x_(c))` | `3090` | `2750` | `2780` | `2600` | `1650` | `1850` | `14720` |
`sum x^2=13010000 ->(A)`
`(sum x_(c)^2)/(r)=1/3(3090^2+2750^2+2780^2+2600^2+1650^2+1850^2)`
`=1/3(9548100+7562500+7728400+6760000+2722500+3422500)`
`=1/3(37744000)`
`=12581333.3333 ->(B)`
`(sum x_(r)^2)/(c)=1/6(5630^2+4800^2+4290^2)`
`=1/6(31696900+23040000+18404100)`
`=1/6(73141000)`
`=12190166.6667 ->(C)`
`(sum x)^2/(n)=(14720)^2/18`
`=216678400/18`
`=12037688.8889 ->(D)`
Sum of squares total
`SS_T=sum x^2 - (sum x)^2/(n)=(A)-(D)`
`=13010000-12037688.8889`
`=972311.1111`
Sum of squares between rows
`SS_R=(sum x_(r)^2)/(c) - (sum x)^2/(n)=(C)-(D)`
`=12190166.6667-12037688.8889`
`=152477.7778`
Sum of squares between columns
`SS_C=(sum x_(c)^2)/(r) - (sum x)^2/(n)=(B)-(D)`
`=12581333.3333-12037688.8889`
`=543644.4444`
Sum of squares Error (residual)
`SS_E=SS_T - SS_R - SS_C`
`=972311.1111-152477.7778-543644.4444`
`=276188.8889`
ANOVA table
| Source of Variation | Sums of Squares
SS | Degrees of freedom
DF | Mean Squares
MS | F | | Between rows | `SS_R = 152477.7778` | `r-1 = 2` | `MS_R=152477.7778/2=76238.8889` | `76238.8889/27618.8889=2.7604` | | Between columns | `SS_C = 543644.4444` | `c-1 = 5` | `MS_C=543644.4444/5=108728.8889` | `108728.8889/27618.8889=3.9368` | | Error (residual) | `SS_E = 276188.8889` | `(r-1)(c-1) = 10` | `MS_E=276188.8889/10=27618.8889` | | | Total | `SS_T = 972311.1111` | `rc-1 = 17` | | |
This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
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