Ans. The given problem is studied under the
Analysis of variance(ANOVA), i.e., we need to calculate the
variation between and within the groups.
![Tt: Total mean](https://latex.codecogs.com/gif.latex?%5Cbar%7Bx_%7Bt%7D%7D%3A%20Total%20%5C%3Amean)
![ar{x_{1}}, ar{x_{2}},ar{x_{3}},ar{x_{4}}: Means :of :different :groups](https://latex.codecogs.com/gif.latex?%5Cbar%7Bx_%7B1%7D%7D%2C%20%5Cbar%7Bx_%7B2%7D%7D%2C%5Cbar%7Bx_%7B3%7D%7D%2C%5Cbar%7Bx_%7B4%7D%7D%3A%20Means%20%5C%3Aof%20%5C%3Adifferent%20%5C%3Agroups)
As we know that ANOVA is used to study the variation between and
within the groups.
![F=rac{Difference:between:groups}{Difference:within:groups}](https://latex.codecogs.com/gif.latex?F%3D%5Cfrac%7BDifference%5C%3Abetween%5C%3Agroups%7D%7BDifference%5C%3Awithin%5C%3Agroups%7D)
![F=rac{MS_{b}}{MS_{w}}](https://latex.codecogs.com/gif.latex?F%3D%5Cfrac%7BMS_%7Bb%7D%7D%7BMS_%7Bw%7D%7D)
where, ![](https://latex.codecogs.com/gif.latex?)
![MS_{w}=rac{SS_{w}}{df_{w}},:::MS_{b}=rac{SS_{b}}{df_{b}}](https://latex.codecogs.com/gif.latex?MS_%7Bw%7D%3D%5Cfrac%7BSS_%7Bw%7D%7D%7Bdf_%7Bw%7D%7D%2C%5C%3A%5C%3A%5C%3AMS_%7Bb%7D%3D%5Cfrac%7BSS_%7Bb%7D%7D%7Bdf_%7Bb%7D%7D)
mean square
between groups
mean square
within groups.
degree of
freedom for within group
degree of
freedom for between group.
Calculation for
![mathbf{SS_{b}=Sigma N_{k}(ar{x_{k}}-ar{x_{t}})^{2}}](https://latex.codecogs.com/gif.latex?%5Cmathbf%7BSS_%7Bb%7D%3D%5CSigma%20N_%7Bk%7D%28%5Cbar%7Bx_%7Bk%7D%7D-%5Cbar%7Bx_%7Bt%7D%7D%29%5E%7B2%7D%7D)
![SS_{b}=N_{1}(ar{x_{1}}-ar{x_{t}})^{2}+N_{2}(ar{x_{2}}-ar{x_{t}})^{2}+N_{3}(ar{x_{3}}-ar{x_{t}})^{2}+N_{4}(ar{x_{4}}-ar{x_{t}})^{2}](https://latex.codecogs.com/gif.latex?SS_%7Bb%7D%3DN_%7B1%7D%28%5Cbar%7Bx_%7B1%7D%7D-%5Cbar%7Bx_%7Bt%7D%7D%29%5E%7B2%7D+N_%7B2%7D%28%5Cbar%7Bx_%7B2%7D%7D-%5Cbar%7Bx_%7Bt%7D%7D%29%5E%7B2%7D+N_%7B3%7D%28%5Cbar%7Bx_%7B3%7D%7D-%5Cbar%7Bx_%7Bt%7D%7D%29%5E%7B2%7D+N_%7B4%7D%28%5Cbar%7Bx_%7B4%7D%7D-%5Cbar%7Bx_%7Bt%7D%7D%29%5E%7B2%7D)
![SS_{b}=9(2.6667-2.8889)^{2}+9(2.2222-2.8889)^{2}+9(3.4444-2.8889)^{2}+9(3.2222-2.8889)^{2}](https://latex.codecogs.com/gif.latex?SS_%7Bb%7D%3D9%282.6667-2.8889%29%5E%7B2%7D+9%282.2222-2.8889%29%5E%7B2%7D+9%283.4444-2.8889%29%5E%7B2%7D+9%283.2222-2.8889%29%5E%7B2%7D)
![SS_{b}=0.4444+4.0004+2.7772+0.9998](https://latex.codecogs.com/gif.latex?SS_%7Bb%7D%3D0.4444+4.0004+2.7772+0.9998)
![SS, - 8.2218](https://latex.codecogs.com/gif.latex?%5Cmathbf%7BSS_%7Bb%7D%3D8.2218%7D)
Calculation for
![2](https://latex.codecogs.com/gif.latex?%5Cmathbf%7BSS_%7BT%7D%3D%28%5CSigma%20x_%7Bk%7D%5E%7B2%7D%29-N%5Cbar%7Bx%7D_%7Bt%7D%5E%7B2%7D%7D)
N=4*9(no. of groups*no of dataset in each group)=36
![SS_{T}=90+72+143+103-36(2.8889)^{2}](https://latex.codecogs.com/gif.latex?SS_%7BT%7D%3D90+72+143+103-36%282.8889%29%5E%7B2%7D)
![SST408-300.4468](https://latex.codecogs.com/gif.latex?SS_%7BT%7D%3D408-300.4468)
![SS_{T}=107.5532](https://latex.codecogs.com/gif.latex?SS_%7BT%7D%3D107.5532)
![SS_{w}=SS_{T}-SS_{b}](https://latex.codecogs.com/gif.latex?SS_%7Bw%7D%3DSS_%7BT%7D-SS_%7Bb%7D)
we have,
![SS_{T}=107.5532,::SS_{b}=8.2218](https://latex.codecogs.com/gif.latex?SS_%7BT%7D%3D107.5532%2C%5C%3A%5C%3ASS_%7Bb%7D%3D8.2218)
![SS_{w}=107.5532-8.2218](https://latex.codecogs.com/gif.latex?SS_%7Bw%7D%3D107.5532-8.2218)
![SS_{w}=99.3314](https://latex.codecogs.com/gif.latex?SS_%7Bw%7D%3D99.3314)
Calculation for degree of freedom
![df_{w}=N-k=36-4=32](https://latex.codecogs.com/gif.latex?df_%7Bw%7D%3DN-k%3D36-4%3D32)
![df_{b}=k-1=4-1=3](https://latex.codecogs.com/gif.latex?df_%7Bb%7D%3Dk-1%3D4-1%3D3)
number of cases in dataset(i.e., 36=4*9 in our case)
number of groups(i.e., 4 in our case)
,
since we have calculated all the desired value:
![SSw 99.3314 dfu MS.-3 3.10410625 3.1041 32](https://latex.codecogs.com/gif.latex?MS_%7Bw%7D%3D%5Cfrac%7BSS_%7Bw%7D%7D%7Bdf_%7Bw%7D%7D%3D%5Cfrac%7B99.3314%7D%7B32%7D%3D3.10410625%5Capprox%203.1041)
![MS,,,ー3.1041 ~. 3.10](https://latex.codecogs.com/gif.latex?MS_%7Bw%7D%3D3.1041%5Capprox%203.10)
![MS, S 8.2218 dfb](https://latex.codecogs.com/gif.latex?MS_%7Bb%7D%3D%5Cfrac%7BSS_%7Bb%7D%7D%7Bdf_%7Bb%7D%7D%3D%5Cfrac%7B8.2218%7D%7B3%7D%3D2.7406)
![MSb = 2.7406 ~ 2.74](https://latex.codecogs.com/gif.latex?MS_%7Bb%7D%3D2.7406%5Capprox%202.74)
So now we can calculate the value of ![mathbf{F:}](https://latex.codecogs.com/gif.latex?%5Cmathbf%7BF%3A%7D)
![F=rac{MS_{b}}{MS_{w}}](https://latex.codecogs.com/gif.latex?F%3D%5Cfrac%7BMS_%7Bb%7D%7D%7BMS_%7Bw%7D%7D)
![F=rac{MS_{b}}{MS_{w}}=rac{2.74}{3.10}](https://latex.codecogs.com/gif.latex?F%3D%5Cfrac%7BMS_%7Bb%7D%7D%7BMS_%7Bw%7D%7D%3D%5Cfrac%7B2.74%7D%7B3.10%7D)
![Е 0.8839 0.88](https://latex.codecogs.com/gif.latex?F%3D0.8839%5Capprox%200.88)
![F=0.88](https://latex.codecogs.com/gif.latex?F%3D0.88)
The table for the calculated values:
Sources of variation |
SS |
df |
MS |
F |
Between group |
8.2218 8.22 |
3 |
2.74 |
![0.8839approx 0.88](https://latex.codecogs.com/gif.latex?0.8839%5Capprox%200.88) |
Within group(error) |
99.3314 99.33 |
32 |
3.10 |
|
Total |
107.5532 107.55 |
35 |
|
|
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