The given data are elaborately expressed as
Grain-Grass | Grass A | Grass B | Grass C |
Grain A | 175 | 225 | 250 |
Grain A | 160 | 215 | 240 |
Grain A | 185 | 230 | 260 |
Grain B | 190 | 245 | 275 |
Grain B | 185 | 240 | 260 |
Grain B | 195 | 255 | 285 |
Grain C | 210 | 255 | 300 |
Grain C | 220 | 245 | 310 |
Grain C | 200 | 265 | 295 |
Grain D | 225 | 275 | 350 |
Grain D | 235 | 270 | 360 |
Grain D | 220 | 280 | 345 |
Above is the ANOVA-2 way with 3 observations per cell and its analysis is as follws:
Anova: Two-Factor With Replication | ||||||
SUMMARY | Grass A | Grass B | Grass C | Total | ||
Grain A | ||||||
Count | 3 | 3 | 3 | 9 | ||
Sum | 520 | 670 | 750 | 1940 | ||
Average | 173.3333 | 223.3333 | 250 | 215.5556 | ||
Variance | 158.3333 | 58.33333 | 100 | 1215.278 | ||
Grain B | ||||||
Count | 3 | 3 | 3 | 9 | ||
Sum | 570 | 740 | 820 | 2130 | ||
Average | 190 | 246.6667 | 273.3333 | 236.6667 | ||
Variance | 25 | 58.33333 | 158.3333 | 1418.75 | ||
Grain C | ||||||
Count | 3 | 3 | 3 | 9 | ||
Sum | 630 | 765 | 905 | 2300 | ||
Average | 210 | 255 | 301.6667 | 255.5556 | ||
Variance | 100 | 100 | 58.33333 | 1640.278 | ||
Grain D | ||||||
Count | 3 | 3 | 3 | 9 | ||
Sum | 680 | 825 | 1055 | 2560 | ||
Average | 226.6667 | 275 | 351.6667 | 284.4444 | ||
Variance | 58.33333 | 25 | 58.33333 | 3015.278 | ||
Total | ||||||
Count | 12 | 12 | 12 | |||
Sum | 2400 | 3000 | 3530 | |||
Average | 200 | 250 | 294.1667 | |||
Variance | 504.5455 | 418.1818 | 1635.606 | |||
ANOVA-TABLE | ||||||
Source of Variation | SS | df | MS | F | P-value | F crit |
Between Grains | 23097.22 | 3 | 7699.074 | 96.4058 | 1.59E-13 | 3.008787 |
Between Grasses | 53272.22 | 2 | 26636.11 | 333.5304 | 3.08E-18 | 3.402826 |
Interaction | 3127.778 | 6 | 521.2963 | 6.527536 | 0.000348 | 2.508189 |
Within | 1916.667 | 24 | 79.86111 | |||
Total | 81413.89 | 35 |
(a) step 1
H0(Interaction): There is no significant interaction between Grass and Grain
Ha(Interaction): There is a significant interaction between Grass and Grain
step 2
Calculated F = 6.5275 (See the above last ANOVA-TABLE)
Step 3
p-value of F for 6.5275 with(6, 24) degree of freedom is 0.000348 (See the above last ANOVA-TABLE)
Step 4
Since p value(0.000348) is much less than the 5%(=0.05) significance level, then we reject H0(Interaction) and accept Ha(Interaction) and conclude that there is a significant interaction between Grass and Grain.
(b) step 1
H0(grains): There is no significant difference in average weight gain for the cows among the four different grains
Ha(grains): There is a significant difference in average weight gain for the cows among the four different grains
step 2
Calculated F = 96.4058(See the above last ANOVA-TABLE)
Step 3
p-value of F for 96.4058 with(3, 24) degree of freedom is 0(See the above last ANOVA-TABLE)
Step 4
Since p value(0) is much less than the 5%(=0.05) significance level, then we reject H0(grains) and accept Ha(grains) and conclude that there is a significant difference in average weight gain for the cows among the four different grains.
(c) step 1
H0(grasses): There is no significant difference in average weight gain for the cows among the three different grasses
Ha(grains): There is a significant difference in average weight gain for the cows among the three different grasses
step 2
Calculated F = 333.5304(See the above last ANOVA-TABLE)
Step 3
p-value of F for 333.5304 with(2, 24) degree of freedom is 0(See the above last ANOVA-TABLE)
Step 4
Since p value(0) is much less than the 5%(=0.05) significance level, then we reject H0(grasses) and accept Ha(grasses) and conclude that there is a significant difference in average weight gain for the cows among the three different grasses.
3) A dairy man thinks that the average weight gain of his cows depends on two factors: the type of grain which they are fed and the type of grass which they are fed. The dairyman has four differe...
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