Question

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 different types of grain from which to choose and three different types of grass from which to choose. He would like to determine if there is a particular combination of grain and grass which would lead to the greatest weight gain on average for his cows. He randomly selects three one- year-old cows and assigns them to each of the possible combinations of grain and grass. After one year he records the weight gain for each cow with the following results :

	Grass A	Grass B	Grass C
Grain A	175,160,185	225,215,230	250,240,260
Grain B	190,185,195	245,240,255	275,260,285
Grain C	210,220,200	255,245,265	300,310,295
Grain D	225,235,220	275,270,280	350,360,345

Perform the following hypothesis tests at 5 % level of significance. Show all four steps and show complete work. Use P- Value Method. Provide the complete ANOVA TABLE :

Is there a significant interaction between the independent variables (factors) : grass and grain?

b) Is there is a significant difference in average weight gain for the cows among the four different types of grain ?

c) Is there a significant difference in average weight gain for the cows among the three different types of grass?

Answer 1

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.