Question

Source	Df	ss	ms	f stat
housing			0.8088
Wtclass		0.06
interaction			0.3078
error total		4.2133	na	na na

(15 points) If in the previous example, the researchers also determine the weight class (medium and large) for each egg. 3 out 6 eggs in each housing system are medium and the other 3 are large.

Source DF SS MS F-Stat

HOUSING 0.8088

WTCLASS 0.06

Interaction 0.3078

Error N.A.

Total 4.2133   N.A. N.A.

1. Fill in the ANOVA table (total 13 missing values)

2. Are there main effects here? Do hypothesis testing at alpha=0.05. Does your conclusion

   consistent with question 2?

3. Is there evidence of interaction in the model? Do hypothesis testing at alpha=0.05. If

   there is interaction, give an example of possible interaction. If no interaction, explain what “no interaction” means

Answer 1

Source	DF	SS	MS	F-Stat
HOUSING	3	2.4264	0.8088	16.10554
WTCLASS	1	0.06	0.06	1.194773
Interaction	3	0.9234	0.3078
Error	16	0.8035	0.05021875	NA
Total	23	4.2133	NA	NA

Level of HOUSING, a = 4

Level of WTCLASS, b = 2

DF for HOUSING = a - 1 = 4 - 1 = 3

DF for WTCLASS = b-1 = 2 - 1 = 1

DF for Interaction = (a - 1) * (b-1) = (4-1) * (2 - 1) = 3

DF Total = Number of observations - 1 = 6 * 4 - 1 = 23

DF Error = DF Total - (DF HOUSING + DF WTCLASS + DF Interaction)

= 23 - (3 + 1 + 3) = 16

SS HOUSING = MS HOUSING * DF HOUSING = 0.8088 * 3 = 2.4264

SS Interaction = MS Interaction * DF Interaction = 0.3078 * 3 = 0.9234

SS Error = SS Total - (SS HOUSING + SS WTCLASS + SS Interaction)

= 4.2133 - (2.4264 + 0.06 + 0.9234)

= 0.8035

MS Error = SS Error / DF Error

= 0.8035 / 16

= 0.05021875

MS WTCLASS = SS WTCLASS / DF WTCLASS

=  0.06 / 1

= 0.06

F HOUSING = MS HOUSING / MS Error = 0.8088 / 0.05021875 = 16.10554

F WTCLASS = MS WTCLASS / MS Error = 0.06 / 0.05021875 = 1.194773

F Interaction = MS Interaction / MS Error = 0.3078 / 0.05021875 = 6.129185

Null Hypothesis H0: There is no main effect of housing on nutritional content.

Alternative Hypothesis Ha: There is main effect of housing on nutritional content.

Critical value of F at df = 3, 16 and alpha=0.05 = 3.24

Since the observed F (16.10554) is greater than the critical value, we reject the null hypothesis H0 and conclude that there is significant main effect of housing on nutritional content.

Null Hypothesis H0: There is no main effect of wtclass on nutritional content.

Alternative Hypothesis Ha: There is main effect of wtclass on nutritional content.

Critical value of F at df = 1, 16 and alpha=0.05 = 4.49

Since the observed F (1.194773) is less than the critical value, we fail to reject the null hypothesis H0 and conclude that there is no significant main effect of wtclass on nutritional content.

Null Hypothesis H0: There is no interaction between housing and wtclass on nutritional content.

Alternative Hypothesis Ha: There is an interaction between housing and wtclass on nutritional content.

Critical value of F at df = 3, 16 and alpha=0.05 = 3.24

Since the observed F (6.129185) is greater than the critical value, we reject the null hypothesis H0 and conclude that there is significant interaction between housing and wtclass on nutritional content.