Question

Male and female Rats (sex) were provided with either food of type 1 or food of type 2 (food) and the subsequent consumption (cons) rate was measured. Was there a difference in the consumption rate of rats based on the food type and their sex? Use the data provided on Blackboard (‘RatsFood.xlsx’to complete the analysis as asked below). For this question – the data are indeed normally distributed (no need to check)… What are the null and alternative hypotheses? What does the data look like? Please provide appropriate figures and summary statistics (measure of central tendency and variance) to write a sentence or two about the data and discuss the number of levels for each factor. Are the variances equal? Two-way ANOVA What kind of Two-Way ANOVA? What was the outcome of the Two-Way ANOVA? Was there an interaction? Evaluate the model fit If the Two-Way ANOVA indicates that you may reject the null hypotheses, where do the differences lie? As Biologist, what can you conclude from this study? cons food sex

cons	food	sex
19.7	1	male
21.2	1	male
21.1	1	male
18.15	1	male
19.43	1	male
19.67	1	male
19.81	1	male
22.38	1	male
20.54	1	male
19.48	1	male
18.56	1	male
19.97	1	male
18.26	1	male
18.72	1	male
20.87	1	male
20.69	1	male
20.13	1	male
21.88	1	male
20.86	1	male
18.8	1	male
21.24	1	female
18.82	1	female
20.31	1	female
20.23	1	female
21.13	1	female
22.34	1	female
22.97	1	female
22.66	1	female
22.92	1	female
20.62	1	female
19.48	1	female
20.68	1	female
18.42	1	female
21.76	1	female
19.63	1	female
20.06	1	female
21.14	1	female
21.07	1	female
20.08	1	female
21.71	1	female
22.48	2	male
20.97	2	male
19.51	2	male
19.08	2	male
20.83	2	male
21.11	2	male
22.07	2	male
19.59	2	male
21.12	2	male
21.96	2	male
20.84	2	male
23.39	2	male
22.65	2	male
21.67	2	male
20.08	2	male
20.96	2	male
20.29	2	male
22.03	2	male
22.31	2	male
21.84	2	male
18.82	2	female
21.83	2	female
19.01	2	female
19.12	2	female
19.7	2	female
19.73	2	female
19.59	2	female
19.45	2	female
21	2	female
20.78	2	female
19.25	2	female
20.13	2	female
19.36	2	female
19.45	2	female
20.7	2	female
20.42	2	female
20.66	2	female
20.59	2	female
19.46	2	female
18.54	2	female

Answer 1

There are three sets of hypothesis in Two-way Anova:

H1o: There is no significant difference in consumption based on food type
H1a: There is significant difference in consumption based on food type

H2o: There is no significant difference in consumption based on the sex
H2a: There is significant difference in consumption based on sex

H3o: There is no interaction effect between the two factors
H3a: There is an interaction effect between the two factors

About the data:

The data points look evenly distributed (normally) about the mean which is just over 20. Hence, it can be said that the data is normally distributed.

The mean of the data is 20.498. Standard deviation = 1.25

There are two factors in the data, food type and sex. In food type, there are two levels - Food 1 & 2. In Sex, there are two levels, Male and Female.

The variances based on Food Type are 1.67 and 1.45 & Sex are 1.70 and 1.40. The variances are assumed to be equal and not of significant difference.

Two-Way Anova: It is a two-way anova with two independent variables as factors (Categorical) and one continuous dependent variable

The results of the Two-Way ANOVA are:

ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Sample	0.300125	1	0.300125	0.236243	0.628331	3.96676
Columns	1.28018	1	1.28018	1.007693	0.318642	3.96676
Interaction	24.48684	1	24.48684	19.2748	3.61E-05	3.96676
Within	96.55093	76	1.270407
Total	122.6181	79

We can see from the above table that in the interaction row, p-value<0.05. Hence, the 3rd null hypothesis can be rejected and can be said that there is an interaction effect between the two factors.

Average	Male	Female
F1	20.01	20.8635
F2	21.239	19.8795

Thus, we can say that, as a biologist, there was a significant effect of Food Type and Sex of the rat on the consumption rate. Food 2 in Males can be said to have a significant higher rate and Food 2 in Females can be said to have a significant lower rate than others.