In a study of the dietary treatment of anemia in cattle, researchers randomly divided 144 cows into four treatment groups. Group A was a control group, and groups B, C, and D received different regimens of dietary supplementation with selenium. After a year of treatment, blood samples were drawn and assayed for selenium. The accompanying table shows the mean selenium concentrations ($mg / d / Li$). The $SS_{E} (= SS_{\text {within }})$ from the ANOVA was 289.940.
| Group | A | B | C | D |
| :---- | :-- | :-- | :-- | :-- |
| Mean | 0.8 | 5.4 | 6.2 | 5.0 |
| $n$ | 36 | 36 | 36 | 36 |
Answer the following questions.
(a) What's the TOTAL degrees of freedom, i.e., $df_{\text {total }}$?
(b) What's the ERROR degrees of freedom, i.e., $df_{\text {error }}$?
A. 143; 140
B. 144; 140
C. 143; 36
D. 143; 4
Answer (1)
Calculate the total degrees of freedom: d f t o t a l = 144 − 1 = 143 .
Calculate the error degrees of freedom: d f error = 144 − 4 = 140 .
The total degrees of freedom is 143.
The error degrees of freedom is 140. The answer is 143 ; 140 .
Explanation
Understand the problem We are given a study where 144 cows are divided into four treatment groups. We need to find the total degrees of freedom ( d f t o t a l ) and the error degrees of freedom ( d f error ).
Calculate total degrees of freedom The total degrees of freedom is calculated as the total number of observations minus 1. In this case, the total number of observations is the number of cows, which is 144. Therefore, d f t o t a l = N − 1 = 144 − 1 = 143
Calculate error degrees of freedom The error degrees of freedom is calculated as the total number of observations minus the number of groups. In this case, the total number of observations is 144, and the number of groups is 4. Therefore, d f error = N − k = 144 − 4 = 140
State the final answer The total degrees of freedom is 143, and the error degrees of freedom is 140.
Examples
In experimental design, understanding degrees of freedom is crucial for interpreting the results of statistical tests like ANOVA. For instance, if you're testing different fertilizers on plant growth, the degrees of freedom help determine if the observed differences in growth are due to the fertilizers or just random chance. Knowing d f t o t a l and d f error allows you to assess the significance of your findings and make informed decisions about which fertilizer is most effective.
