If the number of subjects in the study described in Question 6 is 30, what will be the degrees of freedom for the error term used to test the AB interaction?

Question

BlackTom AI · Accepted Answer

First, recall the structure of a two-way ANOVA with replication: the total number of observations is N = a * b * n, where a and b are the number of levels for factors A and B, and n is the number of subjects per cell. The degrees of freedom for the error term (the within-cell or residual error) in this design are given by N − ab, which simplifies to ab(n − 1) since N = abn. The question states there are 30 subjects in total per study context, and from the provided answer value we can infer that the product of the levels a and b is 10 (because 10 * 29 = 290, and 29 corresponds to n − 1 with n = 30).

Step by step:
- Use the formula for error df in a two-way ANOVA with replication: error df = ab(n − 1).
- Substitute n = 30: error df = ab * 29.
- Given the target result of 290 for the error df, solve  ab * 29 = 290, which leads to ab = 290 / 29 = 10.
- Therefore, the error degrees of freedom are ab(n − 1) = 10 * 29 = 290.

If one were to consider alternative interpretations, you would need the exact values of a and b (the number of levels for each factor) to compute ab directly. However, since the product ab is 10 in this scenario, multiplying by (n − 1) with n = 30 yields 290. In general, this approach helps verify the error df for AB interaction testing under replication: error df = ab(n − 1).

Finally, the key relationships to remember are: total df = N − 1, factor A df = a − 1, factor B df = b − 1, AB interaction df = (a − 1)(b − 1), and error df = N − ab = ab(n − 1).

If the number of subjects in the study described in Question 6 is 30, what will be the degrees of freedom for the error term used to test the AB interaction?数值题

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