Chapter 27, Problem 1E

To determine

Give the ANOVA model for the design in the given form.

Answer to Problem 1E

The ANOVA model for the design in the given form is $y_{ijk}=\mu +{\tau }_j+{\gamma }_k+{\epsilon }_{ijk}$.

Explanation of Solution

It is given that there are two factors, Sex and Athlete, in analyzing TV watching by students.

The ANOVA model:

In the ANOVA model, each observation is broken into sum of the grand mean, the treatment effect, and an error. The general model for ANOVA for the i^th observation is as follows:

$\begin{array}{l}{y}_{ij}=\mu +{\tau }_j+{\epsilon }_{ij},\\ \text{where, }\mu \text{ is the grand mean, }\\ {\tau }_j\text{ is the treatment effect, and}\\ {\epsilon }_{ij}\text{ is the error}\text{.}\end{array}$

Here, the two factors are Sex and Athlete.

Denote $\tau$ as the treatment effect of the i^th observation at level j of factor “Sex” and $\gamma$ as the treatment effect of i^th observation at level k of the factor “Athlete”. There are 2 levels in factor “Sex” and 4 levels in factor “Athelete”. That is, j takes values 1 and 2 and k takes values 1, 2, 3, and 4.

The ANOVA model for the design in the given form is $y_{ijk}=\mu +{\tau }_j+{\gamma }_k+{\epsilon }_{ijk}$.