Chapter 12, Problem 1P

Explain why the F-ratio is expected to be near 1.00 the null hypothesis is true.

To determine

To explain: The reason for the F-ratio being nearby 1 when the null hypothesis is true.

Answer to Problem 1P

When the treatment effect is null, then numerator and denominator in F ratio will have the same source of variability. In such cases, F ratio gets balanced and takes a value near to 1.

Explanation of Solution

The null hypothesis in ANOVA is: There is no treatment effect.

The F ratio is given as:

$F=\frac{\left(\text{systematic treatments effects}\right)+\left(\text{random unsystematic differences}\right)}{\text{random unsystematic differences}}$

When null hypothesis is true which means treatment effects are null or not significant, then F ratio is:

$\begin{array}{c}F=\frac{0+\left(\text{random unsystematic differences}\right)}{\text{random unsystematic differences}}\\ =\frac{\text{random unsystematic differences}}{\text{random unsystematic differences}}\end{array}$

When the treatment effect is null, then numerator and denominator in F ratio will have the same source of variability. Since, F ratio will be 1.