Chapter 10.4, Problem 19E

The Scheffé Test If the null hypothesis is rejected in a one-way ANOVA test of three or more means, then a Scheffé Test can be performed to find which means have a significant difference. In a Scheffé Test, the means are compared two at a time. For instance, with three means you would have these comparisons: ${\bar{x}}_1$ versus ${\bar{x}}_2$, ${\bar{x}}_1$, versus ${\bar{x}}_3$ and ${\bar{x}}_2$ versus ${\bar{x}}_3$ For each comparison, calculate

$\frac{{\left({\bar{x}}_a-{\bar{x}}_b\right)}^2}{\frac{S{S}_W}{{\displaystyle \sum \left(n_i-1\right)}}\left(\frac{1}{n_a}+\frac{1}{n_b}\right)}$

where ${\bar{x}}_a$ and ${\bar{x}}_b$, are the means being compared and n_a and n_b are the corresponding sample sizes. Calculate the critical value by multiplying the critical value of the one-way ANOVA test by k − 1. Then compare the value that is calculated using the formula above with the critical value. The means have a significant difference when the value calculated using the formula above is greater than the critical value.

Use the information above to solve Exercises 19−21

19. Refer to the data in Exercise 5. At a = 0.05, perform a Scheffé Test to determine which means have a significant difference.

5. Toothpaste The table shows the costs per ounce (in dollars) for a sample of toothpastes exhibiting very good slain removal, good stain removal, and fair stain removal. At a = 0.05, can you conclude that at least one mean cost per ounce is different from the others? (Source: Consumer Reports)