Question

40	1	469.5
40	1	625.5
40	2	840.2
40	2	693.3
40	3	376.7
40	3	809
50	1	730.6
50	1	807.8
50	2	786.7
50	2	952.7
50	3	604.7
50	3	1053
60	1	589.2
60	1	819.8
60	2	848.8
60	2	872.7
60	3	874.5
60	3	1093.5

The data in Table B represents the times (in seconds) for men of three different ages (40, 50 and 60) in each of three different fitness classes (1, 2 and 3) to run a 2 km course. For each runner, age is recorded in the first column, fitness category is recorded in the second column, and running time is recorded in the third.

            Two men in each of the nine categories ran the course. You should be interested in determining whether age and/or fitness affect running time. Each data point can be classified according to age of the runner or according to fitness of the runner. The data therefore requires a two‑way analysis of variance. It is possible that differences among ages of runner will depend upon the fitness categories of those two runners. The model for the analysis should include an interaction term.

Source of variation	Degrees of freedom		Sum of squares		Mean square		F		P
Age of runner	2
Fitness of runner	2
Interaction	4
Error	9

14. What is the value of the F test statistic for testing the hypothesis that age, on average, has no effect on running time?

15. What are the numerator degrees of freedom for that F statistic reported in question 14?

16. What are the denominator degrees of freedom for that F statistic reported in question 14?

17. What is the value of the F test statistic for testing the hypothesis that fitness, on average, has no effect on running time?

18. What is the value of the F test statistic for testing the hypothesis that the effect of age (if any) on running time does not depend of the runner's fitness?

                                                                        NOTE

In analysis of variance, the null hypothesis should be rejected whenever the calculated F‑statistic is greater than the critical value for a chosen significance level and appropriate numerator and denominator degrees of freedom. Equivalently, the null hypothesis should be rejected whenever the computed p‑value is less than the chosen significance level. Use α = 0.01 (significance level =1 %) and answer the following two questions.

n Prepare a table of mean running times and answer the following three questions.

Age	Fitness 1	Fitness 2	Fitness 3	Average
40
50
60
Average

21. What was the average running time for all 60‑year olds?

22. What was the average running time for all men in fitness category 3?

23. What was the mean running time of the two 60‑year, category 3 runners?

                                                                                                                                      {Example 25}

Answer 1

Using SPSS:

Two way Anova table:

(14):  The value of the F test statistic for testing the hypothesis that age, on average, has no effect on running time is 2.578.

(15): The numerator degrees of freedom for that F statistic reported in question 14 is 2.

(16): The denominator degrees of freedom for that F statistic reported in question 14 is 9.

(17): The value of the F test statistic for testing the hypothesis that fitness, on average, has no effect on running time is 1.345.

(18): The value of the F test statistic for testing the hypothesis that the effect of age (if any) on running time does not depend of the runner's fitness is 0.453.