Question

8.36 Football and Brain Size: Randomization Test Exercise 8.35 describes a study examining hippocampus volume in the brain between controls who have never played football, football players with no history of concussion, and football players with a history of concussion. The data are available in FootballBrain. Use StatKey or other technology to conduct an ANOVA randomization test to determine whether there is a significant difference in mean hippocampus volume between the three categories. What is the F-statistic for the observed data

Answer 1

Since the calculated F= 2.8657 does not lie in the critical region F (2,2)≥ 19 we conclude that there no difference between the means.

Since the calculated F= 205.0819 lies in the critical region F(1,2) ≥ 18.51 we conclude that there is a difference between the standard deviations of the three groups.

The F- statistic for the observed data is 0.101 as calculated by F- test in excel.

Explanation:

To conduct an ANOVA randomization test we perform anova without replication in excel.

The data for the test is

Mean Standard Deviation Sample Size

Never

Played Football 7602.5 1074 25

FB with no History 6459.2 779.9 25

FB with Concussion 5734.6 593.4 25

Total 6598.8 1133.5 75

Anova: Two-Factor Without Replication

SUMMARY Count Sum Average Variance

Row 1 2 8676.5 4338.25 21310656.13

Row 2 2 7239.1 3619.55 16127224.25

Row 3 2 6328 3164 13215968.72

Column 1 3 19796.3 6598.766667 886871.7433

Column 2 3 2447.3 815.7666667 58708.90333

ANOVA Table

Source Sum of Degrees Mean

of Variation Square of freedom Squares

SS df MS F P-value

B/w Rows 1401945.703 2 700972.8517 2.8657 0.259

B/w Col 50164633.5 1 50164633.5 205.0819 0.00484

Error 489215.59 2 244607.795

Total 52055794.79 5

Set up the null and alternate hypotheses :

H0 ; there's no difference between the means and

there's no difference between the standard deviations

against the claim

Ha: there's a difference between the means and

there's a difference between the standard deviations

The significance level is chosen at 0.05

The test statistics to use are

F1= estimated variance from Between Means/ estimated variance from Error SS

F1= s1²/s3²

F1= estimated variance from Between Standard Deviations / estimated variance from Error SS

F2- s2²/s3²

with υ1= (2,2) and υ2(1,2) degrees of freedom.

The critical regions for F (2,2)≥ 19 for 0.05 significance level

and F(1,2) ≥ 18.51 for 0.05 significance level

Since the calculated F= 2.8657 does not lie in the critical region F (2,2)≥ 19 we conclude that there no difference between the means.

Since the calculated F= 205.0819 lies in the critical region F(1,2) ≥ 18.51 we conclude that there is a difference between the standard deviations of the three groups.

The F- statistic for the observed data is 0.101 as calculated by F- test in excel.