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

Since the calculated F=  2.8657does 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.

Step-by-step explanation:

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

The data for the test is

                            Mean    Standard DeviationSample Size

Never

Played Football7602.5          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    

SUMMARYCountSumAverageVariance

Row 128676.54338.2521310656.13

Row 227239.13619.5516127224.25

Row 326328316413215968.72

Column 1319796.36598.766667886871.7433

Column 232447.3815.766666758708.90333

ANOVATable  

Source           Sum of     Degrees     Mean

of Variation      Square    of freedom  Squares

                            SS     df          MS              FP-value

B/w Rows1401945.7032700972.85172.86570.259

B/w Col      50164633.5150164633.5205.0819  0.00484

Error     489215.59         2244607.795                          

Total52055794.795    

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=  2.8657does 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.

a. 32 degrees

step-by-step explanation:

here we have to use the rule.

m∠e = 1/2 (major arc - minor arc)

m∠e = 1/2 (212 - 148)

m∠e = 1/2 (64)

m∠e = 32 degrees

answer: a. 32 degrees

you