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Essentials of Statistics for the B...

8th Edition
Frederick J Gravetter + 1 other
ISBN: 9781133956570

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Chapter
Section
BuyFindarrow_forward

Essentials of Statistics for the B...

8th Edition
Frederick J Gravetter + 1 other
ISBN: 9781133956570
Textbook Problem

Most sports injuries are immediate and obvious, like a broken leg. However, some can be more subtle, like the neurological damage that may occur when soccer players repeatedly head a soccer ball. To examine long-term effects of repeated heading. Downs and Abwender (2002) examined two different age groups of soccer players and swimmers. The dependent variable was performance on a conceptual thinking task. Following are hypothetical data, similar to the research results.

  1. a. Use a two-factor ANOVA with α= .05 to evaluate the main effects and interaction.
  2. b. Calculate the effects size (η2) for the main effects and the interaction.
  3. c. Briefly describe the outcome of the study.

images

ΣX2 = 6360

a.

To determine

To test: Whether there is any significant difference between the treatment means.

To test: Whether the interaction effect exists between the two factors.

Explanation

Given info:

The data represents the Downs and Abwender of two different age groups of soccer players and swimmers with the dependent variable performance on a conceptual thinking task. The level of significance is α=0.05.

Calculation:

The test hypotheses for main effects of sports are given below:

Null hypothesis:

H0:μ1=μ2=μ3

Alternate hypothesis:

Ha:Atleast one treatment mean of sports significantly differs from the other

The test hypotheses for main effects of age are given below:

Null hypothesis:

H0:μ1=μ2=μ3

Alternate hypothesis:

Ha:Atleast one treatment mean of age significantly differs from the other

The test hypotheses for interaction effects are given below:

Null hypothesis:

H0:The interaction effect does not exists between the two factors

Alternate hypothesis:

Ha:The interaction effect exists between the two factors

Degrees of freedom for within group effects:

dfwithin=19+19+19+19=76

Total degrees of freedom:

dftotal=N1=79

Degrees of freedom for between group effects:

dfbetween=number_of_treatment1=41=3

Sum of squares:

The sum of squares for treatment A is,

SSA=T2rownrowG2N=260240+340240600280=80

Thus, the sum of squares for treatment A is SSA=80

The sum of squares for treatment B is,

SSB=T2columnncolumnG2N=360240+240240+600280180260=180

Thus, the sum of squares for treatment B is SSB=180

The sum of squares for interaction of treatment A and treatment B is,

SA×B=SSbetween_treatmentSSASSB=34080180=80

Thus, the interaction sum of squares is SA×B=80

The sum of squares of total is,

SStotal=X2G2N=6360600280=1860

Thus, thetotal sum of squares is SStotal=1,860

The treatment within sum of squares is,

SSwithin=SSINSIDE=380+390+350+400=1520

Thus, the treatment within sum of squares is SS

b.

To determine

To Explain: The results of ANOVA

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