Question

Aron
	1	2	3	4	Average
8/6/2017	90	138	118	105	112.8
8/19/2017	162	101	120	145	132
9/16/2017	101	129	132	111	118.3
Average	117.7	122.7	123.3	120.3	121

Mjorgan
	1	2	3	4	Average
8/6/2017	115	88	94	102	99.8
8/19/2017	89	75	77	90	82.8
9/16/2017	74	110	117	90	97.8
Average	92.7	91	96	94	93.4

Aron believes there is a distinct difference between not only the games played each night, AND within the date on which they bowled, but also between himself and Mjorgan. Construct a statistical test to determine if this might be the case – where are the real variations in their bowling? Interpret your results in a few sentences.

 

Answer 1

We will be using ANOVA to test the various hypothesis.

a)

For Aron,

Null and Alternative Hypothesis:

H₀: µ_{Game 1} = µ_{Game 2} = µ_{Game 3} = µ_{Game 4}

H₁: Not all Means are equal

Alpha = 0.05

N = 12

n = 3

Degress of Freedom:

df_Between = a – 1 = 4-1 =3

df_Within = N-a = 12-4 = 8

df_Total = N-1 = 12-1 = 11

Critical Values:

Game (df_Between, df_Within) : (3,8) = 4.07

Decision Rule:

If F is greater than 4.07, reject the null hypothesis

Test Statistics:

SS_Betwen = ?(?a_i)²/n - T²/N = 59.33

SS_Within = ?(Y)² - ?(?a_i)²/n = 4798.67

SS_Total = SS_{Betwen ­­}+ SS_Within = 4858

MS = SS/df

F = MS_effect / MS_error

Hence,

F = 19.78/599.83 = 0.03297

Result:

Our F = 0.032972, we fail to reject the null hypothesis

Conclusion:

All means are equal. Ie There is no distinction between games played by Aron.

b)

For Aron,

Null and Alternative Hypothesis:

H₀: µ_{Day 1} = µ_{Day 2} = µ_{Day 3}

H₁: Not all Means are equal

Alpha = 0.05

N = 12

n = 4

Degress of Freedom:

df_Between = a – 1 = 3-1 =2

df_Within = N-a = 12-3 = 9

df_Total = N-1 = 12-1 = 11

Critical Values:

Day (df_Between, df_Within) : (2,9) = 4.26

Decision Rule:

If F is greater than 4.26, reject the null hypothesis

Test Statistics:

SS_Betwen = ?(?a_i)²/n - T²/N = 786.5

SS_Within = ?(Y)² - ?(?a_i)²/n = 4071.5

SS_Total = SS_{Betwen ­­}+ SS_Within = 4858

MS = SS/df

F = MS_effect / MS_error

Hence,

F = 393.25/452.39 = 0.869

Result:

Our F = 0.869, we fail to reject the null hypothesis

Conclusion:

All means are equal. Ie There is no distinction between games played on different days by Aron.

c)

For Aron & Mjorgan

Null and Alternative Hypothesis:

We will have three hypotheses:

H₀: µ_Aron = µ_Mjorgan

H₁: Not all Means are equal

H₀: µ_{Game 1} = µ_{Game 2} = µ_{Game 3} = µ_{Game 4}

H₁: Not all Means are equal

H₀: An interaction is absent

H₁: Interaction is present

Alpha = 0.05

Degress of Freedom:

Df_Player(A) = a-1 = 2-1 = 1

Df_Games (B) = b-1 = 4-1 = 3

df _{Player * Games} (A*B) = (a-1) * (b-1) = 1*3 = 3

df _error = N – ab = 24 – 2*4 = 16

df _total= N – 1 = 24 – 1 = 23

Decision Rule (3):

We have three hypotheses, so we have three decision rules:

Critical Values:

Player (df_Player(A), df _error) : (1,16) = 4.49

Games (df_Games (B),df _error) : (3,16) = 3.24

Player*Games (df _{Player * Games} (A*B), df _error) : (3,16) = 3.24

[Player] If F is greater than 4.49, reject the null hypothesis

[Games] If F is greater than 3.24, reject the null hypothesis

[Interaction] If F is greater than 3.24, reject the null hypothesis

Test Statistics:

SS_Player = ?(?a_i)²/b*n - T²/N = 4565.04

SS_Games = ?(?b_i)²/a*n - T²/N = 62.12

SS_Player*Games = ?(?a_i * b_i)²/n - ?(?a_i)²/b*n - ?(?b_i)²/a*n + T²/N = 37.46

SS_Total = ?(Y)² - T²/N = 11851.96

SS_Error = SS_Total - SS_Player - SS_Games - SS_Player*Games = 7187.33

MS = SS/df

F = MS_effect / MS_error

Hence,

F_Player = 4565.04/449.21 = 10.16

F_Games = 20.71/449.21= 0.05

F_Interaction = 12.49/449.21 = 0.03

Results:

[Player] If F is greater than 4.49, reject the null hypothesis

Our F = 10.16, we reject the null hypothesis.

[Games] If F is greater than 3.24, reject the null hypothesis

Our F = 0.05, we retain the null hypothesis.

[Interaction] If F is greater than 3.24, reject the null hypothesis

Our F = 0.03, we retain the null hypothesis.

Conclusion:

There is distinct difference only between the players.

For a Player, there is no difference in Games or Days