Question

A researcher did a two-way ANOVA to test the effects of course type (two levels: in-class vs. online) and course topic (two levels: statistics vs. cognitive psychology) on the instructors’ stress levels. A summary table (with a few missing values) is provided here.

Source	SS	df	MS	F
Between	625	3		---------------------
Type		1	300	12
Topic				5
Type*Topic			200
Within				---------------------
Total	3525	119	---------------------	---------------------

a. Complete the missing values in the table. You must submit your calculations (if you do not submit your calculation, you will lose points). (10 points).

b. N=? You must submit your calculations (if you do not submit your calculation, you will lose points). (1 point)

c. Is there a significant main effect of course type? Explain your answer (2 points).

d. Is there a significant main effect of course topic? Explain your answer (2 points).

e. Is there a significant interaction effect? If so, what does it mean? Explain your answer (3 points).

f. Which of the following graphs could depict these data? Explanation is optional. (2 points).

Answer 1

a. Complete the missing values in the table.

Given

Source	SS	df	MS	F
Between	625	3		---------------------
Type		1	300	12
Topic				5
Type*Topic			200
Within				---------------------
Total	3525	119	---------------------	---------------------

So we have   TSS = 3525     and    SSR ( Between ) = 625

Thus   SSE ( Within ) = TSS - SSR = 3525 - 625 = 2900

Hence SSE = 2900           ...... (1)

We now that degree of freedom associated with Total sum of square ( TOTAL ) is df = N-1

Given   df(TSS) = 119

Hence N-1 = 119  

N = 120                      ..... ( 2 )

So df(TSS) = 119

df(SSR) = 3

Thus   df(SSE) = df(TSS) - df(SSR) = 119 - 3 = 116      .... ( 3 )

Hence degree of freedom associated with sum of square of error ( within ) is df = 116

MS = SS/df

Thus for SSR ( between ) : -   df=3 and SS=625

Hence MS(SSR) = SS/df = 625 / 3 = 208.3333              .....(4)

Degree of freedom associated with tye , topic , type*topic is 1 ( for each )

Hence MS = SS/df = SS/1      ( since here df=1 )

Thus Sum of square due to type / topic / topic*type are

SSR ( type ) = MS(Type) = 300

SSR ( type*topic ) = MS(type*topic) = 200

SSR ( Topic ) = SSR - SSR(type) - SSR(type*topic)

                    = 625 - 300 - 200 = 125

Hence

SSR ( type ) = MS(Type) = 300

SSR ( Topic ) = MS( Topic) = 125

SSR ( type*topic ) = MS(type*topic) = 200

F = MSR / MSE

We have F = 12 { for type }

Thus   F ( Type ) = 12 = MS(Type)/MSE

                               MSE = MS(Type) / 12 = 300 / 12 = 25

Hence MSE = 12                ....(5)

                               

For between

F = MSR / MSE

and MSR = MS(SSE) = 208.3333        { form 4 }

And MSE = 25

F = MSR / MSE = 208.33 / 25 = 8.32

Hence   F = 8.32

and also F ( type*topic ) = MS( type*topic ) / MSE = 200 / 25 = 8

So final table will be as follow

Source	SS	df	MS	F
Between	625	3	208.333	----8.32-----------------
Type	300	1	300	12
Topic	125	1	125	5
Type*Topic	200	1	200	8
Within	2900	116	25	---------------------
Total	3525	119	---------------------	---------------------

b)

N= 120    { from 2 }

c) Is there a significant main effect of course type

F-Value = 12              ( correspond to type )

F-Critical value is given by       ,where = 0.05

It can be obtained from R

> qf(1-0.05,1,116)
[1] 3.922879

Hence F-Critical value is = 3.922879

Since F-Value = 12 > 3.922879 ( )

i.e F-Values > .

We can conclude that there a significant main effect of course type

Hence answer is yes .

d. Is there a significant main effect of course topic? Explain your answer

F-Value = 5              ( correspond to topic )

F-Critical value will not change and it is given by = 3.922879 .

Here also

F-Value = 5 > 3.922879 ( )

i.e F-Values > .

We can conclude that there a significant main effect of course topic

Hence answer is yes .

e. Is there a significant interaction effect? If so, what does it mean? Explain your answer

F-Value = 8              ( correspond to interaction effect type * topic )

F-Critical value will not change and it is given by = 3.922879 .

Here also

F-Value = 8 > 3.922879 ( )

i.e F-Values > .

We can conclude that there a significant interaction effect of type * topic

Hence answer is yes .

If so, what does it mean?

Interaction effects occur when the effect of one variable depends on the value of another variable.

An interaction effect occurs if there is an interaction between the independent variables ( here type nad topic ) that affect the dependent variable ( instructors’ stress ) .

So here we can say course type (two levels: in-class vs. online) and course topic (two levels: statistics vs. cognitive psychology) significantly affects the instructors’ stress levels combined at different levels of each type and topic.

Note that - for part f) no graph is provided so need some information for that part .

If there is any doubt in notation used in part (1) while completing anova , you can ask for that in comment box