Two Way ANOVA Question Most professors prefer to have their students participate

Question

Two Way ANOVA Question

Most professors prefer to have their students participate actively in class. Ideally, students will ask their professors questions and answer their professor’s questions, making the classroom experience more interesting and useful. Some professors seek ways to encourage their students to participate in class. One particular professor believes that there are a number of external factors that affect student participation. He believes that the time of day and the configuration of seats are two such factors and tested this by organizing an experiment. Six classes of about 30 students each were scheduled for one semester. Two classes were scheduled at 9 AM, two at 1 PM, and 2 at 4 PM. At each of the three times, one of the classes was assigned to a room where the seats were arranged in rows and the other was assigned to a room where the desks were tiered and formed a U. In each of the six classrooms, over five days, student participation was measured by counting the number of times students asked and answered questioned. The data are in StatCrunch (use the stacked data). Use the significance level of 0.05.

1. How many what are the main factors of this experiment?

2. What is the response variable?

3. Identify the levels of each factor.

4. Test to see if there is any interaction present. State the interaction hypotheses and provide the p-value of the F test statistic.

5. Produce interaction plots as we did in class. Describe what you see and discuss this graph in relation to your results in part (d).

6. Provide the ANOVA table from StatCrunch and test each main effect hypothesis (no matter what the result is in part (d). State each hypothesis and provide the p-values.

7. Draw conclusions and interpret your results.

Data

Class 9:00 AM 1:00 PM 4:00PM Classroom Time Participation Rows 10 9 7 Rows 9AM 10 Rows 7 12 12 Rows 9AM 7 Rows 9 12 9 Rows 9AM 9 Rows 6 14 20 Rows 9AM 6 Rows 8 8 7 Rows 9AM 8 U-Shape 15 4 7 U-Shape 9AM 15 U-Shape 18 4 4 U-Shape 9AM 18 U-Shape 11 7 9 U-Shape 9AM 11 U-Shape 13 4 8 U-Shape 9AM 13 U-Shape 13 6 7 U-Shape 9AM 13 Rows 1PM 9 Rows 1PM 12 Rows 1PM 12 Rows 1PM 14 Rows 1PM 8 U-Shape 1PM 4 U-Shape 1PM 4 U-Shape 1PM 7 U-Shape 1PM 4 U-Shape 1PM 6 Rows 4PM 7 Rows 4PM 12 Rows 4PM 9 Rows 4PM 20 Rows 4PM 7 U-Shape 4PM 7 U-Shape 4PM 4 U-Shape 4PM 9 U-Shape 4PM 8 U-Shape 4PM 7

Explanation / Answer

there are n=30 observations and level of significance=0.05

1. there are two major factors- time of the day and configuration of seats

2. here the response variable is the number of times students asked and answered questions.

3. there are 3 levels of the factor "time of the day"- 9 am, 1 pm, 4 pm

and there are 2 levels of the factor "configuration of seats"- rows and U-shaped

4. the interaction hypothesis is that H0: no interaction effect between "time of the day" and "configuration of seats" is present vs alternative H1: interaction effect is present

to test the hypothesis we have the F statistic as

F=MSAB/MSE

ehere MSAB is the mean sum of square of interaction effect

and MSE is the mean sum of squares due to error

under H0 F follows a F distribution with dfs=(number of levels of "time of a day-1)*(number of levels of "configuration seats"-1) and (30-number of levels of "time of a day"*number of levels of "configuration seats")

hence the dfs are =(3-1)(2-1) and (30-3*2)

                          =2 and 24

so F~F2,24

from anova table

F=MSAB/MSE=12.28

hence p value=p=P[F>12.28] where F~F2,24

using MINITAB p=0.000

hence p<0.05

so H0 is rejected and the conclusion is that interaction effect is present

6. the anova table is provided as

Source         DF       SS       MS      F      P
Classroom     1       13.333   13.333   1.58 0.220
Time            2      46.667   23.333   2.77 0.083
Interaction 2      206.667 103.333 12.28 0.000
Error          24     202.000    8.417
Total          29      468.667

from ANOVA table

for the factor "time of day" we have the hypothesis of main effect as

H0: there is no differential effects among the 3 different level of the factor "time of day"

vs H1: not H0

to test this we have

F=MST/MSE which under H0 follows an F distribution with dfs=2,24

and MST=mean sum of suqares due to "time of day"

from the table we have F=2.77

and p value=p=P[F>2.77]=0.083>0.05

hence at 5% level of significance we accept H0 and conclude that there is no differential effects of the levels of "time of day" i.e, this factor have no effect on the response

similarly

for the factor "configuration of seats" we have the hypothesis of main effect as

H0: there is no differential effects among the 2 different level of the factor "configuration of seats"

vs H1: not H0

to test this we have

F=MSC/MSE which under H0 follows an F distribution with dfs=1,24

and MST=mean sum of suqares due to "configuration of seats"

from the table we have F=1.58

and p value=p=P[F>1.58]=0.220>0.05

hence at 5% level of significance we accept H0 and conclude that there is no differential effects of the levels of "configuration of seats" i.e, this factor have no effect on the response

7. hence the final conclusion is that the "time of a day" and "configuration of seats" have no effect on the participation of students

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