Question

Proficiency of several biology sections is evaluated using a special departmental assessment. Three different instructors at two different class times are assessed. The data are below. What can the department conclude with an α of 0.01?

Instructor A Time 1	Instructor B Time 1	Instructor C Time 1	Instructor A Time 2	Instructor B Time 2	Instructor C Time 2
100 88 96 98	85 79 80 90	89 78 89 61	94 81 80 85	72 63 73 77	90 94 96 87

a) What is the appropriate test statistic?
---Select--- na One-Way ANOVA Within-Subjects ANOVA Two-Way ANOVA

b) Compute the appropriate test statistic(s) to make a decision about H₀.
Time: critical value =  ; test statistic =  
Decision:  ---Select--- Reject H0 Fail to reject H0

Instructor: critical value =  ; test statistic =  
Decision:  ---Select--- Reject H0 Fail to reject H0

Interaction: critical value =  ; test statistic =  
Decision:  ---Select--- Reject H0 Fail to reject H0


c) Compute the corresponding effect size(s) and indicate magnitude(s).
Time: η² =  ;  ---Select--- na trivial effect small effect medium effect large effect
Instructor: η² =  ;  ---Select--- na trivial effect small effect medium effect large effect
Interaction: η² =  ;  ---Select--- na trivial effect small effect medium effect large effect


d) Make an interpretation based on the results.

There is a time difference in the assessment.

There is no time difference in the assessment.    

There is an instructor difference in the assessment.

There is no instructor difference in the assessment.    

There is a time by instructor interaction in the assessment.

There is no time by instructor interaction in the assessment.    

Answer 1

Based on the given data, we have to test the effect of Instructors and different Class times on proficiency of biology sections. We have to study the effect of two factors (Categorical) and their interaction on the dependent variable ' Proficiency'.

a) The appropriate test statistic would be that of a
- Two-Way ANOVA

Running a Two way ANOVA with replication (4 times in each group),

We get the output:

From the output obtained,

b) To test H₀. Class time has no effect on Proficiency Vs H_a. Class time has a significant effect on Proficiency

The appropriate test statistic can be expressed as:

with critical / rejection region given by F > F_Critical
Time: Critical value = 8.285 ; Test statistic = 1.313 Since F = 1.313 < 8.285 does not lie in the rejection region, we fail to reject H₀.

Instructor: critical value = 6.013 ; test statistic = 6.357
Since F = 6.357 > 6.013 lie in the rejection region, Reject H₀

Interaction: critical value = 6.013 ; test statistic = 7.152
Since F = 7.152 > 6.013 lie in the rejection region, Reject H_0.

c) The corresponding effect size(s) for the factors and their interaction can be obtained from partial eta square using the formula:

It gives the a proportion of variance in the dependent variable accounted for by the effect. It quantifies how strong the effects are.

By Cohen's rule of thumb, η² = 0.01 implies small effect; η² = 0.06 implies moderate effect and η² = 0.14 implies large effect.

Using excel,

Time: η² = 0.068 ; medium effect
Instructor: η² = 0.414 ; large effect
Interaction: η² = 0.443 ; large effect

d) Based on the results, we may conclude that, There is no time difference in the assessment. (p-value 0.267 > 0.01) There is an instructor difference in the assessment. (p-value 0.008 < 0.01) There is a time by instructor interaction in the assessment. (p-value 0.005 < 0.01)