10.6 - Example: Exam Grade by Professor
10.6 - Example: Exam Grade by ProfessorScenario
Three professors were each teaching one section of a course. They all gave the same final exam and they want to know if there are any differences between their sections’ mean scores.
\(H_0:\mu_1=\mu_2=\mu_3\)
\(H_a: Not\;all\;\mu\;are\;equal\)
Means
| Instructor | N | Mean | StDev | 95% CI |
|---|---|---|---|---|
| Dr. Al | 60 | 68.367 | 17.719 | (63.977, 72.756) |
| Dr. Oh | 87 | 71.448 | 16.702 | (67.803, 75.094) |
| Dr. Pa | 98 | 67.939 | 17.465 | (64.504, 71.373) |
Pooled StDev = 17.2609
The standard deviations for all three classes are all similar.
- Open the Minitab file: ANOVA_Exam_Profs.mpx
- Using Minitab: Stat > ANOVA > One-Way
The result is the following ANOVA source table:
Analysis of Variance
| Source | DF | Adj SS | Adj MS | F-Value | P-Value |
|---|---|---|---|---|---|
| Instructor | 2 | 635.3 | 317.7 | 1.07 | 0.346 |
| Error | 242 | 72101.1 | 297.9 | ||
| Total | 244 | 72736.4 |
F (2, 242) = 1.07
From our ANOVA source table, p = .346
Because \(p > \alpha\), we fail to reject the null hypothesis.
There is not enough evidence to conclude that the mean scores from the three different professors’ sections are different.
Tukey Pairwise Comparisons
There is some debate as to whether pairwise comparisons are appropriate when the overall one-way ANOVA is not statistically significant. Some argue that if the overall ANOVA is not significant then pairwise comparisons are not necessary. Others argue that if the pairwise comparisons were planned before the ANOVA was conducted (i.e., "a priori") then they are appropriate.
The results of our Tukey pairwise comparisons were as follows:
Grouping Information Using the Tukey Method and 95% Confidence
| Instructor | N | Mean | Grouping |
|---|---|---|---|
| Dr. Oh | 87 | 71.448 | A |
| Dr. Al | 60 | 68.367 | A |
| Dr. Pa | 98 | 67.939 | A |
Means that do not share a letter are significantly different.
Tukey Simultaneous Tests for Differences of Means
| Difference of Levels | Difference of Means | SE of Difference | 95% CI | T-Value | Adjusted P-Value |
|---|---|---|---|---|---|
| Dr. Oh-Dr. Al | 3.08 | 2.90 | (-3.70, 9.86) | 1.06 | 0.537 |
| Dr. Pa-Dr. Al | -0.43 | 2.83 | (-7.05, 6.19) | -0.15 | 0.987 |
| Dr. Pa-Dr. Oh | -3.51 | 2.54 | (-9.46, 2.44) | -1.38 | 0.351 |
Individual confidence level = 97.99%
Looking at the first table, all three instructors are in group A. Means that share a letter are not significantly different from one another (i.e., they are in the same group). Because all three instructors share the letter A, there are no significantly different pairs of instructors.
We could also look at the second table which gives us the t-test statistic and adjusted p-value for each possible pairwise comparison. This p-value is adjusted to take into account that multiple tests are being conducted. You can compare these p-values to the standard alpha level of .05. All p-values are greater than .05, therefore no pairs are significantly different from one another.