In: Statistics and Probability
A researcher investigates the factors that are associated with the salaries of professors who teach courses at a major university. The researcher gathers data about the subject area and the salary per course for a random sample of professors.
Salary per course Subject
1700 Humanities
1900 Humanities
1800 Humanities
2100 Humanities
2500 Humanities
2700 Humanities
2900 Humanities
2500 Humanities
2600 Humanities
2800 Humanities
2700 Humanities
2900 Humanities
2500 Social Sciences
2300 Social Sciences
2600 Social Sciences
2400 Social Sciences
2700 Social Sciences
2400 Social Sciences
2600 Social Sciences
2400 Social Sciences
2500 Social Sciences
3500 Social Sciences
3300 Social Sciences
3600 Social Sciences
3400 Social Sciences
2700 Engineering
2800 Engineering
2900 Engineering
3000 Engineering
2800 Engineering
2700 Engineering
3700 Engineering
3600 Engineering
3700 Engineering
3800 Engineering
3900 Engineering
2500 Managament
2600 Managament
2300 Managament
2800 Managament
3300 Managament
3400 Managament
3300 Managament
3500 Managament
3600 Managament
Ans:
One-way ANOVA: Salary versus Subject
Source | DF | SS | MS | F | P |
Subject | 3 | 4167567 | 1389189 | 6.23 | 0.001 |
Error | 41 | 9144878 | 223046 | ||
Total | 44 | 13312444 |
S = 472.3 R-Sq = 31.31% R-Sq(adj) = 26.28%
Conclusion: The p-value for the test statistic F is 0.001. Hence,
we can conclude that at least one mean salary of a subject has a
significant difference from the remaining subjects at the 0.05
significance level.
2. Obtain boxplots and residual plots for these data.
Ans:
From the box-plot, the large part of the areas of the box-plot is overlap between the Engineering, Management, and Social Science subjects but the subject Humanities has at the lower. Hence, the mean salary of the Humanities subject may be different from the other remaining subjects.
From the histogram of the residuals, we know that residuals have a bimodal normal distribution. Hence, the assumption of a univariate normal distribution is violated.
3. Are the assumptions of an ANOVA reasonably satisfied? Explain in reference to the plots.
Ans:
The assumptions of an ANOVA reasonably do not satisfy because of the bimodal from residuals histogram. Out assumption unimodal normal distribution on residuals.
4. If there is a difference in salaries run a Tukey’s test to show how the salaries for the different subject areas compare to each other. Describe what the results of the Tukey’s test tell you.
Ans:
From the above Tuke's test, the confidence intervals of Humanities VS Engineering and Humanities VS Management does not include the value zero. Hence, we can conclude that the salary per course between the Humanities & Engineering and Humanities & Management have a significant difference at the 0.05 significance level.