ASSIGNMENT TITLE - Analysis of Variance
Q1. An employment agency gives an aptitude test to all clients. The results of this test are analysed according to the type of college degree held by the clients. The test results for eight randomly selected clients of each degree type are given in the table below.
(a) Under the assumption of normality, set up the null and alternative hypotheses and carry out the analysis using R. Interpret the practical meaning of the results of this analysis.
(b) Using the summary output from the above analysis, perform multiple comparisons (Bonferroni) of the mean scores by degree at 10% significance level (use appropriate confidence intervals.) What is the individual level for each comparison? Explain which means differ.
(c) Repeat for the Holm method using R. Explain the differences in conclusions if any.
(d) Suppose we are only interested in the differences between liberal and fine arts, and also between engineering and science graduates. Set up appropriate contrasts, provide analysis, report on the p-values and 90% confidence intervals, and interpret the results.
(e) Check the assumptions and comment on the trustworthiness of the results.
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COLLEGE DEGREE TYPE
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Business
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49
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34
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45
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59
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74
|
53
|
51
|
57
|
|
Science
|
77
|
65
|
71
|
85
|
79
|
91
|
60
|
72
|
|
Liberal arts
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50
|
41
|
65
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57
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62
|
47
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59
|
48
|
|
Fine arts
|
37
|
26
|
52
|
39
|
44
|
50
|
41
|
35
|
|
Engineering
|
88
|
70
|
95
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77
|
81
|
89
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62
|
83
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Q2. Consider the table overleaf. Here, A and B are two fertilizer types; M, N, O, and P are four wheat types, the values are wheat yields in bushels per plot.
(a) Set up the hypotheses and perform the tests. Also, perform a graphical analysis to check for interactions. Report the final model. i.e. the model with significant effects only.
(b) Calculate Bonferroni 90% confidence intervals for the differences of the effects due to wheat types (μ.M - μ.N, etc.) What are your conclusions?
(c) Calculate 95% confidence interval for μBO.
(d) Define two pair wise contrasts: Wheat type N versus type O, type O versus type P, and report estimates, 95% confidence intervals and the p-values for each contrast.
(e) Check that the assumptions are satisfied and report your conclusions.
(f) Write a short report summarizing your methods and results.
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Wheat type
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|
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M
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N
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O
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P
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Fertilizer A
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19.4
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24.0
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24.8
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24.3
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20.6
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24.5
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25.4
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23.7
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Fertilizer B
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21.6
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25.6
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26.4
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25.4
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22.1
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26.8
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27.0
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24.5
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