MTH205 Introduction to Statistical Methods
Tutorial 4
Based on Chapter 4
1. Measurements of total solids, in g/L, for seven wastewater treatment sludge specimens were obtained below:
20 5 25 43 24 21 32.
The measurements were rounded to the nearest gram. Perform the Wilcoxon signed-rank test.
(i) Can you conclude that the mean concentration of total solids is greater than 14 g/L? Use the p-value to conclude.
(ii) Can you conclude that the mean concentration of total solids is less than 30 g/L? Use the p-value to conclude.
(iii) It is claimed that the mean concentration of total solids is equal to 18 g/L. Can you
conclude that the claim is false? Use the p-value to conclude.
The distribution of total solids is assumed to be symmetric.
2. In a research on chemical reactions, the benzene conversions (in mole percent) of 24 di↵erent benzenehydroxylation reactions are reported below:
52.3 41.1 28.8 67.8 78.6 72.3 9.1 19.0
30.3 41.0 63.0 80.8 26.8 37.3 38.1 33.6
14.3 30.1 33.4 36.2 34.6 40.0 81.2 59.4
Perform the Wilcoxon signed-rank test.
(i) Can you conclude that the mean conversion is less than 45? Use the p-value to conclude.
(ii) Can you conclude that the mean conversion is greater than 30? Use the p-value to conclude.
(iii) Can you conclude that the mean conversion di↵ers from 55? Use the p-value to conclude. .
The distribution of the conversion is assumed to be symmetric.
3. This exercise shows that the signed-rank test can be used with paired data. Two gauges that measure tire tread depth are being compared. Ten different locations on a tire are measured once by each gauge. The results, in mm, are presented in the table below:
|
Location
|
Gauge 1
|
Gauge 2
|
Difference
|
|
1
|
3.95
|
3.80
|
0.15
|
|
2
|
3.23
|
3.30
|
-0.07
|
|
3
|
3.60
|
3.59
|
0.01
|
|
4
|
3.48
|
3.61
|
-0.13
|
|
5
|
3.89
|
3.88
|
0.01
|
|
6
|
3.76
|
3.73
|
0.03
|
|
7
|
3.45
|
3.56
|
-0.11
|
|
8
|
3.01
|
3.02
|
-0.01
|
|
9
|
3.82
|
3.77
|
0.05
|
|
10
|
3.44
|
3.49
|
-0.05
|
Assume the di↵erences are a sample from a symmetric population with mean μ . Perform the Wilcoxon signed-rank test totest H0 : μ = 0 against H1 : μ ≠ 0. Use the p-value in your conclusion.
4. In a comparison of the effectiveness of distance learning with traditional classroom instruction,
12 students took a business administration course online, while 14 students took it in a classroom. The final exam scores were as follows.
Online: 66 75 85 64 88 77 74 91 72 69 77 83
Classroom: 80 83 64 81 75 80 86 81 51 64 59 85 74 77
(i) Perform the Wilcoxon rank-sum test to determine if the mean score di↵ers between the two types of course. Use p-value in your conclusion.
(ii) What assumptions are necessary for the result of the test to be valid?