MTH205 Introduction to Statistical Methods Tutorial 4Web

Java Python 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