Assignment:
Q1: The total body calcium (expressed as a percentage of the mean normal value) was determined for a random sample of 60 rheumatoid arthritis patients receiving prednisolone treatment and results are tabulated below.
|
Total Body Calcium (%)
|
|
85
|
80
|
90
|
81
|
76
|
67
|
|
77
|
85
|
81
|
83
|
75
|
80
|
|
86
|
82
|
78
|
82
|
71
|
66
|
|
81
|
90
|
82
|
80
|
76
|
69
|
|
79
|
74
|
80
|
80
|
81
|
83
|
|
83
|
86
|
85
|
81
|
80
|
93
|
|
80
|
78
|
73
|
83
|
83
|
80
|
|
79
|
84
|
83
|
82
|
83
|
81
|
|
77
|
86
|
80
|
79
|
78
|
91
|
|
74
|
74
|
85
|
88
|
80
|
96
|
(a) Using a tally mark count, construct a frequency distribution table for these results based on classes of width 5 and beginning with the class with lower class boundary 65.
(b) Using this frequency distribution table, plot the ogive and hence produce graphical estimates of the median and quartiles of total body calcium of these rheumatoid arthritis patients.
(c) Using this frequency distribution table, and employing a suitable coding technique, calculate the mean and variance of this sample of total body calcium determinations.
Q2: In a recent study of coastal pollution, the amount of lead absorbed by the seaweed Fucus spiralis was determined. This seaweed species tends to grow at the high-water mark and is therefore exposed for most of the time. Levels of lead were measured in mg Pb (dry weight) and results are tabulated below.
|
Levels of lead in Fucus spiralis
|
|
15.1
|
11.9
|
13.0
|
15.6
|
10.9
|
14.0
|
16.3
|
|
13.0
|
14.6
|
13.7
|
14.6
|
13.0
|
12.2
|
13.4
|
(a) Plot these results on Normal probability graph paper and hence describe the evidence that these results are Normally distributed.
(b) Use your plot to provide graphical estimates of the population mean, μ^, and population standard deviation σ^
(c) Assuming that the level of lead varies according to a Normal distribution with (true) mean 14.3 mg Pb-1 and (true) standard deviation 1.4 mg Pbg-1,evaluate the probability that in a randomly selected sample of seaweed the amount of lead is determined as being
(i) below 12 mg Pb ,
(ii) above 16 mg Pb .
Q3 (a) An engineering company registers 160 industrial accidents throughout its workforce over the course of one year. Concerned with this Health and Safety record, a survey is carried out in which one of the issues addressed is the time of day when the accidents occurred on the shop floor.
The following observations were obtained for each of the six 2-hour periods over the working day.
|
Time Period
|
06.00-
|
08.00-
|
10.00-
|
12.00-
|
14.00-
|
16.00-
|
|
No. of accidents
|
36
|
30
|
21
|
17
|
20
|
36
|
Are these results consistent with the assumption that the rate at which accidents occur is uniform throughout the 12-hour working day?
What recommendations would you make based on your analysis?
(b) When employees are classified according to gender (male, female) and accident experience (none, one, two or more), the following contingency table is produced.
|
|
No. of accidents
|
|
Gender
|
0
|
1
|
2+
|
|
Male
|
143
|
65
|
26
|
|
Female
|
53
|
31
|
3
|
Investigate the possible association between gender and accident experience within this workforce. State your conclusions clearly.
Q4: Random samples were drawn from each of the following three types of worker: high-exposure-level employees of a chemical pesticide firm, farm labourers exposed to the pesticide for several weeks each year, and workers with no known exposure to the pesticide. Chemical determinations were made on blood samples from each subject with the following results.
|
Employees
|
Farm Labourers
|
Unexposed
|
|
7.8
|
7.0
|
6.4
|
|
8.0
8.2
8.4
10.8
7.4
9.0
9.9
5.1
6.8
|
7.3
7.5
10.2
7.5
6.7
8.2
8.9
|
6.6
6.8
6.9
9.5
5.1
7.5
8.2
4.1
|
Assuming that the chemical determinations within any one subject type are Normally distributed, use ANOVA techniques to investigate whether the mean level of chemical differs significantly over these three subject types.
Q3: A recent survey into Road Safety addressed the particular problem when cyclists are overtaken by cars. The data tabulated below refer to the available travel space (the distance between a cyclist and the roadway centre line) and the separation distance (the distance between the bicycle and the overtaking car as determined by photography). These observations were collected for twelve independent overtaking manoeuvres.
|
Travel Space (m)
|
Separation Distance (m)
|
|
3.94
3.97
3.97
4.18
4.46
4.49
4.65
5.38
6.00
6.25
6.27
6.40
|
1.69
1.90
1.93
2.15
2.40
2.55
2.18
2.62
2.71
2.83
2.65
2.86
|
(a)Plot these results on a scattergram.
(b) Use the method of least squares to fit the (linear) regression equation of separation distance on travel space, and plot the regression line on the scattergram. Hence predict (to the same accuracy as the original data) the separation distance for a cyclist when the travel space is 5 metres.
(c) Test the validity of this regression line.