Q1.(i) The logistics department of a large firm employs three "estimators" whose job it is to decide the likely cost of delivering their products to their customers. This enables them to negotiate with Third Party Logistics (3PL) providers for an acceptable price. In order to check if there is any serious discrepancy between the methods used by the three they were each asked to produce estimates for the same four jobs. The results (in hundreds of pounds) are shown in the table below:
|
|
Job Number
|
|
1
|
2
|
3
|
4
|
|
Estimator
Number
|
1
|
6.2
|
5.0
|
4.6
|
6.6
|
|
2
|
6.3
|
5.4
|
4.9
|
6.8
|
|
3
|
5.9
|
5.4
|
4.4
|
6.3
|
(a) Does the data suggest any discrepancy between the estimators?
(b) Is it possible to identify any interaction between estimators and jobs? Explain your answer.
Q1. (ii) A study of trends in logistics information systems published in the journal Industrial Engineering, found the greatest advances in computerisation were in transportation. It showed 90% of all industries contain open order shipping files in their databases. In a random sample of 10 firms, if x is the number of open order shipping files in their databases, then:
(a) What distribution is most likely to be appropriate for modelling the data?
(b) Find the probability that x = 8.
(c) Find the probability that x > 6.
(d) Find the mean and standard deviation of x.
Q2. (i) An American study on the effect of motor car engine size (in cubic inches) on fuel consumption gave the results shown in the table below.
|
Model/
Make
|
Engine
Size
|
Miles/Qal.
|
|
A
|
350 |
18.90
|
|
B
|
350 |
17.00
|
|
C
|
250 |
20.00
|
|
D
|
351 |
18.25
|
|
E
|
225 |
20.07
|
|
F
|
440 |
11.20
|
|
G
|
231 |
22.12
|
|
H
|
262 |
21.47
|
|
I
|
97 |
30.40
|
|
J
|
350 |
16.50
|
|
K
|
500 |
14.39
|
|
L
|
400 |
16.59
|
|
M
|
318 |
19.73
|
|
N
|
351 |
13.90
|
|
0
|
350 |
16.50
|
(a) Draw the graph relating the two variables.
(b) Does the data support the suggestion that engine size and fuel consumption are related? If so find the relationship between them and state the probable limits of the Regression Coefficient with 95% confidence.
(e) Test the hypothesis that the graph should pass through the origin (i.e. α = 0). Explain what your answer means in practical terms.
Q2. (ii) A company making electronic products has studied the pollution in the atmosphere by taking samples of air flowing into their filters and counting the number of dust particles. The results are shown below.
|
Number of Particles
In Sample
|
Number of
Occasions
|
|
0
|
35
|
|
1
|
46
|
|
2
|
38
|
|
3
|
19
|
|
4
|
4
|
|
5
|
3
|
|
6 or more
|
0
|
(a) What model would be expected to describe this data?
(b) Test to see if your suggestion appears to be valid.
Q3. (i) A company buys springs to use in an assembly. The specification requires the "rate", i.e. the force required to compress the spring by one centimetre, to be within the range 35 Newtons ± 2.5 Newtons. A sample of one hundred springs was selected and the rate was measured using a standard test rig, with the results shown in the table below.
|
Rate (N)
|
Frequency
|
|
32.5
|
3
|
|
33.0
|
7
|
|
33.5
|
15
|
|
34.0
|
13
|
|
34.5
|
20
|
|
35.0
|
18
|
|
35.5
|
17
|
|
36.0
|
3
|
|
36.5
|
3
|
|
37.0
|
1
|
(a) Does it seem that the "rates" are Normally distributed? (Provide clear evidence for your conclusion).
(b) If the distribution were Normal, what proportion of all springs would be expected to lie outside the specification?
(c) From the sample, does it seem that the population mean is on target?
Q3. (ii) (a) Cars on a toll road arrive at the payment booth at an average rate of 80 per hour. What is the probability of no cars arriving in an interval of 2.5 minutes?
(b) At the same toll booth what is the probability that the time to the next car arriving will be between one and two minutes?
Q4. (i) Complaints were made about the level of pollutants in the discharge from a certain factory. The factory refuted the complaints by showing the results of their own analysis of the discharges. However, the Environmental Health Agency claimed the method of analysis used by the firm was faulty. A comparison was made over nine days using two methods of analysis in parallel to check the pollution levels. The results (in ppm.) are shown below:
|
Day#
|
Method A
(firm's
method)
|
Methiod B
(EHA's
method)
|
|
1
|
11
|
18
|
|
2
|
37
|
35
|
|
3
|
35
|
38
|
|
4
|
42
|
36
|
|
5
|
34
|
47
|
|
6
|
35
|
48
|
|
7
|
48
|
57
|
|
8
|
32
|
28
|
|
9
|
33
|
42
|
Does the data suggest the firm's method does underestimate the level of pollution?
Q4. (ii) A company making lamps has drawn a sample from its production line and measured the light output from each. The results, in microamps, are as follows:
9.1 9.8 9.5 10.4 10.7 10.2 9.8 10.0 10.3 10.1 9.6
At a later date second sample is drawn and tested, with the following results:
9.4 9.3 9.8 10.3 9.9 10.5 10.7 10.4 9.7 10.6
(a) Is there any evidence of a change in the performance of the lamps between the two dates?
(b) If there is a difference, calculate the 95% confidence limits for the difference.
(c) If not, calculate the 95% confidence limits for the mean light output.