Q1 A study at Ulster University considered the correlation in fifteen SMEs between their abilities to carry out Continuous Improvement (CI) and their cultures of Innovation. For each of the fifteen firms a score was developed to indicate their capability in the two areas. The results are shown below:
| Firm |
Innovation Score |
Continuous Improvement |
| A |
4.25 |
4.19 |
| B |
4.92 |
4.74 |
| C |
2.82 |
2.78 |
| D |
4.02 |
3.78 |
| E |
2.98 |
2.92 |
| F |
3.28 |
3.1 |
| G |
3.35 |
3.32 |
| I-1 |
3.34 |
3.15 |
| I |
4.19 |
4.08 |
| J |
3.01 |
2.92 |
| K |
2.88 |
2.81 |
| L |
4.37 |
4.39 |
| M |
4.01 |
3.83 |
| N |
3.36 |
3.23 |
| 0 |
2.97 |
2.82 |
(a) Do the results suggest any significant difference between the scores for Innovation and those for CI?
(b) The authors of the report suggest that a firm's ability to Innovate can be predicted from their capability in CI. Does the data support this suggestion?
(c) Comment on the findings in parts (a) and (b). Do your answers indicate any difficulty in interpreting the results of the study?
(d) If a firm had a CI score of 3.5 calculate the 95% confidence limits for their predicted score for Innovation.
(e) Test the hypothesis that the graph should pass through the origin (i.e. α = 0). Explain what this means in practical terms.
Q.2. (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.
Q.2. (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 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.
Q.3. (i) The viscosity of an industrial paint, as tested by the Ford Cup method, is specified to be 34 secs ± 1 sec. Fifteen batches of the paint have been tested, and the results are shown below:
33.75 33.05 34.00 33.81 33.46 34.02 33.58 33.27
33.49 33.20 34.62 33.00 33.54 34.12 33.84
(a) From the sample, does it seem that the population mean is on target?
(b) Find the 95% confidence interval for the population mean.
(c) Using some SIMPLE test or tests, check to see if the data is "reasonably" Normal.
(d) What percentage of batches is likely to be outside the specification in the long run?
Q.3. (ii) The air in a "clean room" used for the production of electronic devices is cleaned by filtering. The air on the outside of the filters is monitored daily to check that ambient conditions are not changing significantly. The table below shows the results of counting the number of dust particles in a standard volume of air over a period of time:
|
Number of particles in sample
|
Number of samples
|
|
0
|
34
|
|
1
|
45
|
|
2
|
37
|
|
3
|
18
|
|
4
|
4
|
|
5
|
2
|
|
≥6
|
0
|
(a) Suggest a model that is likely to be appropriate for modelling this data.
(b) If a sample taken at a later date has 8 particles in a standard sample, does it seem that air quality is deteriorating? (Explain your answer, and choice of test.)
Q4. (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 44 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
|
|
42.0
|
2
|
|
42.5
|
6
|
|
43.0
|
11
|
|
43.5
|
14
|
|
44.0
|
21
|
|
44.5
|
18
|
|
45.0
|
15
|
|
45.5
|
6
|
|
46.0
|
4
|
|
46.5
|
3
|
(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?
(d) Estimate the population mean with 95% confidence.
Q4. (ii) 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)
|
Method
B (EHA's
Method)
|
|
1
|
10 |
18 |
|
2
|
37 |
37 |
|
3
|
35 |
38 |
|
4
|
43 |
36 |
|
5
|
34 |
47 |
|
6
|
36 |
48 |
|
7
|
48 |
57 |
|
8
|
33 |
28 |
|
9
|
33 |
42 |
Does the data suggest the firm's method does underestimate the level of pollution?
Q.5. (i) A firm has five automatic packaging machines that are used to wrap their products before shipping them to customers. The machines are becoming less reliable, and it is found that there is a 10% chance of a machine being unable to start up at the beginning of the working day. It takes a full day for the service mechanic to call and repair the machine in these circumstances. When all five machines are running they have the capacity to wrap two thousand products a day.
If there is insufficient capacity to wrap all of the output in one day the firm sends the surplus out to a sub-contractor. The daily output is found to be Normally distributed with an average of 1500 products and a standard deviation of 125. What proportion of their products does the firm send out to the sub-contractor?
Q.5. (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 three 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?