Q2 (a) A small fastener is used in making a domestic appliance.  The specification is that the length of the fastener must not be greater than 22mm.  A sample of the fasteners, taken from the latest delivery has given the following lengths (in mm).

21.7     21.7     21.9     21.3     21.1     21.9     21.1     21.4     21.3     21.6

What proportion of the delivery do you expect to be outside the specification?  State the assumptions upon which your conclusions are based.

 (b) Estimate the mean length of the fasteners with 95% confidence.

 (c) The head diameters of the sample (in mm). are found to be:

5.2     5.4     5.0     5.3     5.4     5.1     5.0     5.4     5.5     5.2

If the specification is that the diameter should be 5.0 to 5.5mm. what proportion of the delivery will be faulty on both length and diameter ?

(d)   The data in the table below was obtained from a study of gear noise.  Two samples of gears were taken from the factory.  One sample was from a batch considered to be ‘Noisy’, and the other from a batch considered ‘Quiet’.  Each gear was measured for pitch error (in micrometres).  Do noisy gears have greater pitch errors than quiet gears ?

“Noisy” Gears

“Quiet” Gears

10.0

1.3

1.3

0

1.8

0.8

1.6

0

0

0

12.1

2.7

0

0

4.4

0.6

0

0

9.4

15

(Explain your assumptions and your choice of tests.)

Q.3(i)  Warwick Dairies have decided to replace their carton-filling machine with a newer version.  Three alternative makes of machine are available and all of them provide the required accuracy in terms of volume, however it appears there might be a difference in speed between them.  Since this could affect the quantity they can fill per week Warwick conducted a trial to compare the carton-filling time for each of the three makes of machine with the following results (in seconds):

 

(a) Is there any significant difference in performance between the three makes of machine?  If so, which one should they buy?

(b) In a thirty-five hour week, assuming no stoppages or breakdowns, what is the average number of cartons they can expect to fill with 95% confidence?

Q.3(ii) A firm submits twenty bids a week seeking new business from potential customers.  From past records only 10% of bids are successful.  In any one week what is the likelihood they will achieve (i) no new orders, (ii) less than two new orders, (iii) three or more new orders, (iv) more than four new orders?

Q.4(i) It is known that pesticides applied to fruit orchards can remain in the atmosphere causing contamination.  One study reported in the Environmental Science & Technology journal showed measurements taken over a period of eleven days.  Results were collected during the day and during the night with the results (in mg/m3 of air) shown in the table below.

(a) Does the data suggest there is a significant increase in contamination at night?

(bi) If so, what are the 95% confidence limits for the increase?

Explain your choice of test.

Q.4(ii)(a) 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 35 Newtons ± 3 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. 

 

(b)      Does it seem that the “rates” are Normally distributed?  (Provide clear evidence for your conclusion).

(c)   From the sample, does it seem that the population mean is on target?

Q5(i)       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 85% of all industries contain open order shipping files in their databases.  In a random sample of 12 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 = 7.

(c)   Find the probability that x > 8.

(d)   Find the mean and standard deviation of x.

(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

32

1

47

2

37

3

19

4

4

5

2

³6

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 7 particles in a standard sample, does it seem that air quality is deteriorating?  (Explain your answer, and choice of test.)