1. Apply mathematical techniques and modern mathematical software to engineering problems;
2. Apply probability distributions and linear regression to engineering problems.
Question 1: 10 marks
Question 2: 10 marks
Question 3: 15 marks
Question 4: 15 marks
Total: 50 marks

REMEMBER: coursework hand-in now has a hard deadline. Be late, get zero. Plan to hand in at least a day early. Better still, a week.
For full marks, all working must be shown.
Please fit all the working and the answers in the boxes provided. Only printed-out plots should be attached.

 

1 Poisson distribution
It is assumed that the number of people telephoning a certain help line is a Poisson process. The help line was open for calls between 0900 and 1700, Monday – Friday and, over a 1 week period, the number of calls per hour were logged. The result was:

9-10 10-11 11-12 12-13 13-14 14-15 15-16 16-17
Mon. 3 0 1 1 0 2 3 0
Tue. 1 1 0 3 4 1 5 3
Wed. 1 1 2 2 1 0 0 3
Thu. 1 3 2 0 1 5 2 0
Fri. 2 0 2 1 0 2 1 0

Use this data to determine the relative frequency for n calls per hour, with n = 0,1,2,3,4,5. Find the mean number of calls per hour and hence estimate probabilities of receiving n calls in an hour assuming a Poisson distribution end expected frequencies in the observed period of time. Comment on the suitability of the Poisson distribution for this data.
Total question 1: 10 marks
Solution:
The relative frequencies of n calls per hour are
(fill in missing values in row 2 of the table):
r 0 1 2 3 4 5
f
(2 marks)
The mean number of calls per hour is (fill in the value) (1 mark)
If the process is Poisson then

For r = 0, 1, 2, 3, 4, 5 this gives (rounding to 4 dp) the following probabilities:
(fill in missing values in row 2 of table below) (2 marks)
Then the expected frequencies in the observed period of time are (rounded to 1dp)
(fill in missing values in row 3 of table below) (3 marks)

r 0 1 2 3 4 5
P(r)
fexpected

Comment on the agreement between observed frequencies and expected frequencies. Is Poisson distribution adequately describing the given data? (2 marks)
2 Normal distribution
Suppose that an automobile muffler is designed so that its lifetime (in months) is approximately normally distributed with mean 22.4 months and standard deviation 5.1 months. The manufacturer has decided to use a marketing strategy in which the muffler is covered by warranty for 12 months.
(a) Approximately what proportion of the mufflers will fail the warranty?
(b) If a random muffler is selected, what is the probability it will last twice the warranty period?
Round both answers to 2 decimal points
Total question 2: 10 marks
Solution:
COURSEWORK 1
Probability Theory

This is an individual coursework.

Intended Learning Outcomes:
1. Apply mathematical techniques and modern mathematical software to engineering problems;
2. Apply probability distributions and linear regression to engineering problems.
Question 1: 10 marks
Question 2: 10 marks
Question 3: 15 marks
Question 4: 15 marks
Total: 50 marks

REMEMBER: coursework hand-in now has a hard deadline. Be late, get zero. Plan to hand in at least a day early. Better still, a week.
For full marks, all working must be shown.
Please fit all the working and the answers in the boxes provided. Only printed-out plots should be attached.

1 Poisson distribution
It is assumed that the number of people telephoning a certain help line is a Poisson process. The help line was open for calls between 0900 and 1700, Monday – Friday and, over a 1 week period, the number of calls per hour were logged. The result was:

9-10 10-11 11-12 12-13 13-14 14-15 15-16 16-17
Mon. 3 0 1 1 0 2 3 0
Tue. 1 1 0 3 4 1 5 3
Wed. 1 1 2 2 1 0 0 3
Thu. 1 3 2 0 1 5 2 0
Fri. 2 0 2 1 0 2 1 0

Use this data to determine the relative frequency for n calls per hour, with n = 0,1,2,3,4,5. Find the mean number of calls per hour and hence estimate probabilities of receiving n calls in an hour assuming a Poisson distribution end expected frequencies in the observed period of time. Comment on the suitability of the Poisson distribution for this data.
Total question 1: 10 marks
Solution:
The relative frequencies of n calls per hour are
(fill in missing values in row 2 of the table):
r 0 1 2 3 4 5
f
(2 marks)
The mean number of calls per hour is (fill in the value) (1 mark)
If the process is Poisson then

For r = 0, 1, 2, 3, 4, 5 this gives (rounding to 4 dp) the following probabilities:
(fill in missing values in row 2 of table below) (2 marks)
Then the expected frequencies in the observed period of time are (rounded to 1dp)
(fill in missing values in row 3 of table below) (3 marks)

r 0 1 2 3 4 5
P(r)
fexpected

Comment on the agreement between observed frequencies and expected frequencies. Is Poisson distribution adequately describing the given data? (2 marks)

2 Normal distribution
Suppose that an automobile muffler is designed so that its lifetime (in months) is approximately normally distributed with mean 22.4 months and standard deviation 5.1 months. The manufacturer has decided to use a marketing strategy in which the muffler is covered by warranty for 12 months.
(a) Approximately what proportion of the mufflers will fail the warranty?
(b) If a random muffler is selected, what is the probability it will last twice the warranty period?
Round both answers to 2 decimal points
Total question 2: 10 marks
Solution:
Let X represent the random variable measuring the lifetime of a muffler, assumed to be continuous. Using time units of months, the mean lifetime of a muffler is given by (fill in the value) (1 mark)
It follows that Z = will be corresponding standard normal distribution

(fill in expression in X) (2 marks)
(a) We then have

(fill in values of X, Z and the final probability) (3 marks)
Thus % of mufflers will fail within the warranty period (1 mark)
(b) Now we need to find:
(3 marks)

3 Linear regression
This is final energy consumption (FEC) of energy products for transport (published by Department of Energy and Climate Change) in million tons of oil equivalent:
Year 1970 1973 1976 1979 1982 1985 1988 1991 1994
FEC 28.17 32.44 32.03 35.36 35.04 38.50 45.35 47.97 50.25

Year 1997 2000 2003 2005 2006 2007 2008 2009 2010 2011
FEC 53.08 55.46 56.37 58.78 59.49 59.76 58.21 56.06 55.15 55.19
Total question 3: 15 marks
Using Matlab, do the following:
(a) Plot a scatter plot of FEC (y) vs year (x)
(b) Assuming that the trend is linear, calculate the coefficients of linear regression line using the formulae from lectures
(c) Check your calculation using MatLab built-in polynomial fitting command polyfit
(d) For each year from 1970 till 2050 calculate the FEC predicted by the linear regression analysis
(e) Plot linear regression line together with the scatter plot in one window
(f) Predict FEC for transport in 2050
(g) Comment on reliability of the prediction

Solution:
(a) The MATLAB commands used to enter y and x and plot the scatter plot are:
(enter commands in box) (2 marks)

The MATLAB commands used to calculate mean values of x and y are:
(enter commands in box) (1 mark)

The MATLAB commands used to calculate standardised values X and Y and corresponding values of XY and X2 are:
(enter commands in box) (2 marks)

(b) The MATLAB commands used to calculate coefficients of the linear regression are:
(enter commands in box) (3 marks)

Then the best fit line is given by an equation (fill in the coefficients rounded to 4dp):
(1 mark)
(c) The MATLAB command used to calculate linear regression coefficients using MatLab built-in polynomial fitting command polyfit is:
(enter commands in box)

Therefore the best fit line is:
(2 marks)
(d) The MATLAB commands used to calculate vector xr with years from 1970 till 2050 and corresponding values of yr are:
(enter commands in box): (1 mark)

(e) The MATLAB commands used to plot yr versus xr together with the original scatter plot are:
(enter commands in box and attach a separate sheet with the plot): (1 mark)

(f) Then the predicted value of FEC in 2050 is: (1 mark)

(g) This value is feasible/unfeasible (delete as appropriate) because: (1 mark)

4 Manufacturing question
Grindit & Floggit Ltd make customised Widgets at a cost of £150 per unit of which 3% (on average) happen to be out of tolerance.
Up to 115 Widgets can be produced in one continuous run whose set up incurs additional cost of £10,000.
Since manufacturing runs are set up to produce one of many variants of Widget, Grindit & Floggit estimate that Widgets made in excess of a firm order have no residual value.
All Widgets are tested to see if they are within tolerance only after a manufacturing run has been complete.
Grindit & Floggit receive an order for 100 Widgets
(a) What is the chance to fulfil the order by producing exactly 100 Widgets?
(fill in the working and the answer in the box and round to 3 dp) (3 marks)

(b) What is the chance to fulfil the order by producing 105 Widgets?
(fill in the working and the answer in the box and round to 3 dp) (4 marks)

(c) How many Widgets should be produced to fulfil the order with less than 5% risk?
(fill in the working and the answer in the box and round to 3 dp) (6 marks)

(d) What is the cost of production of each of 100 resulting Widgets in the last case? What price do you think Grindit & Floggit should be charging the customer per Widget in the last case if they want to make 10% profit (round to nearest penny)?
(fill in the working and the answer in the box and round to nearest penny) (2 marks)

21.

ORGANIZATIONAL CHANGE

1. What agency or organization did you work for?
2. Give a brief overview of the organization you worked for.
3. What is the mission of the organization at which you did your internship?
4. Who was your supervisor?
5. What were your specific responsibilities?
Part Two
In this section of the paper, you are to detail your experiences. This part of the paper should be eight to ten pages.

1. Describe a typical day in terms of the specific activities you were assigned.

2. Discuss some representative clients. Explain how you helped your clients,
what their reactions to you were, and what your reactions to them were.
What effect did you have on the clients’ life circumstances?

3. List the major problems that you encountered, and discuss how you were able to
work around them to help the clients.

Part Three

Allow at least two pages for this part.

1. Was your supervisor helpful to you? Did he or she answer your questions to
your satisfaction?

2. What did you learn from your internship? Be specific.

3. Would you be interested in working for the agency or organization if you were
offered a job? Why or why not?

4. Did the employees welcome you and see you as someone who could help them?

5. How did the clients treat you?

6. What would you do differently if you were going to do it again?

7. Did you enjoy your internship? Explain?

8. What new skills did you learn?

9. Were you ever treated badly by the clients and/or the staff? If the
answer is yes, how did you handle it?

10. How will you use what you have learned to make a better life for yourself?