IMAT1503 2018 Exam PDF

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Sheet 1 of 10 EXAMINATION PAPER Code: IMAT1503 Session - 2017/2018 Faculty of Technology Module Title – Mathematics and Business Management Module Code – IMAT1503 Date – Thursday 24 May 2018 Time Allowed – 2 hours Start 12:30 Finish 14:30 _________________________________________________________ Instructions and information for candidates: Attempt ALL questions ALL questions are worth EQUAL marks Total marks achievable = 100 Restricted Open book: You may take into the examination room and refer to “ONE coloured hand-written A4 Information Sheet”. Both sides of the sheet may be written upon. The Information Sheet must be handed in with your script at the end of the examination. Statistical tables 1 and 5 are available at the end of this paper. Programmable calculators are permitted during this examination provided they are ‘reset’ using the reset button found on the underneath of some calculators, ‘cancelled’ (by battery removal) or otherwise checked and proved not to carry textual information, or formulae required by the examination, other than normal scientific/statistical functions.

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Sheet 2 of 10

Q1. a) Public transportation and automobile are two methods an employee has of getting to work each day. Samples of times recorded for each method are shown. Times are in minutes.

Public 28 Transport Automobiles 29

29

32

37

33

25

29

32

41

34

31

33

32

34

30

31

32

35

33

i) Calculate median, mode and range for each method. (3 marks) ii) Calculate the sample mean time to get to work for each method. (4 marks) iii) Calculate the sample standard deviation for each method. (5 marks) iv) Based on your results from (ii) and (iii), which method of transportation should be preferred? Explain your answer. (3 marks) b) A survey is carried out on behalf of an electricity company to investigate the quarterly bills paid by domestic consumers. As part of this investigation, the company took a random sample of the bills of 600 domestic consumers in one town. The average of this sample is £127.62. Assuming that the standard deviation of all bills in this town is £52, Calculate 95% confidence interval for the population mean, µ. (5 marks)

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Sheet 3 of 10

Q2. a) The company that manufactures a component for a machine claims that the lifetime of the component is normally distributed with an average lifetime of 30 hours and a standard deviation of 4 hours. i) What is the probability of the component selected will last more than 37 hours? (3 marks) ii) What is the probability of the component selected will last between 26 and 32 hours? (5 Marks) b) A car assembly-line operation has a scheduled mean completion time of 2.2 minutes. Because of the effect of completion time on both earlier and later assembly operations, it is important to maintain the 2.2 minute standard. A random sample of 45 times shows a sample mean of 2.39 minutes, with a sample standard deviation of 0.40 minutes. Test at the 5% level and then if appropriate the 1% level to find if the operation is meeting the 2.2 minute standard. i)

State the null and alternative hypotheses. (2 marks)

ii)

Calculate the value of the test statistic. (4 marks)

iii)

State the critical value for your hypothesis test at the 5% level and decide if you should reject the null hypothesis. (2 marks)

iv) State the critical value for your hypothesis test at the 1% level and decide if you should reject the null hypothesis. (2 marks) v)

Clearly state the conclusions of the test. (2 marks)

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Sheet 4 of 10

Q3. a) City Cab Company identified 10 primary pickup and drop locations for cab riders. In an effect to minimise travel time and improve customer service and the utilisation of the company’s fleet of cabs, management would like the cab drivers to take the shortest route between locations whenever possible. Using the following network of roads and streets, what is the route driver beginning at location 1 should take to reach location 10? The travel times in minutes are shown on the arcs of the network.

Use the shortest route algorithm to minimise the time from location 1 to location 10. You are required to sketch the diagram to show all your working and then list the distance between nodes and total length. (8 marks) b) i) First prize in a competition is to receive an immediate payment of £4000 or a payment of £1000 a year for each of the next five years, the first payment being made in one years’ time. If the discount rate is 8%, which option has the higher value? (6 marks) ii) If £2500 is invested now and £3421.42 is the value of the investment after 8 years, then determine the annual interest rate when interest is compounded annually. (6 marks)

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Sheet 5 of 10

Q4. a) Consider the following Linear Programming problem: Max

8x + 7y

S. T.

15x + 5y ≤ 75 10x + 6y ≤ 60 x+y ≤8

Material Constraint Machining Hours Constraint Requirement Constraint

x, y ≥0 i)

Draw a graph to show each constraint and the feasible region. (6 marks)

ii)

Identify the optimal solution on your graph. What are the values of x and y as the optimal solution? (1 mark)

iii)

What is the optimal value of the objective function? (1 mark)

b) The Air Conditioning manufactures three home air conditioners: an Economic model, a Standard model, and a Deluxe model. The profits per unit are £63, £95, and £135, respectively. The following linear programming problem was solved using The Management Scientist to maximise profit (the output is shown on next page). Max 63E + 95S + 135D

profit (£)

such that 1E + 1S + 1D ≤ 200 1E + 2S + 4D ≤ 320 8E + 12S + 14D ≤ 2400

Fan motors Cooling coils Manufacturing time (hours)

E, S, D ≥ 0

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Sheet 6 of 10

The Management Scientist Output OPTIMAL SOLUTION Objective Function Value =

16440.000

Variable -------------E S D

Value --------------80.000 120.000 0.000

Reduced Costs -----------------0.000 0.000 24.000

Constraint -------------1 2 3

Slack/Surplus ----------------0.000 0.000 320.000

Dual Prices -----------------31.000 32.000 0.000

OBJECTIVE COEFFICIENT RANGES Variable Lower Limit -------------------------E 47.500 S 87.000 D No Lower Limit

Current Value ----------------63.000 95.000 135.000

Upper Limit --------------75.000 126.000 159.000

RIGHT HAND SIDE RANGES Constraint -----------1 2 3

Lower Limit --------------160.000 200.000 2080.000

Current Value --------------200.000 320.000 2400.000

Upper Limit --------------280.000 400.000 No Upper Limit

i) What is the value of the objective function? (1 mark) ii) How many of each models should be produced? (3 marks) iii) How much of each resource (Fan motors, Cooling coils and Machining time (hrs)) is actually used? Justify your answer. (3 marks) iv) What is the value to the company of one more Fan motor? (1 mark)

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v) What is the value to the company of one more Cooling coil? (1 mark) vi) How does this affect the optimal solution and optimal value if the company wanted to add 10 more Fan motors and 10 more Cooling coils? (3 marks)

Q5. a) Eighty applicants for a job were assessed as either good or poor for their oral and written communication skills. The resulting assessments are given in the contingency table below: Written Skills Good Oral Skills Total

i)

Total

Poor

Good

17

21

38

Poor

12

30

42

29

51

80

Find the probability that an applicant has good written communication skills. (2 marks)

ii) Find the probability that an applicant has good oral skills given that they have good written skills. (3 marks) iii) Find the probability that an applicant has poor written skills given that they have good oral skills. (3 marks)

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Sheet 8 of 10

b) The company has collected following data on their employees’ ages and salaries. Ages (Years - x)

23

32

49

57

48

48

39

25

57

33

Salary (£000s - y)

14

17

32

36

30

28

25

17

32

19

i) Draw a scatter diagram of above data (3 marks)

ii) From above data, if ∑x = 411, ∑y = 250, ∑x2 = 18295, ∑y2 = 6788, ∑xy = 11121 and n = 10. Calculate coefficient of correlation r, to 3 decimal places using below formula:

(4 marks) iii) Interpret your answers to i) and ii). Is it reasonable to calculate the regression equation, give two reasons for your answers? (2 Marks) c) Describe in words the extent of association between two variables when: i) r = -0.79, ii) r = -0.33 iii) r = 0 (3 marks)

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Sheet 9 of 10

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Sheet 10 of 10...