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PRIFYSGOL ABERYSTWYTH UNIVERSITY
SCHOOL OF MANAGEMENT AND BUSINESS
SEMESTER 1 EXAMINATIONS, JANUARY 2009
MBM6010 :) Quantitative Methods (Statistics) MMM7010:)
Time allowed - 2 hours
Full marks will be given for complete answers to all questions in Section A and to two questions in Section B. In Section B credit will be given for the best two answers.
Candidates are permitted to use only the Casio FX83ES or the FX85ES calculator which must be made available on request for inspection by invigilators who are authorised to remove any other calculators.
Statistical Tables will be provided.
FORMULAE
The following results or formulae may be quoted and used in any question, provided you make clear that you are doing so.
Binomial Distribution P (R = r) = nCrpr(1 p)n r^ for r = 0; 1 ; 2 ; :::; n;
where nCr =
n! r!(n r)!
; and n! = n(n 1)(n 2)::: 3 : 2 : 1
ER = np and Var R = np(1 p):
Poisson Distribution
P (N = n) =
e t(t)n n!
; for n = 0; 1 ; 2 ; :::
EN = t = Var N:
Section A
(Answer all questions. Each question is worth 5 marks.)
- The number of sales of a particular model by a motorcycle dealer in twelve months were
27 15 18 31 23 25 19 22 25 19 20 27
Calculate the sample mean and its estimated standard error.
- As part of an investigation into the costing of contracts undertaken by your organisation, you have gathered data on the actual time taken, in units of 100 hours, for twenty contracts to be fulÖlled and the original estimate of time required. After Ötting a linear regression model by least squares you obtain the following output: The regression estimate is
actual = 9:17 + 1: 08 estimate
Predictor Coef. SE Coef. T P Constant 9.171 8.588 1.07 0. Estimate 1.0807 0.4462 2.42 0. S = 4: 760 ; R-Sq = 24.6%, R-Sq(adj.) = 20.4%.
(a) What actual time does your equation predict for a project origi- nally estimated to last 2,500 hours? (b) Is the slope signiÖcantly di§erent from zero? Give a reason for your answer.
- In a survey of 120 customers of a city centre bookshop, 40 said they bought books only as gifts (G) for others. Of these, 30 only ever bought hardback books (H) while of the remaining customers, 20 bought only hardbacks. Find
(i) P (G and H); (ii) P (G but not H); (iii) P (not G j H); (iv) P (not H j not G):
Section B
Answer any 2 questions. Each question is worth 30 marks
9.a. A travel company specialises in íAdventure Experienceíholidays in three locations, A, B and C. The box and whisker plots in Appendix Q9a, summarise data on the numbers of bookings at each location for every week in 2008. State whether the following statements are True or False, giving reasons for your answers. i. Location A is the most popular while C is the least. [3] ii. Bookings for A were the least variable. [2] iii. Every location had at least one booking every week. [2] iv. Bookings never exceeded 40 per week for any of the locations. [2] v. For both B and C, bookings were 10 per week or less for at least half the time. [3] vi. In roughly 3 weeks in 4, bookings for A were 10 or more. [3] vii. It would be reasonable to assume that bookings for each lo- cation are Normally distributed. [3] (a) The total time in seconds taken by an industrial robot to paint a component consists of four independent Normally distributed stages, T 1 ; T 2 ; T 3 ; T 4 : Their means and standard deviations are given in the following table.
Stage 1 2 3 4 Mean 52 15 90 19 Standard Deviation 2 0.5 3 0.
i. Calculate the expected value and standard deviation of the total time taken. [5] ii. Calculate the probability that the total time required lies be- tween 2 minutes 55seconds and 3 minutes 5 seconds. [7]
10.a. A company running an online shopping website uses three di§erent agents to deliver orders. In response to customer complaints, the company gathers information about the number of orders which were (i)delivered within the promised 48 hours; (ii) delivered late after more than 48 hours and (iii) not delivered to the customer. The data are contained in the following table and represented in the barchart in Appendix Q10a.
Delivered Delivered late, Not within 48 hours over 48 hours Delivered Agent 1 65 82 15 Agent 2 90 45 59 Agent 3 127 79 42
State whether the following statements are True or False, giving reasons for your answers.
(a) i. Agent 2 is better at meeting the 48 hour delivery target than Agent 1 but not as good as Agent 3. [3] ii. Agent 3 is the least likely of the three to fail to deliver an order. [3] iii. Agent 2 has the worst record for late delivery. [3] iv. Agent 3 is more likely than not to deliver on time. [3] b. An electronics manufacturer is investigating the relationship be- tween the number of fully functioning items produced by assembly line workers, the length in hours of their work shift and their tar- get number of items produced per hour. These variables were measured for twenty workers on the same day and are contained in the variables Items, Shift length and Target. Appendix Q10b. contains some Minitab output analysing this data. i. Write down the equation Ötted by Minitab: One of the esti- mated coe¢ cients is negative. What does this tell you about the e§ect of this variable? [4] ii. How successful is the equation in explaining the variation in the number of items produced? Justify your answer. [3] iii. Are the coe¢ cients in the Ötted equation signiÖcantly di§erent from zero? Give reasons for your answer. [4]
i. 1 or more damaged? [3] ii. no more than 2 damaged? [3] (b) A representative for a computer software company reckons to make a sale as a result of 70% of the presentations she makes. If she makes 10 presentations in a certain week, what is the probability that she makes i. at least 7 sales, [5] ii. between 4 and 8 sales (inclusive). [4] If she makes 85 presentations in the following 8 weeks, what is the probability that she makes between 55 and 65 sales (inclusive)? [12]
13.a. For a random sample X 1 ; X 2 ; ::::; Xn from the Normal N (; ^2 ) distribution, state the distribution of X; the sample mean. [3] (a) A breakfast cereal manufacturer uses a Ölling machine to Öll pack- ets of cornáakes. The packets are supposed to contain 500 g but in fact the contents vary according to a normal distribution with mean = 498 g and standard deviation = 2 g. i. What is the probability that an individual packet contains less than 495 g? [5] ii. What is the probability that the average content of all packets in a box of 10 is less than 495 g? [7] (b) The formula for a 95% conÖdence interval for is given to be
x t s=
p n:
i. Explain brieáy what are meant by the symbols x; s and t: Find the value of t if n = 20: [10] ii. The machine is now adjusted, twenty packets Ölled, and the contents weighed. The average weight is 501 g with sample standard deviation 1 : 9 g. Calculate a 95% conÖdence interval for , the expected content of a packet after the adjustment. [5]
MBM6010/MMM7010: Quantitative Methods (Statistics) – January 2009
Appendix Q9(a)
A B C
0
10
20
30
40
50
60
70
Location
Bookings
Bookings for 'Adventure Experience'
(52 weeks - 2003)
Appendix Q10(a)
Within 48 Hrs Over 48 Hrs Not delivered
1 2 3
0
50
100
Agency
Percentage of Orders
Efficiency of 3 Delivery Agents
Delivery Status of 956 Customer Orders Delivery Status
