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Prifysgol Aberystwyth University
ARHOLIADAU / EXAMINATIONS
YSGOL RHEOLAETH A BUSNES
SCHOOL OF MANAGEMENT AND BUSINESS
IONAWR / JANUARY 2010
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 three questions in Section B. In Section B credit will be given for the best three 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.
The results or formulae overleaf may be quoted and used in any question provided you make clear you are doing so.
FORMULAE
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):
ESE(pb) =
r bp(1 pb) n
Poisson Distribution
P (N = n) =
e t(t)n n!
; for n = 0; 1 ; 2 ; ::: EN = t = Var N:
ESE(b) =
s b t
ConÖdence Interval for the Mean
x t
s p n
- In a box of 24 chocolate eggs, 5 are damaged. Four eggs are picked at random and inspected. Find the probability that the number of damaged eggs found is
(i) four, (ii) two, (iii) at least one.
- (a) If R s Binomial, Bin(50; 0 :2); Önd (i) P (R 16); (ii) P (8 R 14): (b) If X s Normal, N (2: 5 ; 0 :16), Önd P (1: 5 < X < 4):
- Breakdowns on a production line are Poisson distributed at a mean rate of 3.2 per hour. Using the formula for the Poisson distribution, Önd the probability of
(i) 3 breakdowns in 1 hour, (ii) no breakdowns in 1^12 hours, (iii) at least one breakdown in 1^12 hours.
- In order to assess the impact of a reduction in sta¢ ng levels, the Human Resources manager of a food retailer records the number of customer complaints in 23 branches during the year following the change and calculates the increase in complaints compared to the previous year. Using Minitab to analyse the increases, she obtains the output below. One-Sample T: Increase
Test of mu = 0 vs mu > 0
Variable N Mean St Dev SE Mean T P Increase 23 27.3 75.6 15.8 1.73 0.
(a) How are the Ögures labelled T and P related to those labelled Mean and SE Mean? (b) Is there signiÖcant evidence of an increase in complaints? Explain.
Section B Each question carries 20 marks
- (a) An electrical goods shop summarises one dayís purchases as fol- lows:
Value of purchase (£ ) Number of Purchases Under 50 78 50 99 201 100 199 375 200 499 416 500 1000 37
i. Calculate the percentage of total purchases in each of the intervals given in the table. What proportion of the purchases that are £ 50 or more, cost less than £ 500? [4 marks] ii. Appendix Q9a contains a histogram of the data using the above intervals. A colleague argues that the histogram must be wrong as its shape disagrees with the percentages calcu- lated in part i. Would you agree or disagree? Explain your reasons. [6 marks]
(b) A project to rewire a University building consists of independent stages, speciÖcally three working periods in between which are two lengthy periods of inactivity. These have Normally distrib- uted lengths (in units of 100 hours) whose means and standard deviations are given below:
Stage 1 2 3 4 5 Mean Length (100 hours) 2 4 1 : 2 4 : 8 2 : 4 Standard Deviation 0 : 5 1 : 5 0 : 3 1 : 8 0 : 6
i. Find the expected value and standard deviation of the total time required. [4 marks] ii. Calculate the probability that the total time required lies be- tween 1151 and 2047 hours. [6 marks]
- a. The number of meals sold per day at a fast food restaurant, are in- dependent and Normally distributed with mean 480 and standard deviation 150. What is the probability that i. the number sold in one day is more than 678? [3 marks] ii. the average number sold over a period of 14 days is greater than 465? [6 marks] iii. the total number sold over a period of 30 days is between 13,500 and 15,000? [6 marks] b. In order to assess the impact of a promotional campaign, the daily increases in meals sold for 14 similar restaurants were recorded and analysed using Minitab to give the output below.
Variable N Mean St Dev Increase 14 52 87
Assuming the increases are independent and Normally distributed, calculate a 90% conÖdence interval for the increase. [5 marks]
- a. A consulting Örm advises businesses applying for development grants. If decisions are taken independently with a constant prob- ability of success p, what will be the distribution of R, the number of successful applications out of n? Suppose that past experience has suggested that the consultancy has a 60% success rate but, out of 10 recent applications, only 2 have succeeded. They wish to decide whether their success rate has dropped. Carry out a test of the null hypothesis, H 0 : p = 0: 6 , against an alternative which you should state. Say whether you would reject the null hypothesis and at what level of signiÖcance. What conclusion should the Örm reach? [10 marks] b. i. Each month three employees are chosen at random to do overtime work. If four of the twelve employees are women, calculate the probability that the number of women chosen is a. none, b. one, c. two. [5 marks] ii. In the following month the same procedure is repeated. Cal- culate the probability that, over the two month period, at most one woman is chosen. [5 marks]
- a. An airline has a dedicated repair facility which carries out routine maintenance and minor and major repairs and has been given a target of 24 hours to return each aeroplane to service. 50% of aircraft sent to the facility are there for routine maintenance, 40% for minor repairs and the remaining 10% for major repairs. On examination, some aircraft are found to require additional repairs which may extend the total time required to over 24 hours. The percentage requiring additional repairs is given below for each ini- tial category of repairs along with the probability that the total work required then takes longer than 24 hours. If no additional repairs are required routine maintenance and minor repairs take under 24 hours but 75% of major repairs take over 24 hours.
Class of Repairs Percentage requiring Probability of taking additional repairs over 24 hours Routine Maintenance 50 0. Minor Repairs 40 0. Major Repairs 20 0.
i. Draw a three stage tree diagram to represent this situation.[ marks] ii. What proportion of all aircraft (a) require additional repairs; (b) take over 24 hours? [4 marks] iii. What proportion of aircraft which are repaired within the 24 hour target, were initially having only routine maintenance? [3 marks]
b. In a 26 week period a taxi company takes 79 bookings for its long- distance service to Heathrow Airport. Assuming these bookings occur randomly at a constant rate calculate a 95% conÖdence in- terval for the rate of bookings per week. [ marks]
Appendix Q10a
Game
Number of Orders (1000s)
A B C
8 7 6 5 4 3 2 1 0
Number of Orders (1000s) for 3 Computer Games
Appendix Q10b
Regression Analysis: Sales versus Price
The regression equation is Sales = 48.6 - 1.12 Price
Predictor Coef SE Coef T P Constant 48.644 2.799 17.38 0. Price -1.1181 0.4045 -2.76 0.
S = 5.18281 R-Sq = 29.8% R-Sq(adj) = 25.9%
Regression Analysis: Sales versus Value rating
The regression equation is Sales = 32.8 + 1.82 Value rating
Predictor Coef SE Coef T P Constant 32.791 2.175 15.08 0. Value rating 1.8164 0.4036 4.50 0.
S = 4.24311 R-Sq = 53.0% R-Sq(adj) = 50.3%
Regression Analysis: Sales versus Price, Value rating
The regression equation is Sales = 27.0 + 0.509 Price + 2.34 Value rating
Predictor Coef SE Coef T P Constant 27.033 7.440 3.63 0. Price 0.5093 0.6287 0.81 0. Value rating 2.3420 0.7662 3.06 0.
S = 4.28423 R-Sq = 54.7% R-Sq(adj) = 49.4%
Sales
Price
Value rating
4 8 12 3 6 9 50
40
30 12
8
4
Sales of Wine versus Price and Value Rating 20 supermarket 'own label' brands
