QUESTION ONE (a) Beta Glassworks Ltd. manufactures bottles for the beverage industry. The company specializes in producing bottles of three different capacities: 0.2 litre, 0.3 litre and 0.5 litre. The three types of bottles are made by two machines A and B. The number of hours required for each type of bottle on each machine is as shown below: Machine A Machine B Type of bottle 0.2 litre 0.45 0.30 0.3 litre 0.75 0.15 Machine A has 3.825 hours available and machine B has 2.025 hours available. The management has decided that 1,500 bottles of 0.2 litre must be produced. Required: Determine how many bottles of 0.3 litre and 0.5 litre should be produced in addition to the 1,500 bottles of 0.2 litre. (6 marks) (b) Distinguish between input-output analysis and Markov analysis. marks) Input Motor vehicles 0.17 0.5 litre 0.60 0.75 (c) A miniature economy has three industries: motor vehicle, electricity and steel. These industries are interdependent., such that, the output of one industry is the input of another. The following table shows the input ratios of each industry. Output Electricity 0.25 Motor vehicles 0.25 Electricity 0.25 Steel 0.50 0.33 The Leontif inverse matrix is computed as below: Steel 0.25 0.33 0.33 (I-A) X Required: (1) (ii) (111) 3.08 1.98 2.15 2.64 3.41 2.70 3.96 3.19 4.46 Define the term "input ratio." (2 marks) Interpret the input column and output row for motor vehicles using the input ratios table. (2 marks) Determine the primary inputs required by each industry, if the final demand is Sh. 216 million, Sh. 240 million and Sh. 360 million for motor vehicles, electricity and steel industries respectively. (6 marks) (iv) The assumptions made in making the analysis above. (2 marks) (Total: 20 marks) QUESTION TWO Leisure Publishers Ltd. recently published 20 romantic novels by 20 different authors. Sales ranged from just over 5,000 copies for one novel to about 24,000 copies for another novel. Before publishing, each novel had been assessed by a reader who had given it a rating between 1 and 10. The managing director suspects that the main influence on sales is the cover of the book. The illustrations on the front covers were drawn either by artist A or artist B. The short description on the back cover of the novel was written by either editor C or editor D. A multiple regression analysis was done using the following variables: Y sales (millions of shillings). X 1 if front cover is by artist A. 2 if front cover is by artist B. readers' rating 1 if the short description of the novel is by editor C. 2 if the short description of the novel is by editor D. The computer analysis produced the following results: Correlation coefficient r = 0.921265 Standard error of estimate = 2.04485 Analysis of variance 1 2 3 Regression Residue Individual analysis of variables: Variable Coefficient Constant 15.7588 -6.25485 0.0851136 5.86599 1 Correlation Coefficients Required: Degrees freedom (a) (b) marks) 3 16 (e) -0.307729 1 of Sum squares 375.37 66.903 of Mean QUESTION THREE (a) (1) (ii) marks) square 125.12 1.1814 Standard error 2.54389 0.961897 0.298272 0.922233 0 0.123094 1 Fratio 29.923 F Value 38.375 42.284 0.081428 40.457741 -0.674104 0.310838 0.627329 1 The regression equation. Does the regression analysis provide useful information? Explain. (3 marks) (3 (c) Explain whether the covers were more important for sales than known quality of the novels. (4 marks) (d) State with 95% confidence the difference in sales of a novel if its cover illustrations were done by artist B instead of artist A. (5 marks) State with 95% confidence the difference in sales of a novel if its short description was by editor D and not editor C. (5 marks) (Total: 20 marks) State the Baye's theorem (1 mark) What is the importance of Baye's theorem to a business manager? (2 (b) Excel Audit Consultants (EAC) have engaged three audit trainees; Odhiambo, Koech and Mwai. After three months, the audit supervisor notices that 3%, 5% and 4% of the accountants audited by Odhiambo, Koech and Mwai respectively have some procedural errors. The supervisor had distributed the accounts such that Odhiambo audited 40%. Koech audited 30% and Mwai audited 30% of the accounts. Required: The proportion of audited accounts with procedural errors. (6 marks) The probability that an account with an error is audited by either Koech or Mwai. (5 marks) (1) (ii) (c) The finance manager of Apar Construction Ltd. has two investment proposals before him and he wishes to choose one of them. He will choose the project that offers him a higher expected net present value (NPV). In case both projects have the same NPV, he will prefer the project that has a lesser risk. The data for the two projects is given below: Possible outcomes Project X: Pessimistic Most likely Optimistic Project Y: Pessimistic Most likely Optimistic Probability of possible outcomes 0.25 0.50 0.25 0.25 0.50 0.25 Net present value Sh. million 1,500 5,000 9,000 - 10,000 5,000 20,500 Required: The project the finance manager would choose, given that risk is measured by the standard deviation. (6 marks) (Total: 20 marks) QUESTION FOUR (a) Explain the difference between the paired t-test and the two-sample t-test. (4 marks) (b) Trendy Tyres Ltd. has introduced a new brand of tyres, which in their advertisements claim to be superior to their only competitor brand, the Roadmaster Tyres. The brand manager of Roadmaster Tyres disputes this claim which he says is an advertisement gimmick. The brand manager of the two companies agree to run a road test for the brands. Ten (10) saloon cars of uniform weight and identical specifications are to be used for the test. Each car is fitted with both brands of tyres: One brand at the front and the other brand at the rear. The cars cover a distance of 5,000 kilometres and the trend wear is recorded as follows: 12345nora: 6 7 8 9 10 Trendy tyres Centimetres 1.08 1.06 1.24 1.20 1.17 1.21 1.18 1.10 1.22 1.60 Roadmaster tyres Centimetres 1.12 1.09 1.16 1.24 1.23 1.25 1.20 1.15 1.19 1.13 Required: (1) Determine whether Trendy Tyres Ltd.'s claim is true using x=0.01. (15 marks) (ii) What are the assumptions you have made in (1) above? mark) (Total: 20 marks) QUESTION FIVE (a) State the principal components of a time series. marks) (b) (i) Explain the difference between multiplicative and additive models as used in time series. (2 marks) (ii) State the conditions under which each model is used. (2 marks) (c) The table below shows the sales of new cars by quarters during a period of three years: Year 2001 2002 2003 Quarter 1 Sh. "million" 55.0 54.4 59.3 Quarter 2 Sh. "million" 76.5 65.9 83.2 Required: (1) (ii) Quarter 3 Sh. "million" 61.2 52.7 78.5 Quarter 4 Sh. "million" 77.8 81.4 93.0 Required: (1) Explain the purpose of the seasonal index marks) (ii) The seasonal index for each quarter assuming an additive model. marks) (Total: 20 marks) SECTION II QUESTION SIX (a) A company manufactures mountain and racing bikes. Each mountain bike is sold at Sh. 6.375 and a racing bike is sold at Sh. 9,000. A racing bike requires 6 kilogrammes of aluminium while a mountain bike requires 4 kilogrammes of aluminium. Aluminium cost Sh. 750 per kilogramme. It costs the company Sh. 375 per hour to assemble each bike (irrespective of model). It takes 1 hour to assemble a racing bike and 2 hours to assemble a mountain bike. There are 4,800 kilogrammes of aluminium available and a total of 1,600 hours available for assembly. Formulate the linear programming problem. Graphically solve the problem formulated in (1) above. (2 (12 (6 marks) (10 marks) (b) Is it appropriate to use non-graphical linear programming methods to solve the (2 problem in (a) above? Why? marks) State some of the simplifying assumptions you have made to solve the problem in (a) above. (2 marks) (Total: 20 marks)