Time allowed: Three hours. Total marks: 60 marks.

There are FIVE questions in this examination; each starts on a fresh page. Answer ALL five questions; start each answer on a fresh page. All questions carry equal marks. The value of each sub-question is indicated in brackets. On the front of your answer book, write the number of each question you have attempted. Statistical tables and useful formulae are attached to this examination paper. Electronic calculators may be used. The examination paper may be retained by the candidate. Answers must be written in black or blue ink. Pencils may be used only for drawing, sketching or graphical work. Show the working steps in your answers.

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Question 1 (a) Consider the following binomial distribution :

Answer the following based on the above distribution: (i) What can you say about the skewness of this distribution? Explain. [1 mark] (ii) (iii) What are the mean and variance for the number of successes? [2 marks] The experiment that underlies this distribution is whether a project is profitable or not. Success is defined as a project to be profitable. If undertaking a series of these projects is to be well approximated by a binomial random variable, what are the two important assumptions that need to be satisfied? [1 mark]

(b) Windows Vista is a very unstable operating system. You are running your business using a PC with Windows Vista and from experience you expect your computer to crash 3 times a day. If this process is modeled as a Poisson random variable then answer the following: (i) (ii) (iii) What is the probability that your computer works smoothly through the day without crashing at all? [1 mark] If your computer crashes more than five times in one day, you get a free software upgrade. What is the probability of that? [1 mark] If, for every time your computer crashes, you lose $6 in business revenue, what are the mean and standard deviation of your loss per day? [2 marks]

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(c)

(i)

(ii)

ˆ To estimate a population parameter we have constructed two estimators ~ ~ ˆ and . Explain what it means to say that is efficient relative to .[1 mark] Production of a gadget takes two tasks that have to be completed in succession. The time it takes to complete task one is distributed normally with mean 35min and standard deviation 6min. The time it takes to complete task two (that starts right after task one is completed) is independent of time it takes for task one and is also distributed normally with mean 22min and standard deviation 3min. What is the probability that the production of a gadget takes no more than one hour? [2 marks]

(iii) After production is completed, every gadget goes into quality assurance. Experience shows that the quality of about 10% of gadgets is rejected. If 360 gadgets enter quality assurance a day, what is the probability that at least 320 gadgets pass the quality test in one day? [1 mark]

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Question 2

(a) Explain briefly the importance of the Central Limit Theorem for statistical inference. [2 marks] (b) The marketing department of a large store company is interested in finding out how much customers spend when they visit a city centre store. Accordingly, it randomly samples 26 customers who are leaving the store and asks them about their expenditures. The data reveals a sample mean of $125.40 and a sample standard deviation of $27.30. Stating carefully any assumptions you make, construct a 90% confidence interval for the mean value of expenditure by all customers. [3 marks] (c) The store manager knows that a similar survey was conducted last year and that the mean expenditure then was $116. Assuming that consumer price inflation during the year was 3% and that $116 was previously the population mean expenditure, examine whether or not real consumption expenditure at the...