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Question One

a) Find the stationary points of. Classify each of them as maxima, local, minima or points of inflexion.

b) Use differentiation to find the maximum or minimum points of; . Use differentiation again to find out whether it is a maximum or a minimum. How does this confirm what know about the shape of a quadratic function?

c) Find turning points of. Are these maxima, minima or points of inflexion?

d) A firms output is, where L is the number of labour hours in thousands. How many hours maximize output? Find also the value that maximizes the marginal product for labour.

Question Two

A textile company uses three service companies, Alpha, Beta and Gamma, to repair machinery when it breaks down. When a piece of machinery breaks down there is a 20%chance that it is sent to Alpha and a 40% chance that it is sent to each of Beta and Gamma. A study of these companys’ service times over the last two years has revealed the following:

The probability that broken machinery sent to Alpha is returned within a week is 0.5. The probability that broken machinery sent to Beta is returned within a week is 0.6 and the probability of the same return time for Gamma is 0.4.

Machinery returned within a week by Alpha or Beta has an 80% probability of being satisfactorily repaired, whereas machinery returned within a week by Gamma has a 90% probability of being satisfactorily repaired.

Machinery sent to Alpha or Beta, which takes longer than a week to repair has a 90% probability of being satisfactorily repaired, whereas machinery sent to Gamma and taking longer than a week is always satisfactorily repaired. If a piece of machinery has taken over a week and is not satisfactorily repaired what is the probability that it was sent to Alpha?

Question Three

a) Petrol consumption for all small cars is normally distributed with µ=30.5m.p.g and ?=4.5 m.p.g. a manufacturer wants to make a car that is more economical than 95% of small cars. What must be its m.p.g. Use a computer.

b) In a study of efficiency of a lie-detector test, 1000 people were given the test. Of these 500 lied and 500 told the truth. The lie-detector said that 185 of those who were telling the truth were liars and that 120 of the liars told the truth. Consider a single person who is about to take a lie-detector test.

Required:

i) Write down the null and alternative hypotheses for such a test

ii) Suppose the lie-detector says that he is lying. Is this evidence that you should reject the null hypothesis or not?

iii) Is the lie-detector test a good one or not?

c) Legislation dictates that bottles of wine should contain an average volume of exactly 0.7litre. A sample of 6 bottles from a wine importer gives a mean of 0.697 and a standard deviation of 0.01. Test whether the wine importer is under filling bottles. State any assumptions that are necessary for the test.

Question Four

The lane used by 8 athletes in a 400m race was noted, along with the athletes’ times in seconds. From the information below, calculate the correlation between the lane number and the athletes’ times.

Lane number

1

2

3

4

Time (s)

56.1

57.3

55.7

56.5

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