**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 σ=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 |
5 |
6 |
7 |
8 |

Time (s) | 56.1 |
57.3 |
55.7 |
56.5 |
58.0 |
58.8 |
55.9 |
57.8 |

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