N11127 QUANTITATIVE METHODS 1B SPRING SEMESTER 2021-22
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A LEVEL 1 MODULE, SPRING SEMESTER 2021-22
QUANTITATIVE METHODS 1B
1. A natural monopoly faces the following demand curve and total cost function:
(a) Find the maximum profit the company can make and demonstrate that it is a maximum using second-order conditions. [9 marks]
(b) The government wishes to increase output in this industry to 30. Find the minimum efficient scale for this industry, and hence find the number of additional firms which should be allowed to enter it. [6 marks]
2. A night club is planning to introduce a discounted student price for entry. It faces the following inverse demand and total cost functions:
where P1 and P2, and Q1 and Q2 represent the price and quantity for non-students and students respectively.
(a) Find the profit maximising price and quantities for each group and demonstrate that it is a maximum using second order conditions. [13 marks]
(b) Find the marginal revenue and marginal cost for both students and non-students. Without further calculation, if students demand rises, explain why the price of non-student tickets should increase. [7 marks]
3. An insurance company is investigating the link between the number of dogs owned by a household and the probability that the house is burgled in a given year. Based on previous data, they estimate the probability distribution to be as follows:
|
|
|
|
Number of dogs (X) |
||
|
|
|
|
0 |
1 |
2 |
|
|
|
0 |
0.6 |
0.11 |
0.04 |
|
Number of burglaries (Y) |
1 |
0.15 |
0.03 |
0.01 |
|
|
|
|
2 |
0.05 |
0.01 |
0 |
(a) Find the marginal distribution of the number of dogs owned by a household. [2 marks]
(b) Find the mean and variance of the number of dogs owned by a household (X). [5 marks]
(c) Given that a house has 1 dog, what is the probability that it is burgled twice in a given year? [2 marks]
(d) Find the covariance between X and Y and comment on the relationship between these variables. [6 marks]
4. Eric owns 1000 shares in Company A, which have a value of 200p each. The value of shares in Company A at the end of the year follows a normal distribution with a mean of 228 and a standard deviation of 24.
(a) What is the expected value of Eric’s shares at the end of the year? [3 marks]
(b) What is the probability that Eric’s shares will be worth more than £2400 at the end of the year? [4 marks]
(c) Eric has a further £1000 to invest, and decides to buy 1000 shares in Company B, which have a value of 100p each. The value of shares in Company B at the end of the year follows a normal distribution with a mean of 108 and a standard deviation of 10, and is independent of the value of Company A’s shares. What is the expected value and standard deviation of Eric’s total investment in both Company A and B at the end of the year? [4 marks]
(d) What is the probability that Eric’s total investment is worth less than £3000 at the end of the year? [4 marks]
(e) Eric thinks his current investments are two risky, and wants to reduce the probability of making a loss. He’s planning to sell all his shares in Company A and buy an additional 2000 shares in Company B, as it has lower standard deviation. Do you think this is a good idea? What would you advise Eric to do if he wants to minimize the probability of making a loss? Give reasons for your answer. [5 marks]
5. An engineering company is trying to estimate the average time needed to complete maintenance on a recently deployed military jet engine. So far, they have performed the task 17 times, with a sample mean of 13.4 hours and a sample standard deviation of 4.8 hours.
(a) Construct a 90% confidence interval for the true mean maintenance time. [6 marks]
(b) What assumption have you made in your answer for (a)? [2 marks]
(c) Without further calculation state whether a 95% confidence interval would be wider or narrower than your answer to (a). Give reasons for your answer. [2 marks]
(d) One of the engineers tells you that he believes the distribution of maintenance times is skewed. If he is correct, explain why this would make a confidence interval difficult to calculate. [5 marks]
6. A sales company is piloting a new technique for its telesales team. It has trained 1% of its staff in the new technique, and they have made 112 sales from 600 calls. Its previous technique has a 16% success rate in telesales.
(a) Conduct a hypothesis test to see whether there is sufficient evidence that the new technique has a higher success rate than the old technique, at the 5% level. [9 marks]
(b) One of the directors of the company raises concerns about significant costs associated with retraining all staff in the new technique. Assuming the company decides to retrain all staff if it finds there is sufficient evidence that the new technique is better, what do Type 1 and Type 2 error represent in this case? Comment on whether you think 5% is the best significance level to use here, and explain what advice you would give the company on whether to retrain their staff in the new sales technique. [6 marks]
2023-08-15