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BUSI1074-E1

A LEVEL 1 MODULE, AUTUMN SEMESTER 2021-2022

QUANTITATIVE METHODS 1B (EXAM)

Question 1

A firm has successfully separated its customers into two groups. Customers cannot resale the product or switch from one group to the other. The firm knows its own cost structure and the demand curves of both groups. Total costs depend on the total production (sales) and not on how total sales are divided among groups. The corresponding demand equations are given by

respectively. The total cost function is  where .

a) Find the quantities produced and the prices that the firm should charge its trade and individual customers to maximize total profit. [7 marks]

b) Define and calculate the price elasticities of demand at these prices. For which group will the firm set a higher price? Explain. [5 marks]

c) Find and classify the inflection points of function . [5 marks]  [17 marks Total]

Question 2

A firm’s production function is given by

with  and .

a) Find the values of the marginal products  and  at this point. [4 marks]

b) Use the results of part (a) to estimate the overall effect on  when  increases by 11 units and  decreases by 12 units. [3 marks]

c) Find the derivative of function . [8 marks]

d) Find the equation of the tangent line to  at . [6 marks] [21 marks Total]

Question 3

There are 6 boys who enter a boat with 8 seats, 4 on each side. In how many ways can

a) they sit anywhere? [3 marks]

b) two boys A and B sit on the left side and another boy W sit on the right side? [5 marks] [8 marks Total]

Question 4

a) A doctor is called to see a sick child. The doctor has prior information that 80% of sick children in that neighbourhood have the flu, while the other 20% are sick with 1 measles. Let F stand for an event of a child being sick with flu and M stand for an event of a child being sick with measles. Assume for simplicity that there is no other malady in that neighbourhood. A well-known symptom of measles is a rash (the event of having which we denote R). Assume that the probability of having a rash if one has measles is  = 0.98. However, occasionally children with flu also develop rash, and the probability of having a rash if one has flu is  = 0.07. Upon examining the child, the doctor finds a rash. What is the probability that the child has measles?

(Use the appropriate notation for probability theory). [8 marks Total] 

Question 5

Two random variables X and Y show the following joint probability distribution:

a) Find the cumulative probability F(1,3) (the exact value is required for each result). [3 marks]

b) Find the marginal distribution of Y (the exact value is required for each result). [5 marks] [8 marks Total]

Question 6

Let’s say that 10% of all business startups in the IT industry report that they generate a profit in their first year. A sample of 55 new IT business startups is selected.

a) Find the probability that exactly seven startups will generate a profit in their first year. [2 Marks]

b) What probability distribution have you used? Why? [2 Marks]

c) Also use other appropriate approximate distributions and compare the results. [3 marks]

d) Finally, suppose that the startups’ annual revenues are normally distributed, with a mean of 8 $mil. and a standard deviation of 0.8 $mil. Calculate the probability that a startup gets revenues between 6 $mil and 9 $mil. [3 marks] [10 marks Total]

Question 7

Imagine that we want to test whether or not girls, on average, score lower than 600 on the SAT verbal section. Suppose we also happen to know that the standard deviation for girls’ SAT verbal section scores is 100. We collected the data using a random sample of 64 girls and their verbal section scores, with an average score of which equal to 585.

a) Perform a hypothesis test using an 8% level of significance. [5 marks]

b) What distribution did you need to use to determine the critical value and why? [2 marks]

c) Calculate the p-value of the test statistic and interpret your result. [3 marks]

d) Define the Type 2 Error and calculate the value of β if the true mean is 580. [6 marks]

e) Define and calculate the power of the test in this case. [2 marks] [18 marks Total]

Question 8

A health-care actuary has been investigating the cost of maintaining the cancer patients within its plan. These people have typically been running up costs at the rate of $1240 per month. A sample of 15 cases for November (the first 15 for which complete records were available) reveals an average cost of $1080, with a standard deviation of $180. 

State the null and alternative hypothesis. [2 marks]

Calculate the test statistic. [3 marks]

What distribution do we need to use to determine the critical value and why? [2 marks]

Is there any evidence of a significant change (at a 2% level of significance)? Explain. [3 marks] [10 marks Total]