Mathematical and statistical exercises

Your answers must be set out neatly and concisely and should be accompanied by any associated workings, so that, if necessary, your answers can be verified. If you provide your workings, then we can also give you partial credit for answers that are incorrect. The marking of each question will be on the basis of logical structure and method (70%), the accuracy of results (20%) and presentation (10%).

Question 1 (15 marks):

A manufacturer produces three products A, B, and C. The profit for each unit of A, B, and C sold are $2, $3, and $5, respectively.  Fixed cost are $50,000 per year, and the variable costs of producing each unit of A, B, and C are $3, $6, and $4, respectively.  Next year, a total of 27,550 units of all three products is to be produced and sold, and a total profit of $109,910 is to be realized.  If total cost is to be $174,100, how many units of each of the products should be produced next year? (Use the matrix operations to solve this question). 

Question 2 (15 marks):

An investment firm offers 3 stock portfolios, A, B, and C. The number of blocks of each type of stock in each of these portfolios is summarized in the following table:

A client wants 165 blocks of high-risk stock, 132 blocks of moderate-risk stock, and 138 blocks of low-risk stock. Use the matrix operations to find how many of each portfolio should be suggested.

Question 3 (15 marks)

1. Suppose that the equation r = 432q– 3q² gives the total revenue r (in dollars) that a manufacturer receives when q units of a product are sold. Determine the marginal revenue when q = 155 and interpret your result.
2. If a manufacturer's average cost is given by AC = 0.002q²- 0.24q + 776.  Find:

     (a)  the marginal cost function;

     (b)  evaluate it when 60 units are produced.

Question 4 (15 marks)

1. The demand equation for a monopolist's product is p= 200 - 0.98q, where p is the price per unit (in dollars) of producing q units. If the total cost c (in dollars) of producing q units is given by C = 0.02q² + 2q + 8000.  Find the level of production at which profit is maximized.
2. Suppose that the saving function of a country is given by: 

where the national income (I) and the national saving (S) are measured in billions of dollars.  Determine the country’s marginal propensities to consume and its marginal propensities to save when the national income is $121 billions.

Question 5 (40 marks):       

A bank branch located in a commercial district of a city has the business objective of developing an improved process for serving customers during the noon-to-1:00 p.m. lunch period. The waiting time, in minutes, is defined as the time the customer enters the line to when he or she reaches the teller window. Data collected from a sample of 25 customers during this hour are stored as follows:

4.30     5.55     3.02     5.13     3.07    

2.34     3.54     3.20     4.50     6.15

1.38     5.12     6.26     6.19     4.89

3.34     3.74     5.10     4.50     4.80

5.13     2.17     3.82     4.05     4.80

1. Compute the mean, median, and mode.
2. Compute the variance, standard deviation, range, coefficient of variation, and Z scores.
3. Are there any outliers? Explain.
4. As a customer walks into the branch office during lunch hour, she asks the branch manager how long she can expect to wait. The branch manager replies, “Almost certainly less than five minutes.” On the basis of the result of (1) and (2), evaluate the accuracy of this statement.