The long Life Insurance Company receives applications to buy insurance from its salespeople, who are specially trained in selling insurance to new customers. After the applications are received, they are processed through a computer. The computer is programmed so that it prints messages whenever it runs through an item that is not consistent with company policies. The company is concerned with the accuracy of the training that its salespeople received, and it contemplates recalling them for more training if the quality of their performance is blow certain limits. Five samples of 20 applications received from specific market areas were collected and inspected with the following results:

(1) Find the sample size and average error rate.

(2) Compute three-sigma control limits.

(3) Draw the control chart(s).

(4) There are two new samples were taken each with 20 applications. Four and five applications were found to have mistakes, respectively. Without re-computing the control limits, is the training process out of control?

(5) Is there a need for recalling the sales force? Explain.

Let us choose seven arbitrary distinct positive integers, not exceeding 24. Show that there will be at least two subsets chosen from these seven numbers with equal total sums. (Keep in mind that sets, and hence subsets, have no repeated elements.) Hint: How many subsets can you form altogether? What is the largest total sum of such a subset?

Question 2:

Consider the following data on 10 students:

(a) Calculate the values of β0 ̂ and β1 ̂ for the simple linear regression model given by:

food =β̂0 +β̂1 incomei + ei

2. (b) Interpret those values in the context of the variable definitions and units of measurement.

3. (c) Using the results from part (a), calculate the error (ei) for each of the 10 observations.

(c) Calculate and interpret the standard error of tĥe regression (se).

(d) Calculate and interpret the standard error of the β1 ̂ estimate (s β̂1)

(e) Test the null hypothesis that income has no effect on food expenditures. What do you conclude?

The Perotti Pharma Company is investigating the relationship between advertising expenditures and the sales of some over-the-counter (OTC) drugs.

The following data represents a sample of 10 common OTC drugs. Note that AD = Advertising dollars in millions and S = Sales in millions $.

1. What is the equation of the regression line?

2. Interpret the slope in the context of the problem.

3. Find the coefficient of determination.

4. Interpret the meaning of R² in the context of the problem.

5. State the hypotheses to test for the significance of the regression equation.

6. Is there a significant relationship between dependent and independent variables at alpha=0.05? Why?

7. Create a 95% confidence interval for Sales if Advertising dollars = $50 million and interpret its meaning.

8. Paste the table with the results of regression analysis.

Consider the he kinetics data below. Note that two reactions were studied. Use the information to calculate the EQUILIBRIUM CONSTANT for

            2 NOCl (g) -> 2NO(g) + Cl₂  

                                                                        [NO]               [Cl₂]                initial rate

                                                                        0.10M             0.10M             0.54M/s

                                                                        0.10M             0.20M             1.05M/s

2NO(g) + Cl₂(g) -> 2NOCl(g)                    0.10M             0.30M             1.62M/s

                                                                        0.10M             0.40M             2.16M/s

                                                                        0.20M             0.50M             10.8 M/s

                                                                        0.30M             0.50M             24.3 M/s

                                                                        0.40M             0.50M             43.2 M/s

[NOCl]            time

5.00M             0s

4.55M             5.0 x 10⁵s

2NOCl(g) -> 2NO(g) +   Cl₂(g)                   4.174M           1.0 x 10⁶s

3.855M           1.5 x 10⁶s

3.582M           2.0 x 10⁶s

Use the pumping lemma to prove that the following languages are not regular.

(a)L2 = {y = 10 × x | x and y are binary integers with no leading 0s, and y is two times x}. (The alphabet for this languages is {0, 1, ×, =}.) For example, 1010 = 10 × 101 is in L2, but 1010 = 10 × 1 is not.

(b)Let Σ2 = {[ 0 0 ] , [ 0 1 ] , [ 1 0 ] , [ 1 1 ]} . Consider each row to be a binary number and let L3 = w ∈ Σ ∗ 2 | the bottom row of w is the square of the top row of w . For example, [ 0 1 ] [ 0 0 ] [ 1 0 ] [ 0 0 ] [ 0 0 ] ∈ L3, but [ 0 1 ] [ 0 0 ] [ 0 0 ] [ 1 0 ] [ 0 0 ] [ 0 0 ] ∈/ L3.

A cable company offers two basic packages: sports and kids, and a combined package. There are three different types of users: parents, sports fans, and generalists. Assume that the cable company cannot discriminate among the three groups and must charge all customers the same price. The following table shows the maximum price that each type of consumer is willing to pay for each package. Sports Package Kids Package Parents 10 50 Sports fans 50 10 Generalists 40 40

If the cable company sells the packages separately, how much price should it charge for each package? What will be total revenue? What will happen if this price is slightly raised?

b. Rather than raising the single package price to increase revenue, suppose now the firm opts to offer mixed bundling where the customers can buy either a single package or a bundled package. Show that the cable company will earn a higher revenue with mixed bundling than the single pricing method in part a.

Find the probability of rolling a sum of two dice that is a 7 or a 12. Round answer to 4 decimal places.

PROBLEM SET 7

Question 1

Erika, Ayami and Alison are partners in a fast growing food distribution business in Courtenay. To maximize their profit in the long-run by using the right mix of inputs, what should be their decision when labour or capital varies? (2 lines)

2 Marks

Question 2

You recently replace Antonia, the manager of Division 3, despite Antonia’s strong external sales record. Division 3 produces frozen drinks. Frozen drinks are relatively simple to make. Ignoring the ingredients, it requires only labour and a blender. As you begin reviewing the company’s production information, you learn that labour is paid $8 per hour and the last worker produced 100 drinks per hour. The company rents blenders for $15 per hour and the marginal product of capital is 120 drinks per hour. What do you think Antonia could have done to keep his job? Explain (3 lines)

2 Marks

Question 3

A firm has two plants, one in the United States and one in Canada, and it cannot change the size of the plants or the amount of capital equipment. The wage in Canada is $5. The wage in the U.S. is $20. Given current employment, the marginal product of the last worker in Canada is 100, and the marginal product of the last worker in the U.S. is 500.

a. Is the firm maximizing output relative to its labor cost? Show how you know. (2 lines)

b. If it is not, what should the firm do? (2 lines)

3 Marks

Question 4

Explain why one of diseconomies of scale and diminishing marginal product of the variable factors is a short-run concept and the other a long run concept.

3 Marks