# mathematics

1. This problem is in reference to students who may or may not take advantage of the opportunities provided in QMB such as homework. Some of the students pass the course, and some of them do not pass. Research indicates that 40% of the students do the assigned homework. Of the students who do homework, there is an 80% chance they will pass the course. The probability of not passing if the student does not do the home work is 90%. What is the probability of a student not doing homework or passing?

2.Suppose that for a certain football game the probability that the home team will be ahead at half-time is 0.60 and the probability that the home team will be ahead at half- time as well as at the final gun is 0.45. What is the probability that the home team will win this game given that it is ahead at the half?

3. A company markets two products (Product A and Product B) through mail order. The company will market them in sequence with the first mail order offer for product A. It f feels that there is a 30% chance that any customer will purchase product A. Product B is offered some months later. It is felt, for product B, that there is a 30% chance of selling product B to a customer if the customer purchased product A and a 5% chance of selling product B to a customer who did not purchase product A.

What is the probability of not selling product B to a particular customer?

4. A quality control department finds that it accepted only 5% of all bad items and it rejected only 1% of good items. A supplier has just delivered a shipment of a certain item. Past records show that only 90% of the parts of that supplier are good. If the department accepts an item, what is the probability that the item is bad? Round your answer to five decimal places.

5. The probability that house sales will increase over the next six months is estimated at 0.25. It is also estimated that the probability is 0.74 that 30 year fixed-loan mortgage rates will increase over this period. Economists estimate that the probability is 0.89 that either housing sales or interest rates will increase.

The probability that both house sales and interest rates will increase is estimated at:

6. Over the last 100 business days, Harry had 20 customers on 30 of those days, 25 customers on 20 days, 35 customers on 30 days, 40 customers on 10 days, and 45 customers on 10 days. What is the variance of the number of Harry’s customers?

7. What is the probability that exactly 1 out of 10 cars experience a breakdown if the probability of a breakdown is 30%?

8.An accounts receivable auditor is examining accounts for a client. The accounts receivable balance can be considered as a continuous random variable that exhibits normal distribution characteristics. The mean amount due per account is $5000. The standard deviation is $1000. The auditor selects an account at random.

The probability that the account selected by the Auditor has a balance which is outside of the range between $6500 and $7000 is:

9. Suppose the length of time (in days) between sales for an automobile salesperson is modeled as an exponential distribution with a mean of 2 days. What is the probability the salesperson goes more than 5 days without a sale?

10. During lunch time, customers arrive at Joe’s Lunch counter according to a Poisson distribution with an average of 2 per 30 second period. What is the probability of having more than two arrivals in a two-minute period?

Use the following to answer questions 11-12

The new owner of a beauty shop is trying to decide whether to hire one, two, or three beauticians. She estimates that profits next year (in thousands of dollars) will vary with demand for her services and has estimated demand in three categories low, medium and high.

11. If she uses the optimistic criterion, how many beauticians will she decide to hire?

12. If she uses the minimax regret criterion, how many beauticians will she decide to hire?

**Use the following decision tree to respond to the next 5 questions:**

13. What is the expected value at node 4?

14.What is the expected value of perfect information?

15.What is the expected value at node 2?

16. What is the best decision strategy for the manager?

17. What is the expected value of sample information?

18. A statistician working for a car manufacturer developed a statistical model for predicting delivery time (the number of days between ordering a car and actual delivery) of a particular model for which there is a range of factory-fitted options. Use the output below to forecast the difference in delivery times for cars with 6 options and 9 options, rounded to two decimal places.

19. Given the table below what is the MAD and the MSE?

20. The Acme Computer Company has recorded sales of one of its products for a six-week period: Using the three-week simple moving-average method, forecast sales for week 7.

21 Shown below are data that reflect the number of daily traffic accidents at a dangerous city intersection. The regression equation is: number of accidents = 5.3 + 0.5 t

What is the forecasted number of accidents for day 6?

22. The following table contains the number of consumer complaints received in a Publix supermarket in Florida. Use exponential smoothing with a constant of α = 0.33 to forecast the number of complaints in March, round to the nearest whole number of complaints.

23. According to the regression model, the correlation between Halliburton’s stock price and the S&P500 index value is:

24. The regression model ABOVE is:

25. According to the regression model ABOVE, when the S&P 500 index value is 1,000, the forecasted Halliburton stock price is:

26. Corporate AAA bond interest rates for 12 consecutive months are 9.5%, 9.3%, 9.4%, 9.6%, 9.8%, 9.7%, 9.8%, 10.5%, 9.9%, 9.7%, 9.6% and 9.6%. The three-month weighted moving average forecast (with weights 0.2, 0.4 and 0.4 – from oldest to most recent respectively)for the next month, rounded to two decimal places, is:

27. What is the optimal production quantity of the Fliptop model?

28. If all the constraint inequalities in the original problem were ≥, then the following is true:

29. Let M be the number of units to make and B be the number of units to buy of a certain product. If it costs $2 to make a unit and $3 to buy a unit and 4000 units are needed, the objective function of the LP model to minimize the cost of production would be,

30. The Quiet Meadow Studio sells photographs and prints. It cost $20 to purchase each photograph and it takes 2 hours to frame it. It costs $25 to purchase each print and it takes 5 hours to frame it. The store has at most $400 to spend and at most 60 hours to frame.

It makes $30 profit on each photograph and $50 profit on each print. Determine the maximum profit.

31. Answer the following question using the Excel output above. Which constraint(s) are binding?

32. Answer the following question using the Excel output above. Keeping within the confines of the problem, the profit on Map A has increased by one dollar. Determine the new objective function value.

33. Quentin Magic Brown manufactures sports shoes and wants to maximize the company’s profits. The company makes two types of sport shoe, Airwalkers and Bouncy Basketball shoes. The company earns $10 profit on each pair of Airwalkers and $18 profit on each pair of Bouncy Basketball shoes.

The manufacturing process includes cutting the materials on a machine and having workers assemble the pieces. Each pair of Airwalkers requires 3 minutes of cutting time and the Bouncy Basketball shoes require 2 minutes. The machines that cut the material can run at most 1200 minutes a week.

Each worker takes 7 hours to assemble a pair of Airwalkers and 8 hours to assemble a pair of Bouncy Basketball shoes; the maximum number of hours available is 3500 per week.

Determine the maximum profit for this problem?

Questions 34-37 apply to the Excel output for the Quantum Mo-Botics model is below. The company makes three types of machines and has limitations with regards to the amount of skilled and unskilled labor hours available and time on the assembly line.

34. Answer the following question using the Excel output above, determine the new objective function value if the profit on the second variable, Robotic, increases by $1000?

35. Answer the following question using the Excel output above. Keeping within the confines of the problem, you are required to hire a full time (40 hours) person who is qualified to work in any department. Select the constraint where you will gain the most profit and determine the additional profit to be gained?

36. Answer the following question using the Excel output above. Keeping within the confines of the problem, how many more hours of skilled workers could you add to the department?

37 Using the Excel output above, how much is each additional unit of unskilled labor worth?

38. An ice cream plant make’s Chocolate and Strawberry ice cream.

There is $40 profit for a case of Chocolate and $32 for a case of Strawberry and has the following constraints:

39. Using the decision table below

40. In a survey about soft drinks it was found that 5% of the respondents like diet soft drinks. 12% of them liked the brand Coke. Of all the Diet soda drinkers 40% liked Coke.