5. A Pearson correlation statistic is only valid when the relationship between the two quantitative (continuous) variables is ____________.

1. Explain why it is true that the slope of a line is related to the Pearson correlation statistic, r.
   
2. Create a scatterplot to investigate the association between the amount of fluoride in domestic water (ppm) and the number of dental caries in permanent teeth per 100 children for 21 cities. The data are below.
   1. Create the scatterplot
   2. Describe the association you see in your scatterplot.
   3. The value of r is -0.86, -0.36, 0.36, or 0.86?

3. CityID	FLUORIDE (ppm)	CARIES
   1	1.9	236
   2	2.6	246
   3	1.8	252
   4	1.2	258
   5	1.2	281
   6	1.2	303
   7	1.3	323
   8	0.9	343
   9	0.6	412
   10	0.5	444
   11	0.4	556
   12	0.3	652
   13	0.0	673
   14	0.2	703
   15	0.1	706
   16	0.0	722
   17	0.2	733
   18	0.1	772
   19	0.0	810
   20	0.1	823
   21	0.1	1037

Investment advisors estimated the stock market returns for four market segments: computers, financial, manufacturing, and pharmaceuticals. Annual return projections vary depending on whether the general economic conditions are improving, stable, or declining. The anticipated annual return percentages for each market segment under each economic condition are as follows:

a. Assume that an individual investor wants to select one market segment for a new investment. A forecast shows stable to declining economic conditions with the following probabilities: improving (0.2), stable (0.5), and declining (0.3). What is the preferred market segment for the investor, and what is the expected return percentage?

b. At a later date, a revised forecast shows a potential for an improvement in economic conditions. New probabilities are as follows: improving (0.4), stable (0.4), and declining (0.2). What is the preferred market segment for the investor based on these new probabilities? What is the expected return percentage?

A computer company manufactures 3 types of computers which are personal computers, workbooks and gamebooks. Each personal computer is produced in 0.2 hours, assembled in 0.1 hours, and inspected in 0.05 hours. Each workbook is produced in 0.2 hours, assembled in 0.3 hours, and inspected in 0.075 hours. Each gamebook is produced in 0.4 hours, assembled in 0.4 hours, and inspected in 0.15 hours. At most 600 hours of production, 480 hours of assembly and 160 hours of inspection can be done. The company profits 60$ from a personal computer, 90$ from a workbook, 100$ from a gamebook. Besides, due to the lack of demand, at least 60 personal computer should be ready in the inventory. Formulate an LP that can be used to maximize the profit of computer company. Explain each decision variable, objective function, and constraint with at least one sentence. Write down the standard form and add artificial variables if needed. Find a basic feasible solution to start simplex. Solve the LP by using simplex method step by step. Write down the optimal solution.

1. Harley's employer gave him $5,000 to cover expenses for an upcoming sales trip. He is not required to provide any receipts for his expenses, and if his actual expenses are less than $5,000, he does not have to return any money to his employer. His actual expenses for the trip were as follows:

Airfare and rental car $2,000

Meals       $1,500

Hotel $1,000

Tickets to theater for a client and him $200

Total $4,700

What the income tax consequences to Harley and his employer?

2. Which of the following is a deductible business expense?

3. Which of the following business expenses would not be deductible because it is not ordinary, necessary, or reasonable?

1. You are given the following payoff table with profits (in $).

Assume the following probability information is given, where I1 and I2 are the outcomes of the sample information available:

1. [2] Find the values of P(I1) and P(I2).

2. [2] Determine the values of P(s1 | I1), P(s2 | I1), P(s1 | I2), and P(s2 | I2).

[4] Determine the optimal strategy based on the sample information I1 and I2. What is the expected value of your solution?

Suppose that a set of three traffic lights along one section of road operate independently (i.e., no communication or special timing between the lights). Since this is a fairly main road, the lights are green with a probability of 0.7 and red with probability 0.3. As you go through this stretch of road, What's the probability that :

At least one of the lights is red?

Males and females are observed to react differently to a given set of circumstances. It has been observed that 70% of the females react positively to these circumstances, whereas only 40% of males react positively. A group of 20 people, 15 female and 5 male, was subjected to these circumstances, and the subjects were asked to describe their reactions on a written questionaire. A response picked at random from the 20 was negative. what is the probability that it was that of a male?(using bayes' rule)

An ideal gas is contained in a piston-cylinder device and undergoes a power cycle as follows: 1-2 isentropic compression from an initial temperature T1 5 208C with a compression ratio r 5 5 2-3 constant pressure heat addition 3-1 constant volume heat rejection The gas has constant specific heats with cv 5 0.7 kJ/kg·K and R 5 0.3 kJ/kg·K. (a) Sketch the P-v and T-s diagrams for the cycle. (b) Determine the heat and work interactions for each pro- cess, in kJ/kg. (c) Determine the cycle thermal efficiency. (d) Obtain the expression for the cycle thermal efficiency as a function of the compression ratio r and ratio of specific heats k.

Probability

Let A, B and C be Boolean variables denoting three independent events with P(A=1) = 0.7, P(B=1) = 0.3, and P(C=1) = 0.1. Let D be the event that at least one of A and B occurs, i.e., D = A OR B. Let E be the event that at least one of B and C occurs, i.e., E = B OR C. Let F be the event that exactly one of A and B occurs, i.e., F = A XOR B.

(a) (4 pts) Express F as a set of clauses (i.e., as a conjunction of disjunctions) involving A and B.

(b) (9 pts) Suppose that A and E are true. What is the probability that B is true? Show your work.

(c) (9 pts) Suppose that D and E are true. What is the probability that A is true? Show your work.

A person tried by a 3-judge panel is declared guilty if at least 2 judges cast votes of guilty. Suppose that when the defendant is in fact guilty, each judge will independently vote guilty with probability 0.7, whereas when the defendant is in fact innocent, this probability drops to 0.2. If 70 percent of defendants are guilty, compute the conditional probability that judge number 3 votes guilty given that

(a) judges 1 and 2 vote guilty;

(b) judges 1 and 2 cast 1 guilty and 1 not guilty vote;

(c) judges 1 and 2 both cast not guilty votes.

Let Ei , i = 1, 2, 3 denote the event that judge i casts a guilty vote. Are these events independent? Are they conditionally independent? Explain.

Central Adventures

Fatima Hopkins, the CEO of Central Adventures, is having difficulties with all three of her top management level employees. With one manager making questionable decisions, another threatening to leave, and the third likely ‘in the red’, Fatima is hoping there is a simple answer to all her difficulties, and needs some advice from her accountant on how to proceed.

Central Adventures owns and operates three amusement parks in Michigan: Central Funland, Central Waterworld, and Central Treetops. Central Adventures has a decentralized organizational structure, where each park is run as an investment center. Each park manager meets with the CEO at least once annually to review their performance, as measured by their park’s ROI. The park manager then receives a bonus equal to 10% of their base salary for every ROI percentage point above the required rate.

Central Funland is an outdoor theme park, with twelve roller coaster rides and several other attractions. This park has first opened 1965, and most of the rides have been in operation for 20+ years. Attendance at this park has been relatively stable over the past ten years. The park manager of Funland, Janet Lieberman, recently shared with Fatima a proposal to replace one of their older rides with a new roller coaster, a hybrid steel and wood rollercoaster with a 90 degree, 200 foot drop and three inversions. The proposal indicated that the ride would cost $8,000,000 with an estimated life of 20 years. In addition, this new style of coaster would require additional maintenance, costing $125,000 each year. However, it projected that this new attraction would boost attendance, earning the park an additional $1,190,000 per year in revenues. Janet ultimately decided not to invest in this new attraction.

Central Waterworld is an indoor water park, operating year-round. Run by park manager David Copperfield, Waterworld was built in 2016 and has increased attendance by 20% every year since. David recently sent you an email complaining that, based on the current bonus payout schedule, Janet Lieberman’s bonus last year was significantly higher than his. He points to the increasing attendance, and says that his park is being punished for having opened so recently (his park assets are much more recent than the roller coasters at Funland). He currently has an employment offer from another company at the same pay rate, which he says he will accept if his performance is not appropriately acknowledged.

Central Treetops includes a high ropes course and has a series of ziplines that criss-cross over the Chippewa River. For many years, it was a popular venue for corporate team-building activities, so it is equipped with a main indoor facility with cafeteria and overnight guest rooms. This park has lost popularity in recent years, and has been ‘in the red’ for the past two years. If the park is not profitable this year, you will need to decide whether to close it - permanently. Central Adventures has a $86,000 mortgage payment on the land and buildings for Treetops, which would still need to be paid if the park is closed. Incidentally, you recently had a conversation with the regional head of the YMCA, who would like to open a summer camp in the central Michigan region. If you decided to close Treetops, you are fairly certain that you could lease that land to the YMCA for $250,000 annually.

A partial report of this year’s financial results for Central Adventures shows the following:

In addition to the information above, there are $2,542,920 in corporate costs, which are currently allocated evenly between the three parks. These costs are primarily due to employee benefits costs, which are billed at the corporate level. If the Treetops park is closed, the allocated corporate costs would decrease by $12,000. Central Adventures has a required rate of return of 12 percent (set at the company’s weighted-average cost of capital) and are subject to 18% income taxes.

Fatima needs to see this year’s performance results before she can make any decisions. Is David’s complaint about the performance evaluation metrics valid? Is that also affecting management decisions in the form of Janet’s rejection of the proposed new rollercoaster? And is the company better off without Treetops? She sets off to the company accountant’s office to help get some answers.

a. Create a multilevel income statement for Central Adventures.

b.   Calculate the current annual ROI, residual income and EVA for the three parks.

c. Did Janet Lieberman (the Funland park manager) make the ‘right’ decision (i.e., was it in Central Adventure’s overall best interest for Funland to reject the new rollercoaster)? Explain your answer. Provide the appropriate financial analysis(es) to support your conclusion.

d.   Is David Copperfield’s (the Waterworld park manager) complaint valid? Or would a different performance metric tell the same story?

e. Provide a recommendation on whether to close Treetops. Provide the appropriate financial analysis to support your conclusion.

f.    Provide a recommendation on a different allocation base for corporate overhead.