The Normal Distribution (20 points). There are 6,000 households in the town of Rockbottom, MA. The annual household incomes are normally distributed with a mean of $70,000 and a standard deviation of $20,000.
Approximately how many households in Rockbottom have annual incomes greater than $130,000 ? Approximately how many households in Rockbottom have annual incomes less than $30,000 ?
Probability Calculations (10 points). Given: P(A)=0.58,P(B│A)0.35,and P(B│A ̅ )=0.05. Find: P(A ̅ ),P(A and B),P(A ̅and B),P(B ̅│A),P(B ̅│A ̅ ),P(B),P(B ̅ ),P(A│B),P(A ̅│B), P(A│B ̅ ),and P(A ̅│B ̅ ).
The Binomial Distribution (10 points). In the Hard-Knox High School (2500 students), 18% of the students smoke cigarettes. A) If 2 are selected at random, use a Venn diagram with 2 circles; 1 representing the probability that the first student smokes and 1 representing the probability that the other student smokes. Determine the probability that at least 1 of them smokes cigarettes. B) Repeat the above analysis when 3 students are selected at random. Note: These trials would be independent given the large population of students. Decision Trees (40 points). Assume that you have a client that is a paper manufacturer and they have expressed concern that the government will pass a new regulation banning the use of chlorine-based technologies within their manufacturing process. If the firm were certain that the regulation would pass they would take proactive remedial action now (at a cost of $500K) and introduce a newer, compliant technology. They don’t want to spend the $500K if they don’t have to. If they don’t act now, believing that the government will not change the regulation, and end up being wrong then they will find themselves changing their manufacturing process from a reactive position and the cost for getting in compliance will be $850K, much greater than the $500K. Currently, the company’s best guess that the government will pass the new regulation is 20%.
As a consultant in this area your first instinct is to see if you can get more information on the likelihood of the government moving forward and making a change. Some research informs you that there are two firms available with expertise in forecasting regulatory policy as it relates to the papermaking industry. Firm A charges $80K for a study and their record is perfect. In the past, whenever Firm A has predicted that the government would introduce a more stringent regulation the prediction always came true and for those times that Firm A predicted that no new regulation would occur, none has occurred. If Firm A were to predict a change then your client would immediately undertake proactive remedial action and if Firm A predicted no change they would maintain the current manufacturing process.
Another firm, Firm B, has a much lower fee, charging only $20K but their predictive performance has not been perfect. In the past, Firm B has predicted government policy for passing new regulations with a ninety percent accuracy (i.e. for those times the government introduced a tougher regulation then Firm B has predicted it about 90% of the time and missed it about 10% of the time). For those times when there was concern over the government changing the regulation but ended up not doing so, Firm B had correctly predicted no change 80% of the time.
Your client has stated that if Firm B is hired and if B predicts that the government will introduce a new regulation then before taking action they would hire Firm A to do an assessment before proceeding, just to be absolutely sure. If Firm B were to predict no change then you would advise the client to take their chances and take no action.
The client wants you to make some recommendations. They are unsure whether they should do nothing, re-mediate now, or hire Firm A or B to get better information. They need your advice. Please use decision analysis tools discussed in class to determine what action the firm should pursue in order to minimize expected loss; either manual analysis or software is acceptable. Please also calculate the minimum expected loss. Tornado Diagrams (20 points). Your best friends from high school have decided to start the “Spic & Span Housecleaning Business.” They are trying to calculate the sensitivity of their projected profits to various parameters. They have determined that their Profit = Revenues – Costs. They have also determined:
Monthly Revenue = (Price per job) x (Number of jobs per month) Number of Jobs per month = (Number of Clients) x (Number of jobs per month per client) Number of Clients = (0.01 x Promotion Costs) + 10 Monthly Costs = Monthly Fixed Costs + Promotion Costs + Monthly Variable Costs Monthly Variable Costs = $5 x (Number of Jobs per month)
They have constructed the following table of their anticipated parameter values, along with maxima and minima:
Parameter Probable Maximum Minimum Price per Job $70 $150 $35 Promotion Costs $1000 $5000 0 Number of jobs per month per client 2 5 1 Monthly Fixed Costs $500 $1000 $400
First develop an equation that represents the monthly profit as a function of the 4 parameters listed in the table. Then, construct a Tornado diagram for this situation that demonstrates the most (and least) sensitive parameters.
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