small medium large standard $20M $30M $50M horizontal -$20M $40M $90M (a) (5 pts) Find the mean payoffs of the two different drilling strategies.

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1 Business Statistics Homework #1 1. An oil company wants to drill in a new location. A preliminary geological study suggests that there is a 20% chance of finding a small amount of oil, a 50% chance of a moderate amount and a 30% chance of a large amount of oil. The company has a choice of either a standard drill that simply burrows deep into the earth or a more sophisticated drill that is capable of horizontal drilling and can therefore extract more but is far more expensive. The following table provides the payoff table in millions of dollars under different states of the world and drilling conditions small medium large standard $20M $30M $50M horizontal -$20M $40M $90M (a) (5 pts) Find the mean payoffs of the two different drilling strategies. (b) (5 pts) Find the variance in payoffs of each strategy. (c) (2 pts) Which strategy would you advocate for and why? (d) (3 pts) How much are you willing to pay for a geological evaluation that would determine with certainty the quantity of oil at the site prior to drilling? 1

2 2. Answer the following statements TRUE or FALSE, providing a succinct explanation of your reasoning. (2 pts each). (a) The median is the value that minimizes squared error prediction loss. (b) If P(A and B) 0.4 then P (A) (c) If X is normally distributed with mean 2 and standard deviation 3, then P( 2.5 X 6.5) is a number between 95% and 99.7%. (d) If the odds in favor of A are 3:5 then P(A) =

3 3. A cosmetic company is going global and is deciding between two cities for their international launch: one in India or one in Brazil. To guide the decision they are considering two fundamental quantities, the profit-per-unit and your captured-market size (in millions of units). In both countries the major determinant of market size will be dictated by whether or not the dominant local chain agrees to carry your product and the profits-per-unit will depend on if you can place the factory in a low rent versus high rent district. On the basis of these two factors you come up with the following scenarios and associated probabilities: 12M 40M 10M 30M INDIA $2 35% 5% BRAZIL $1.50 5% 80% $ % 50% $1 10% 5% (a) (5 pts) Calculate the expected market size for the each city. (b) (5 pts) Calculate the variance of the market size for each city. (c) (5 pts) Calculate the expected profit-per-unit for the each city. (d) (5 pts) Calculate the variance of the profit-per-unit for each city. (e) (5 pts) Compute the expected total profit for each city. (f) (5 pts) Compute the variance of the total profit for each city. 3

4 (g) (2 pts) Which city do you prefer for the product launch and why? (h) (7 pts) Now suppose that, prior to the launch, you find out that your parent company refuses to do business with chain groceries, limiting your market in either country to the smaller size. Compute the expected total profit for each city under this scenario. Which option you prefer now? 4. The following table shows survey results relating household income and level of education of the head-of-household. The cells of this table may be interpreted as the probability that a randomly selected household satisfies the criteria denoted by the cell s position in the table. Under $50K $50-75K Over $75K Not HS graduate HS graduate Some College College degree (a) What is the probability of the head-of-household having some college? (b) If the head-of-household has a bachelor s degree or more, what is the probability that his/her income is not over $75K? (c) Knowing that a household income is between $50K and $75K, what is the probability that the household head has a bachelor s degree or more? (d) Are education level and household income independent? 5. Imagine rolling two fair 4-sided dice. What is the probability that the two dice show the same number? 6. Cooper Realty is a small real estate company located in Albany, New York, specializing primarily in residential listings. They have recently become interested in determining the likelihood of one of their listings being sold within a certain number of days. Based 4

5 on historical data, they produced the following figures based on the past 800 homes sold. Days Listed until Sold Under Over 90 Total Under $50K $50-$100K $100 - $150K Over $150K (a) What is the probability that a randomly selected home is listed over 90 days before being sold? (b) What is the probability that a randomly selected initial asking price is under $50K? (c) What is the probability of both the previous two events happening? Are these two events independent? (d) Assuming that a contract has just been signed to list a home that has an initial asking price less than $100K, what is the probability that the home will take Cooper Realty more than 90 days to sell? 7. For each i = 1,..., 4 let X i be a normal random variable with mean µ = 3 and variance σ 2 = 4. For j = 1,..., 6 let Y j be binomial random variables with parameters p = 0.6 and n = 5. Define the random variable C to be the sum of the X i s minus the sum of the Y j s. (a) Use summation notation to express the random variable C. (b) Compute E(C), the expectation of C. (c) Compute V(C), the variance of C. (d) Compute the standard deviation of C. 8. One hundred people were asked Do you smoke? Their responses are summarized in the following table. (a) To interpret these numbers as probabilities what must we do so that they normalize correctly? (b) What is the probability that a randomly selected person from this group is a smoker? 5

6 Yes No Total Male Female Total (c) What is the probability that a randomly selected smoker from this group is a female? (d) Write down the conditional distribution (in the form of a table) of the variable X which denotes (smoker) or (non-smoker), given Y, which denotes (male) or (female). 9. At a certain private college students concentrate in one of three areas: pre-med, pre-law, or theater-arts. Only 15% of the students are theater-arts majors, with the remaining population divided equally among pre-med and pre-law. The most popular intramural sport at this college is ultimate frisbee, but participation differs between concentrations. Only 10% of pre-med students play on a frisbee squad, while 20% of pre-law students do, and 80% of theater-arts students do. (a) Suppose that your new summer intern shows up wearing an ultimate frisbee t- shirt with the school logo on it (which you take as certain evidence that she played ultimate frisbee there in college). What is the probability that this student was a theater-arts major? (b) Suppose we know that pre-med students go on to have a loan-adjusted average post-graduation salary of $60K, pre-law students go on to have a loan-adjusted average post-graduation salary of $80K and theater-arts students have a loanadjusted average post-graduate salary of -$5K. Given that we think the student did play ultimate frisbee in college, what is the student s expected salary? 6