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Penn STAT 101 - STAT 101 EXAM

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Statistics 101 Exam II Fall 2004 November 22, 2004The exam is open book and notes. Calculators are permitted.Explanations or computations must accompany answers to receive credit.1. The human resource department for a large corporation interviews prospective candidates and rates them on the following scale: 1 – “not- hirable” 2- “hirable” 3- “blue-chip candidate.” It is determined that the probability for ratings 1, 2, and 3 are respectively .15, .50 and .35. Let X be the rating of a randomly chosen candidate. Note: You may use the facts that the expectation for the rating of each rater is 2.2 with variance of .46 without verification. In fact, each candidate is rated by two individuals. The joint probability distribution is indicated below: y=Rater 2 1 2 3 ----------------------------------- 1 .10 .05 0x=Rater 1 2 .05 .30 .15 3 0 .15 .20A. (7) What is the correlation between the two ratings? Interpret the answer in the context of this problem.B. (8) What are the mean and standard deviation for the average of the two ratings? Would each of the mean and standard deviation be lower or higher if the ratings were independent? Explain.C. (7) If the two raters give the same rating (i.e., they either both give a rating of 1 or both give a rating of 2 or both a rating of 3), what is the expected value of the rating?2. The likelihood rating that a person purchases a new video game (on a scale of 0- definitely will not, to 10 - surely will) is modeled to have the following density function: f(x) = 3x2/ 1000 if 0 <x < 10 0 otherwiseA. (6) It is determined that a rating of 9 or above is needed to classify a person as aprobable buyer. What proportion of people will be probable buyers?B. (9) Suppose 100 people are sampled. i) Write an expression for the probability that 20 or more will be probable buyers. Do not evaluate this expression. ii) Find an approximate probability for your answer to Bi). C. (9) We now consider a wider range of f(x) of the form: f(x) =constant xk if 0 < x < 10 0 otherwise i) Verify that the constant is (k+1)/10k+1 (e.g., if k=2 then the constant is 3/1000). ii) If we know that the mean rating is 8, what must be the value of k?3. A venture capital company (VCC) can invest in a venture on round 1. The venture will either be successful or go bankrupt. For an investor in round 1, the profit if the venture is successful is 14 (million dollars), and the loss if the venture goes bankrupt is 1 (million dollars). The probability that a venture is successful is only .2 and the probability that the venture goes bankrupt is .8A. (5) Should VCC invest in the venture? Choose any suitable criterion.B. (7) VCC has the opportunity of investing in 25 such ventures. Each venture behaves according to the above description. Assume that the outcomes of these 25 ventures are independent.What is the approximate probability that the total profit that will be made from these 25 ventures is negative? C. (10) Assume again that VCC is going to invest in only one such venture. It has the opportunity to wait until the second round. By then the outcome of the venture will be better (although not perfectly) determined. If the venture is going to be successful, it always reaches the second round. If the venture is going to go bankrupt, then there is onlya 50% chance that it reaches the second round. i) If a venture reaches the second round what is the probability that the venture will be successful? ii) There is still a loss of 1 million if the venture goes bankrupt. But since the company is entering on a later round, it only makes 5 million if the venture is successful. Based on expected profit computations, is it better to buy on round 1 or round 2? Show your work. 4. University employees may invest in the retirement plan TIAA/CREF. CREF is a stock fund and TIAA is a fixed assets fund. CREF has a mean return of 8% and standard deviation of 20%. TIAA has a mean return of 4% and essentially no risk (assume its standard deviation is zero).An employee is allowed to specify what fraction P she wants to put in CREF, and the remaining 1-P goes to TIAA. For example, P=.5 corresponds to splitting your retirement equally between TIAA and CREF. Assume that the performances of TIAA and CREF are independent.A. (9) i) For what values of P would the average return be at least 5%? ii) Choose one such arrangement (i.e., one value of P) from Ai) and find the standard deviation for the return of the plan that invests P in CREF and 1-P in TIAA.B. (9) The university allows each employee to rebalance (move money from either TIAA into CREF or vice versa each fiscal year). By rebalancing, the employee can achieve the long run return that was illustrated in class with the dice simulation. Be careful. The means and standard deviations must now be phrased as fractions: the respective means for CREF and TIAA are .08 and .04; the respective standard deviations are .20 and 0. i) If an employee puts half of her money in each fund, what would be the average long run return? ii) Write an expression for the average long run return in terms of P and find the value of P that maximizes the average long run return. 5. Strategic decisions in companies are generally made by the top three officers of the firm: CEO (chief executive officer); CFO (chief financial officer) and COO(chief operation officer). Assume that each of these officers has a probability of .3 of recommending a strategic decision and their recommendations are independent. There are two kinds of firms.(HI) Horizontally Integrated: In the horizontally integrated firm, all three officers vote


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