2006-Fall-Midterm

2006-Fall-Midterm - More PastPaper:...

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Unformatted text preview: More PastPaper: http://ihome.ust.hk/~cs_gxx ISMT111 Business Statistics Midterm Examination For sections 3 & 4 only 23rd October 2006 Directions 1) Answer ALL FIVE questions. Marks are shown in square brackets. 2) There are 4 pages in this examination paper, which includes a normal table. Check to make sure you have a complete set and notify the invigilator immediately if part of it is missing. 3) Key formulas are provided separately. 4) Calculator may be used in this examination. ht tp :// ih om e. us t.h k/ ~c s_ gx x/ 5) You are given TWO HOURS to complete this examination. Do not begin until you are told to do so. More PastPaper: http://ihome.ust.hk/~cs_gxx Question 1: [15 Marks] Question 2: (a) Why probability can explain the variation in data sets? [18 Marks] ht tp :// ih om e. us t.h k/ ~c s_ (b) Given that the S&P 500 finished higher after the first five days of trading, what is the conditional probability that it finished higher for the year? (c) Are the two events first week performance and annual performance independent? (d) A financial analyst proposed another method to forecast the annual performance of S&P 500. The method, when applying to the data from 1950 to 2003, gives correct forecasts 28 of 39 years S&P 500 ended up higher, while for those years S&P 500 ended up lower the method is correct 13 of 15 years. For 2007, the method forecasts that the S&P 500 will end up higher, what is the probability that the forecast is correct? gx x/ The following table gives the first week and annual performance of the stock market index S&P 500 from 1950 to 2003. In 34 of the 54 years from 1950 to 2003, the S&P 500 finished higher after the first 5 days of trading (first week). In 29 of those 34 years S&P 500 finished higher for the year. Answer the following questions to see if a good first week is a good forecast for the upcoming year. S&P 500 annual performance First week Higher Lower Higher 29 5 Lower 10 10 More PastPaper: http://ihome.ust.hk/~cs_gxx Question 3: [20 Marks] Suppose the skill level X of students in a large statistics course is normally distributed. A student can answer a question correctly in an examination if he (or she) has skill level higher than that required by the question. (a) Suppose X has mean 1.2 and standard deviation (s.d.) 0.3. For a question which requires a skill level of 1.6, what is the proportion of students answer it correctly? (b) What is the range of skill levels for the middle 50% of students? (c) Suppose we do not know the mean and s.d. of skill levels but know that 33% of students answer correctly a question which requires a skill level of 1.8 and 67% of students answer correctly a question which requires a skill level of 1.2. Please find the mean and s.d. of skill levels for students. (d) If we only know that the distribution of X is symmetric and it is know that 50% of the students answer correctly a question which requires skill level 1.6 and 25% of the students answer correctly a question of skill level 2.3. Please find the inter-quartile range of X. Question 4: [18 Marks] The president of a company has an advisory committee of 7 members. From the past experience, each committee member voted for, against, and abstain on a proposal with probability 0.3, 0.6 and 0.1, respectively. A positive recommendation from the committee needs at least 4 members vote for the proposal. (a) What is the probability that there are exactly four members vote for a proposal? (b) If one member, Albert, decides to vote for a proposal, from his standpoint, what is the probability that the committee will be positive about the proposal? Question 5: [22 Marks] Suppose you are a consultant providing professional service to a particular industry. A major firm is interested in your service. There are three options for calculating your fee based on the performance rating. From past experience, you estimated the following probability distribution. Probability 1 2 3 4 5 0.1 0.3 0.3 0.2 0.1 ht tp :// ih om e. us t.h k/ ~c s_ (a) For option 1, the fee equals to the performance rating multiply by $120,000. Please calculate the expectation and variance of your fee. (b) Under option 2, you will be paid $300,000 in the beginning of the contract period. If your performance rating is higher than 3, you will receive a bonus of $100,000 at the end of the contract. Please calculate the expectation and variance under option 2. (c) According to option 3, you will be paid a fee of $750,000 up front, but if your performance rating is lower than 5, for each point lower your client will get a rebate of $200,000 from you. Please calculate the expectation and variance according to option 3. (d) If you choose the option with highest expectation, which one should you take? In what sense it is the optimal strategy? gx x/ Performance Rating ...
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This note was uploaded on 02/27/2010 for the course ISOM ISOM111 taught by Professor Anthonychan during the Spring '09 term at HKUST.

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