Midterm Exam Business Statistics Fall 2001 Russell

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1 Name Midterm Exam Business Statistics Fall 001 Russell Do not turn over this page until you are told to do so. You will have 1 hour and 0 minutes to complete the exam. There are a total of 100 points divided into three parts. The true and false questions are worth 10 points, the multiple choice are 3 points each for a total of 30 points and the long answer questions are worth 60 points. You can use one side of an 8.5x11 sheet of notes during the exam. No other notes are permitted. Please write clearly and provide answers in the space provided. If you need additional space use the back of the exam pages and clearly organie your work. Students in my class are required to adhere to the standards of conduct in the GSB Honor Code and the GSB Standards of Scholarship. The GSB Honor Code also requires students to sign the following GSB Honor pledge, "I pledge my honor that I have not violated the Honor Code during this examination." Please sign here to acknowledge

2 I. True or False Clearly indicate the best answer by circling T or F indicating that the statement is true or false respectively. If neither T nor F is clearly indicated the problem will be marked as incorrect. Each problem is worth 1 point. 1. T F The covariance must always be larger than the correlation.. T F If X and Y are any two random variables then Var(X+Y)=Var(X-Y). 3. T F The central limit theorem says that sums of a large number iid random variables are approximately normally distributed. 4. T F If X is a random variable denoting your score on this exam and Y is a random variable denoting your score on the final exam then if the covariance between X and Y is.1 then X and Y must be dependent. 5. T F X and Y are as in problem 4 above. If we also assume that X and Y are identically distributed with mean 85 and variance 5 then the variance of the average of the midterm and final score for a student is T F For any bivariate discrete independent random variables the joint probabilities can be recovered from the marginal probabilities. 7. T F Converting a bivariate pair of random variables from dollars to thousands of dollars has no impact on the correlation between them. 8. T F If the probability that the San Diego Chargers win any given home game is.8 and independent of their other games, the probability that they loose all 4 of their next home games is T F The empirical rule says the for mound shaped data, roughly 95% of the data should lie within standard deviations of the mean. 10. T F If the probability that a student gets this question correct is.8 and the probability that the student gets this question correct and scores above a 85 overall on the exam is.7 then the probability a student scores above 85 on the exam given that the student got this question correct is 87.5%.

3 II. Multiple choice: Clearly circle the answer that is best. Each problem is worth 3 points for a total of 30. No partial credit will be given in this section. If no answer is clearly circled the problem will be marked as incorrect. 1. Let X be Binomial(5,.5) and Y be Binomial(10,.8) and X and Y are independent then if Z=X+Y has mean and variance given by a. µ =10.5 and b. µ =15 and c. µ =15 and d. µ =10.5 and σ =7.65 σ =.89 σ =7.65 σ =.89. Suppose that it costs $100 per day to open and run a sandwhich shop. Further suppose that it costs $1.0 to make each sandwhich and they sell for $5.50. If the total demand for sandwhiches on a given day is N ( µ,σ ) the shop to break even? a. 79 b. 18. c. 75 d. 15, what does µ have to be for 3. If weekly peanut butter sales at a grocery store are iid and positively skewed (ie a long right tail) then the distribution of annual peanut butter sales will be a. Also positively skewed. b. May still be a little skewed but should look approximately symmetric c. Will be exactly Normally distributed d. Not enough information is given. 4. If I play a slot machine 50 times and the probability that I win on any given attempt is.01, what is the probability that I don t win a single attempt? a..5 b..75 c..605 d.. 3

4 5. If the annual return on an asset is given by a N(.,.1 ) then the probability that the return is negative is a..5 b..5 c..05 d Suppose the mean and standard deviation of asset X is.1 and.6 respectively and that the mean and standard deviation of asset Y is.18 and.31 respectively. The covariance between the two assets is -.. What is the mean and variance of a portfolio that invests 80% of wealth in asset X and 0% in asset Y? a..116 and.1606 respectively b..116 and.1736 respectively c and.1148 respectively d and.1148 respectively. Consider this scatter plot for X1 and X in the following questions. 30 X X Which of the following is correct: a. X1 has a higher variance and higher mean than X. b. X has a higher variance and higher mean than X1. c. X1 has a higher variance, but X has a higher mean. d. X has a higher variance, but X1 has a higher mean. e. Need more information. 4

5 8. Which of the following is correct: a. The correlation between X1 and X is about.7 and the covariance is about 5. b. The correlation between X1 and X is about.5 and the covariance is about 10. c. The correlation between X1 and X is about.7 and the covariance is about 6. d. The correlation between X1 and X is about.9 and the covariance is about 5. e. The correlation between X1 and X is about.9 and the covariance is about Suppose that X is a Binomial(n,p). Then the variance of n X is: a. np( 1 p) b. p( 1 p) p ( 1 p) c. n p(1 p) d. n 10. A sample of sie 10 taken from a normally distributed population with variance 4 yields a sample average of A 95% confidence interval for µ is given by: a. 14.5, b. 18.5, c. 19, 135 d ,

6 III. Long answer questions. Try to do work in the space provided under each question and be show all work in order to facilitate partial credit. Be sure to clearly indicate your final answer. 1. (5 points) Consider the following joint probability table. X 0 1 X ?.3 a. What is the probability that X1=X? b. What is the probability that X1=1 and X=0? c. What is the distribution of X1? d. What is the distribution of X? e. What is the mean and variance of X1? f. What is the mean and variance of X? g. What is the covariance between X1 and X? h. What is the mean and variance of X1-X? i. What is the distribution of X1 given X=1? j. Are X1 and X independent? (be sure to state why or why not). 6

7 . (0 points) Suppose that annual returns for an asset are iid N(.15,. ). a. What is the probability that the return next year will be positive? b. What is the probability that the return next year will be between.15 and.5? c. What is the probability that the return will be positive in each of the next 5 years? d. Conditional on the return being positive this year, what is the probability that it will also be positive next year? Cumulative Distribution Function Normal with mean = 0 and standard deviation = P( Z <= )

8 3. (15 points) Let P0 be some number. Let Di be iid N(0, ) For i>=1 let Pi = Pi-1+Di. a. Express P in terms of P0 and D values. b. Express Pn in terms of P0 and D values. c. Recall that the sum of independent normally distributed random variables is also normally distributed. Express Pn as a function of P0 and a Normally distributed N µ σ find µ and random variable Xn. What is the distribution of Xn? Ie X n ~ ( ) σ n., n 8

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Business Statistics Midterm Exam Fall 5 Russell Name Do not turn over this page until you are told to do so. You will have hour and 3 minutes to complete the exam. There are a total of points divided into