Business Statistics Midterm Exam Fall 2015 Russell. Please sign here to acknowledge

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1 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 three parts. The true and false questions are worth points, the multiple choice are points each for a total of points and the long answer questions are worth 7 points. You can use one side of an 8.5x sheet of notes during the exam. No other notes are permitted. You may use a calculator. Please write clearly and provide answers in the space provided. If you need additional space use the back of the exam pages and clearly organize your work. Students in my class are required to adhere to the standards of conduct in the Booth Honor Code and the Booth Standards of Scholarship. The Booth Honor Code also requires students to sign the following Booth Honor pledge, "I pledge my honor that I have not violated the Honor Code during this examination. I also understand that discussing the contents of this exam before all students have completed the exam would be a violation of the Honor Code". 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 point for a total of points.. T F Probability is a statement regarding the long run frequency of occurrence of an event.. T F The sample covariance can be made arbitrarily large or small by simply changing the units that we use to measure the data. 3. T F The empirical rule says that it is always the case that about 95% of the data should lie within standard deviations of the mean. 4. T F The random walk model for prices is an example of an IID model. 5. T F The Binomial distribution is defined as the sum of n IID Bernoulli(p) variables. 6. T F If you play slot machines and whether or not you win is an iid Bernoulli(.), the probability that you don t get any winners is T F Standardizing values from a data set is only useful if we think the data are Normally distributed. 8. T F For a random variable Y with cumulative distribution function (cdf) F(y) and probability density function (pdf) f(y), F(y) is the same as the area under f(y) to the left of the value y. 9. T F The median is less sensitive to outliers (extreme values) than the mean.. T F If you guessed the answer to all of these questions (including this one), we could model the number of correct answers as a Binomial(,.5).

3 II. Multiple choice: Clearly circle the answer that is best. Each problem is worth points for a total of. No partial credit will be given in this section. If no answer is clearly circled the problem will be marked as incorrect. The heights of a group of students is Normally distributed with a mean of 68 inches and standard deviation of 4. Below is the Cumulative Distribution Function (CDF) function F(h) for this distribution. Use the CDF to answer the next four problems. Height (h) F(h) What is the chance that a randomly selected student is shorter than 6 inches? a..5 b..67 c..4 d..3 e. None of the above.. What is the chance that a random student is somewhere between 65 inches and 7 inches tall? a..5 b..493 c..547 d..5 e. None of the above.

4 3. What is the chance that a randomly selected student is taller than 7 inches? a..69 b..5 c..7 d..39 e. None of the above. 4. About 4% of the students are taller than a. 75 inches b. 6 inches c. 68 inches d. 76 inches e. None of the above. 5. One hundred students took a test on which the mean score was 73 with a variance of 64. A grade of A was given to all who scored 85 or better. Approximately, how many A s were there, assuming scores were normally distributed? (Choose the closest number.) a. 4 b. 7 c. 58 d. 3 e. 6. A firm s employees were surveyed to determine their feelings toward a new dental plan and a new life insurance plan. The results showed that 8% favored the insurance plan, while only 35% favored the dental plan. Of those who favored the insurance plan, 3% also favored the dental plan. What percentage of the employees favored both plans? a. 5% b. 8.35% c..5% d. 4.3% e. None of the above

5 The histograms below are labeled X, X, X3, and X4. Match the histogram to the Normal distributions below. Histogram of x Histogram of x 5 5 Frequency Frequency x 3 - x 3 Histogram of x3 Histogram of x Frequency 8 Frequency x3 4 - x N(,) 8. N(,) 9. N(,). N(,)

6 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 and complete all computations. This section is worth 7 points.. ( points) Consider the sample variance covariance matrix for Apple and HP daily returns data. The sample average returns are. and.7 for Apple and HP respectively. Apple HP Apple.6.6 HP.6.5 a. What is the correlation between Apple and HP? b. What is the expected return on a portfolio that invests.8 in Apple and. in HP? c. What is the variance of a portfolio that invests.8 in Apple and. in HP? d. What is the variance of a portfolio that invests. in Apple and. in HP?

7 . (6 points) Consider the following series of 3 observations: 3 Time Series Plot of Y Y Index Histogram of Y Normal Mean StDev 3.53 N 3 5 Frequency Y a. Does the series look IID? Explain why or why not.

8 b. The blue curve is a best fit Normal distribution. The histogram looks a little like a Normal distribution, but there are signs that it is not Normal. If you used the best fit Normal as a model for Y, what kinds of mistakes would you make? Explain. 3. (6 points) Consider the following plot of 3 observations: 6 P a. Does this series look IID? Explain why or why not.

9 This plot is of the differenced series D t =P t P t : b. Does this plot look IID? Explain. D c. What does that tell you about the model for P?

10 This is the histogram of D. There are 3 observations. Histogram of D 8 Frequency D d. The last value in the sample is 3, find a good model for P 3 given P 3 =3.

11 4. (36 points) Consider the following joint probability distribution for a trade direction indicator variable. Let X i denote whether i th trade is a buy or a sell. X i = denotes a buy and X i = denotes a sell. Let X i+ denote a buy sell indicator for the i+ subsequent trade. Consider the following joint model for X i and X i+ : X i X i a. What is the mean of X i? b. What is the variance of X i? c. Are consecutive trade directions (buy/sell) dependent or independent? d. Explain why X i and X i+ are identically distributed and state the distribution (model).

12 e. What is the probability of a buy? f. What is the probability of two buys in a row? g. Given that trade i is a buy, what is the probability that trade i+ is also a buy? h. Given that trade i is a buy, what is the probability that trade i+ is a sell? i. What is the conditional distribution of trade i+ given trade i is a buy?

13 j. Given that trade i is a buy, what is the expected value of X i+? Now, let s also assume that the trade process is Markov so that. X X X X X X Pr,,... Pr i i i i i k. Consider the sequence of trades X =, X =, X 3 =, X 4 =, X 5 =, X 6 =, X 7 =. Find the probability of this outcome. l. Consider the sequence of trades,,,,,,. Find the probability of this outcome conditional on X =.

Midterm Exam Business Statistics Fall 2001 Russell

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