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Practice exam 3

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Department
Administration
Course
ADM2303
Professor
David Wright
Semester
Fall

Description
University of Ottawa School of Management ADM 2303: STATISTICS FOR MANAGEMENT I FINAL EXAMINATION December 9, 2006 LAST NAME (PRINT) FIRST NAME (PRINT) STUDENT #. Section (Circle One): A B C D F G A Wednesday, 13:00 – 14:30 Friday, 11:30 – 13:00 Nash B Wednesday, 10:00 – 11:30 Friday, 08:30 – 10:00 Nash C Monday, 16:00 – 17:30 Wednesday, 14:30 – 16:00 Khoroshilov D Monday, 08:30 – 10:00 Thursday, 10:00 – 11:30 Wright F Tuesday 10:00 – 11:30 Thursday, 08:30 – 10:00 Wright G Monday, 19:00 – 22:00 Phansalker Time: 3 hours Total marks: 60 NOTE THAT YOUR NAME & STUDENT NO. GO ON THIS SHEET AND 4 ANSWER SHEETS Put your name on THIS sheet too! 1. It has the Academic Integrity Statement 2. Any notes here MAY help you in cases where you are close to pass/fail boundary. Answers without explanation of how they are found may get no marks. ALL ANSWERS (INCLUDING BRIEF EXPLANATIONS) GO ON THE ANSWER SHEET. The exam question sheets willnot be marked, though space on the back of sheets is provided here for your rough work. Deposit question sheets in the box provided to allow for verification if needed. Note that there are marks for explaining your answers, so make sure you include brief explanations on the answer sheet. There are marks for identifying probability distributions. Calculators, 1 double-sided sheet of notes, on 8.5" by 11" paper (no stick-ons!) are allowed. In using tables, you do not need to interpolate, but take the nearest table value. Statement of Academic Integrity The School of Management does not condone academic fraud, an act by a student that may result in a false academic evaluation of that student or of another student. Without limiting the generality of this definition, academic fraud occurs when a student commits any of the following offences: plagiarism or cheating of any kind,use of books, notes, mathematical tables, dictionaries or other study aid unless an explicit written note to the contrary appears on the exam, to have in his/her possession cameras, radios (radios with head sets), tape recorders, pagers, cell phones, or any other communication device which has not been previously authorized in writing. Note: an examination answer sheet without the signed statement will not be graded and will receive a final exam grade of zero. Signature: ADM 2303 Final Examination Dec 9 2006 1 Qu.#1 (Total 10 marks) A warehouse has 2 loading bays at which trucks arrive, at an average rate of 0.6 trucks per hour, to deliver goods. If there are more than 2 trucks at the warehouse at any time, the 3 , and subsequent trucks must wait for loading since there are only 2 loading bays. This congestion is inefficient and to alleviate it, the warehouse manager is considering two options (a) building a 3 rd loading bay at a cost of $325,000, or (b) purchasing mechanized loading equipment for $245,000 to speed up the loading of the trucks. a. [2] What is the standard deviation of the number of trucks arriving per hour? b. [4] At present the warehouse uses trucks that take 2 hours to load. The manager decides to build the 3 rd loading bay if the probability of more than 2 trucks arriving during the 2 hour loading period is greater than 10%. Should the 3 rdloading bay be built? State your reasons clearly. c. [4] Alternatively, mechanized loading equipment would make it possible to load a truck in rd 1.7 hours, so that the 3 loading bay would need to be built only if the probability of 2 trucks arriving during the 1.7 hour loading period is greater than 10%. Should the mechanized loading equipment be bought instead of building a 3 loading bay? State your reasons clearly. Qu.#2 (Total 10 marks) A mining company explores a site for copper and zinc using exploratory drilling to obtain samples. The probability that a sample contains copper is 0.6 and the probability that a sample contains zinc is 0.5. The probability that a sample contains both zinc and copper is 0.4. a) [1] What is the probability that a sample containing zinc also contains copper? b) [2] Is the presence of copper and zinc in samples (i) mutually exclusive (ii) independent? State your reasons clearly. c) [2] What is the probability that 5 consecutive samples contain either copper or zinc? d) [5] In a group of 12 samples, 7 have been found to contain copper and the other 5 do not contain copper. We randomly select 6 of these 12 for further analysis. What is the probability that more than 4 out of these 6 contain copper? ADM 2303 Final Examination Dec 9 2006 2 Qu.#3 (Total 10 marks) A simple binary option pricing model assumes that over a short period of time (i.e, over an hour) a stock price can either go up or down by the same amount. Assume the stock price of ABC Inc. can either increase by $0.03 with probability 0.55 or decrease by $0.03 with probability 0.45 each hour. The stock price of XYZ Inc. can either increase by $0.04 with probability 0.6 or decrease by $0.04 with probability 0.4. Stock price movement is time-independent (i.e. past price change does not affect future price change). In fact, the joint distribution of ABC and XYZ stock prices movement each hour is given below: ABC up by $0.03 ABC down by $0.03 XYZ up by $0.04 0.5 0.1 XYZ down by $0.04 0.05 0.35 a) [2] What is the average hourly price increase and its standard deviation for ABC stock? b) [1] What is the correlation between ABC and XYZ hourly stock price changes? c) [2] What is the probability that ABC stock price will be the same after 6 hours of trade? More sophisticated models of asset pricing assume that the change is stock prices follow a normal distribution. As an example, assume that the hourly change of AAA Inc. stock price follows a normal distribution with mean $0.05 and standard deviation $0.5 while the hourly change of ZZZ Inc. stock price follows a normal distribution with mean $0.06 and standard deviation $0.7. The correlation of AAA and ZZZ stock prices is 0.8. d) [3] Find the average hourly increase in the value of a portfolio of 100 stock of AAA and 200 of ZZZ and its standard deviation. e) [2] Regardless of your answer in (d), assume that the mean hourly increase of this portfolio is $20 and the standard deviation is $100 and that the portfolio follows a Gaussian or Normal distribution. Find the probability that the value your portfolio will NOT decrease over a 1-hour period. Qu.#4 (Total 10 marks) The use of the Internet has led to a rise in uninformed online investments. As a result, the probability of investing in a “good” stock is only 0.20. An online investor invests in 10 stocks, one at a time. Find the probability that: a. [2] At least 1 investment is good. b. [2] No more than three investments are good. c. [2] The seventh investment was the first good investment. d. [4] If a novice online investor invested in 100 investments, what is the probability that he/she would have 25 or more good investments? ADM 2303 Final Examination Dec 9 2006 3 Qu.#5 (Total 11 marks) a. The income of adults last year in Ontario was found to have a mean of 60 (thousand $ or k$) and a standard deviation of 15 (k$), but the distribution shape was not specified. i. [3] A simple random sample of 100 adults was obtained this year. What is the probability that the sample mean is between 57 (k$) and 63 (k$) if this year's distribution is the same as last year's? Explain your answer BRIEFLY. ii. [1] If the sample mean actually was found to be more than 63 (k$), does this imply that the average income has increased compared to last year? Explain BRIEFLY. b. [3] The height of a certain population of males is assumed to be normally distributed with a mean height of 175 cm and a standard deviation of 7.5 cm. If a sample of size 9 were taken, find the probability that the sample mean would be less than 168.5 (cm). Explain your answer BRIEFLY. c. [4] A statistician decides that while heights of a particular male population may be normally distributed, the standard deviation of the population given in part ‘b’ should not be used. However, there is an actual sample (given below). What would be probability that the sample mean would be less than 170 cm if we still believe the population mean to be 175 cm? Explain your answer BRIEFLY. The sample values (in cm) are 158 161 165 170 175 179 182 166 174. Qu.#6 (Total 9 marks) SuperYog is a supplier of yogurt. One of the popular sizes is the 125 gram plastic cup with a foil top. This is sealed by applying a little heat to the rim of the cup after filling so that the plastic liner on the foil bonds to the plastic of the cup. To open the cup, one has to pull on a tab of foil that is left uncut for this purpose. Unfortunately, as we all know, the seal is sometimes stronger than the foil and we are "foiled" in our attempt to open the yogurt since the foil rips. A technician at SuperYog devises a jig to measure the pulling force and determines that a force of over 5.5 newtons will result in the foil ripping. It is also evident that a force under 2.5 newtons implies an improper seal, which may lead to contaminated product or to leakage in transport. The technician takes 50 hourly samples of 5 cups each hour and measures the pull force to open them. a) [1] The descriptive statistics and a histogram of the data are below. Is the sealing of the cups satisfactory? Explain BRIEFLY using the data given. Descriptive Statistics: pullfrce Variable N Mean StDev Minimum Q1 Median Q3 Maximum yogpull 250 4.1163 0.8975 2.1800 3.4575 4.0450 4.6800 7.0400 ADM 2303 Final Examination Dec 9 2006 4 Histogram of yogpull N = 250 Midpoint Count 2.000 0 2.200 1 * 2.400 3 *** 2.600 2 ** 2.800 8 ******** 3.000 16 **************** 3.200 17 ***************** 3.400 19 ******************* 3.600 19 ******************* 3.800 31 ******************************* 4.000 13 ************* 4.200 20 ******************** 4.400 22 ********************** 4.600 18 ****************** 4.800 15 *************** 5.000 20 ******************** 5.200 4 **** 5.400 3 *** 5.600 4 **** 5.800 3 *** 6.000 5 ***** 6.200 2 ** 6.400 0 6.600 1 * 6.800 2 ** 7.000 2 ** ADM 2303 Final Examination Dec 9 2006 5 b) [2] Using the data in (a) and the graph below, is the data consistent with a Normal distribution of force needed to open the cups. State your reasons briefly. The X axis is measured in Newtons. ADM 2303 Final Examination Dec 9 2006 6 c) [2] The technician uses the data to create an Xbar-R chart for the column of data and setting the sample size to 5. Interpret this chart in relation to the cup sealing process? d) [2] SuperYog engages a quality management consultant to review their practices. The consultant uses the SAME data, but supplies a population mean force needed to open cups o 4 newtons with standard deviation 0.5 newtons in accord with the specifications. Still using 5 cups per sample, the following chart is drawn. Interpret the chart and give your conclusion as to whether this chart or the one above is the most appropriate for SuperYog to use? ADM 2303 Final Examination Dec 9 2006 7 e) [2] A different sealing machine is set up and tried. It gives the following histogram and sequential time plot of data over 50 hours, with samples of 5 cups per hour tested. Should SuperYog consider using this machine? Explain BRIEFLY. Histogram of mach2 N = 250 Number of observations represented by each * = 2 Midpoint Count 0.5 1 * 1.0 22 *********** 1.5 35 ****************** 2.0 40 ******************** 2.5 55 **************************** 3.0 46 *********************** 3.5 34 ***************** 4.0 14 ******* 4.5 3 ** ADM 2303 Final Examination Dec 9 2006 8 Standard Normal (Gaussian) Table The table gives the area under the Standard Normal Distribution curve between 0 and the given value of z The value of z is given by the leftmost column and top row of the table. Cumulative probability is given by 0.5 plus table value. z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359 0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753 0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141 0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517 0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879 0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224 0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.2549 0.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852 0.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133 0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389 1 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621 1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830 1.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015 1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177 1.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319 1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.4441 1.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545 1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633 1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706 1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767 2 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817 2.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.4857 2.2 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.4890 2.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.4916 2.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.4936 2.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952 2.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.4964 2.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4974 2.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.4981 2.9 0.4981 0.4982 0.4982 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986 3 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990 3.1 0.4990 0.4991 0.4991 0.4991 0.4992 0.4992 0.4992 0.4992 0.4993 0.4993 3.2 0.4993 0.4993 0.4994 0.4994 0.4994 0.4994 0.4994 0.4995 0.4995 0.4995 3.3 0.4995 0.4995 0.4995 0.4996 0.4996 0.4996 0.4996 0.4996 0.4996 0.4997 3.4 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4998 3.5 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 3.6 0.4998 0.4998 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 3.7 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 3.8 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 3.9 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 4 0.5000
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