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Administration

ADM2303

David Wright

Fall

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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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