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ECO251 QBA1
THIRD EXAM
Dec 1, 2004
Name: _____________________
Student Number: _____________________
Class Time (Circle) 10am 11am 1pm 2pm
Part I: 8 points.
z follows the standardized Normal distribution z ~ N 0,1 .
Find the following. Make diagrams! (0.5 extra credit for each useful diagrams)
1. P2.12  z  1.67 
2. P1.67  z  1.45 
3. Pz  1.45 
4. P1.45  z  3.64 
251x0472 11/22/04
Part II: (22+ points) Do all the following: All questions are 2 points each except as marked. Exam is
normed on 50 points including take-home. (Showing your work can give partial credit on some
problems! In open-ended questions it is expected. Please indicate clearly what sections of the
problem you are answering and what formulas you are using. Neatness counts!) Remember that you
may not be able to finish this section, so ration your time on each problem. [Numbers in brackets are a
cumulative total]
1.
The covariance
a) must be between -1 and +1.
b) must be positive.
c) can be positive or negative.
d) must be between zero and +1
2.
The portfolio expected return of two investments
a) will be higher when the covariance is zero.
b) will be higher when the covariance is negative.
c) will be higher when the covariance is positive.
d) does not depend on the covariance.
3.
An average of 16 customers arrive at a checkout counter every minute and we want to find the
probability that more than 22 will show in a given minute, the Poisson distribution will be used
with a standard deviation of
a) 2
b) 4
c) 16
d) here is not enough information.
4.
(Mansfield) From experience, the director of a computer center knows that any one of the twentyfive PCs in the room is broken 4% of the time. Whether or not any given PC is broken does not
depend on how much it is used; they just seem to break down randomly. What type of probability
distribution would be used to figure out the probability that more than eight of the PCs will be
broken down at the same time?
a) binomial distribution.
b) Poisson distribution.
c) hypergeometric distribution.
d) none of the above.
5.
You receive two tickets for a sold-out concert. You call four potential dates for the evening. You
think that there is a chance of 13 that each of the individuals would call back and agree to go to a
concert with you. You will be very embarrassed if more than one accepts. What is the probability
that exactly one out of the four says ‘yes?’ (Extra credit: Do this last! What is the probability that
you are embarrassed?) Show your work! (3)
[11]
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Questions 6-10 are based on exhibit 1. Show your work if you expect full credit!
Exhibit 1: The table below shows average Fahrenheit temperature and yield in lbs./acre for an industrial crop.
F
Y
63
10
70
15
.
75
17
79
16
80
20
The following calculations are done for you. One more column is needed.
Row
x
x2
y
1
2
3
4
5
63
70
75
79
80
367
3969
4900
5625
6241
6400
27135
10
15
17
16
20
78
y2
100
225
289
256
400
1270
6.
Find the sample standard deviation of Fahrenheit temperatures, x (2)
7.
Find the sample covariance between Fahrenheit temperature and yield.(3)
8.
Find the sample correlation between Fahrenheit temperature and yield. (2) [18]
9.
If the conversion formula for Celsius temperature is C = 5/9(F-32), find the covariance between
Celsius temperature and yield. (Hint: You are finding the sample covariance between two random
variables, one of which is w  5 9 x  1609 and the other of which is v  1y  0 . (2)
10. If the conversion formula for Celsius temperature is C = 5/9(F-32), find the correlation between
Celsius temperature and yield. (2) [22]
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11. We have twenty units of equipment in a bin of which five are defective. Pull three out at random.
What is the probability that exactly one will be defective if we:
a. sample without replacement? (3)
b. sample with replacement? (2)
c. If we sample with replacement, what is he probability that the first item that we find
that is defective is the third item that we pick? (2) [29]
To get full credit for this problem, identify the distribution that you are using and show your
work!
12. If ten people are selected at random from a large (continuous) population what is the
probability that more than half (which is not exactly six) make more than the median income? To
get full credit for this problem, identify the distribution that you are using and show your
work! (2) [31]
13. If x is a random variable with a mean of 7 and a standard deviation of 6, find
a. Cov8x  4, 9 x  3 (3)
b. Corr8x  4, 9 x  3 (2) [36]
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ECO251 QBA1
THIRD EXAM
Dec 1, 2004
TAKE HOME SECTION
Name: _________________________
Student Number: _________________________
Throughout this exam show your work! Please indicate clearly what sections of the problem you are
answering and what formulas you are using.
Part III. Do all the Following (20 Points) Show your work!
1. As everyone knows, a jorcillator has two components, a phillinx and a flubberall. It seems that the
jorcillator works as long as one component works. For example, if the Phillinx failed last year and the
flubberall fails this year, the phillinx fails this year. This problem, which is a simplified version of Problem
H4 has three time periods, last year, this year and Ever After (after this year). You have been in Tibet for
the last year, so you have no idea on what actually happened last year.
The probability of the phillinx failing is given by a standardized Normal Distribution, so , if z represents
the life of the Phillinx, the probability of the phillinx failing last year is Pz  0 , the probability of it
failing this year is P0  z  1 and the probability of it lasting into the Ever After is Pz  1 .
The probability of the Flubberall failing is given by a Continuous Uniform Distribution, where c  1 and
d  2  f , where f is the fourth digit of your student number, divided by 10. For example if Seymour’s
student number is 012345, his value of d  2  .3  2.3 . If the fourth digit of your student number is zero,
use d  3.1 . If x represents the life of the Phillinx, the probability of the phillinx failing last year is
Px  0 , the probability of it failing this year is P0  x  1 and the probability of it lasting into the Ever
After is Px  1 .
Failure of components is assumed to be independent, so that, if the probability of the Phillinx failing in the
first year is .1 and the probability of the Flubberall failing last year is .7, the probability of both components
failing last year is (.1) (.7) =.07 (This is not necessarily the probability that the jorcillator failed last year!)
a) What is the probability the Flubberall failed last year? This year? After this year? (1.5)
b) What is the probability the Phillinx failed last year? This year? After this year? (1.5)
c) What is the probability that the jorcillator failed last year? (2)
d) What is the probability that the jorcillator will fail this year? (2)
e) What is the probability that the jorcillator will fail in the Ever After.? (2)
f) Using the probabilities that you found in c) – e) , associate the number -0.5 with last year, 0.5
with this year and 2.0 with Ever after and find the expected value and standard deviation of the life of
the jorcillator. (2)
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2. (Bowerman and O’Connell, modified) An air traffic control center experiences a week with 3 errors
(planes placed too close together either vertically or horizontally).
On the basis of records of the center, you decide that the center has an average number of errors of d 10 per
week, where d is the number you used in the previous problem.
a) (4) Compute a Poisson table with the following values P0 , P 1 , P2 , P3 and P4 . The easiest
way to do this is to assume that P5 and any probabilities above it are essentially zero and to assume that
after you have computed P0 through P3 , that P4 is whatever is required to make this a valid
distribution.
Example: If d  4.0, you would assume that the mean is 0.40 and would compute e 0.40  .67032 . Using
that in the Poisson formula, you would find the following P0  .6703 , P1  .2681 , P2  .0536 and
P3  .0072 . In order to get a valid distribution, you need P4  .0008 . (If you check this against the
Poisson table you will find that the actual value of P4 is .0007, but the difference won’t affect your
results.) You can see if you are on the right track for your numbers by comparing your results with the
Poisson tables for parameters of 0.3 and 0.4. Your results should be close.
b) (3)To test the hypothesis that the mean is still the value of d 10 that you are using, you compute the
probability that, if the hypothesis is true you would get results as extreme or more extreme than you
actually got. In this case that would be Px  3 . If the probability is less than 1%, you can say that you
strongly doubt that the hypothesis is correct and demand an investigation. If the probability is between 1%
and 5% you can say that you somewhat doubt that the hypothesis is correct and suggest that the controllers
clean up their act. If the probability is above 5%, you would have to say that you cannot doubt the
hypothesis and you would have to shut up. In view of the probability you computed, what would you do?
c) (2)Using the table that you have made, repeat the analysis assuming that during the week there were 2
errors instead of 3.
3. (Extra Credit) (Bowerman and O’Connell, modified) a) A young nurse observes that 8 of the last 30000
patients of a hospital went into a coma during anesthesia. Upon investigation she finds that nationally 6 out
of 100000 patients go into comas after anesthesia (This is 0.006 per cent!). Assume that the national
probability is correct and that the Binomial distribution is appropriate, what is the mean number of patients
out of 30000 that will go into a coma? Find the probability of 8 or more people out of 30000 going into a
coma. Explain whether we have a binomial table that will do this. If not, show that this is an appropriate
place to use a Poisson distribution and use the Poisson tables to find the probability that x  8. Use the
approach in the previous problem to see if the number of comas is consistent with the national rate.
b) How would you change your opinion if the number that went into a coma was 6?
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