251x0343 12/08/03
ECO 251 QBA1
FINAL EXAM
DEC 13, 2003
Name
Class ________________
Exam is normed on 75 points. There are actually 128 possible points.
Part I. Do all the Following (14 Points) Make Diagrams! Show your work!
x ~ N 4, 5 .
1. P5.20  x  5.20 
2. P1  x  22 
3. Px  4.85 
4. F 5.20  (Cumulative Probability)
5. P0  x  11 .3
6. x.065 (Find z .065 first)
7. A symmetrical region around the mean with a probability of 22%
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II. (10 points+-2 point penalty for not trying part a .) Show your work!
The following numbers represent a random sample of 5 countries from The World Guide – 2003/2004 and
give per capita income (PCI) in thousands of US dollars x  and Human Development Index (HDI-based on
life expectancy, access to education and income)  y  .
1
2
3
4
5
Kazakhstan
Greece
Estonia
Samoa
Burundi
x
y
5.9
16.5
10.6
5.0
0.6
0.742
0.881
0.812
0.701
0.309
These sums have been calculated for you.
 y  3.445 s x2  36.697, x  7.72, and
x2
34.81
272.25
112.36
25.00
0.36
 x  38.6 ,  x
2
 444 .78 ,
y  .6890 . Please calculate the following:
a. The sample standard deviation of y (4)
b. The sample covariance between x and y . (3)
c. The sample correlation between x and y . (2)
d. Given the size and sign of the correlation, what conclusion might you draw on the relation
between PCI and HDI if this were the only evidence available? (1)
e. Assume that the HDI in all 5 countries rose by an additive amount 0.002 and the PCI rose by a
multiplicative factor of 1.2. What would be the new values of x , s x , s xy and rxy . Use only the values you
computed in a-c and rules for functions of x and y to get your results. If you state the results without
explaining why, or change x and recompute the results, you will receive no credit. (4).
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III. Do at least 4 of the following 6 Problems (at least 12 each) (or do sections adding to at least 48 points Anything extra you do helps, and grades wrap around) . Show your work! Please indicate clearly what
sections of the problem you are answering! If you are following a rule like E ax  aEx  please state it! If
you are using a formula, state it! If you answer a 'yes' or 'no' question, explain why! If you are using the
Poisson or Binomial table, state things like n , p or the mean. Avoid crossing out answers that you think
are inappropriate - you might get partial credit. Choose the problems that you do carefully – most of us are
unlikely to be able to do more than half of the entire possible credit in this section!)
1. A company manufactures small motors for use in snow blowers. A sample of 16 is taken from a batch of
5000 motors and the sample mean horsepower is found to be 2.85 hp. Find a 98% confidence interval for
the mean horsepower under the following circumstances.
a. The sample standard deviation is found to be 0.08. (4)
b. The sample standard deviation is found to be 0.08, but the sample of 16 is taken from a batch
of only 101motors. (4)
c. The sample of 16 still has a sample mean of 2.85, but it is taken from a batch of 5000 motors
with a population standard deviation that is assumed to be 0.1. (4)
d. Assume that the population mean is actually 2.94 hp, the population standard deviation is
assumed to be 0.1, the sample is of size 16, and the batch is of 5000 motors, what is the
probability that the sample mean will lie between 2.80 and 2.90? (3)
e. For the same distribution as in d), what is the probability that an individual machine has a
horsepower between 2.80 and 2.90? (2)
f. Under the circumstances of d), find an 87% confidence interval for the mean. (Hint: see page
1) (2)
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2. The following joint probability describes x and y.
x
y
1
2
Px 
1
2
3
4
.08 .16 .24 .32 


.02 .04 .06 .08 
P y 
Let A1 be the event that x  1 and B 1 be the event that that y  1. Find the following:
a) P A1  B1  (1)
b) P A1  B1  (1)


c) P A1 B1 (1)


d) P A1  B1 (1)
e)  x , the mean or expected value of x (2)
f)  x , the standard deviation of x (2)
g)  xy , the covariance of x and y (3)
h)  xy , the correlation between x and y (1)
i) The mean and variance of x  y (2)
j) The probability that x  y  4. (3)
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3. Assume that a population is 10% defective.
a. If the population consists of 80 items and you take a sample of ten, what is the chance of at
least one defective item?(3)
b. If the population consists of many items and you take a sample of 10, what is the chance of at
least one defective item?(2)
c. If the population consists of many items and you take a sample of 80 items, what is the chance
of at least five defective items? Show that you can use a Poisson distribution here and use it. (2)
d. Show that you can use a Normal distribution here and use it. (3)
e. We are starting a Bayes’ Rule problem, but it starts real easy. When a process is out of control
(OC ) its operation is described by a process in which the probability that an item is defective is
p  .1. In other words, if we take a sample of 12 the probability of one defective item is described
by the binomial distribution with p  .1 and n  12 . What is this probability ? (2) Call this

probability P x  1 OC

f. When the process is in control, the probability of exactly one defective item in a sample of 12 is described
by a binomial distribution with p  .01 and n  12 . What is this probability? (1) Call this


probability P x  1 OC
g. Now assume that the (prior) probability of the process being in control is .95 and the
probability of the process being out of control is .05, and that you take a sample of 12 and find
exactly one defective item. Find P x  1 OC and P x  1 OC . Then use Bayes' rule to figure
out the probability that the process is out of control when you take a sample and find exactly one
is defective. (5)
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



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4. A salesperson makes sales to 70% of her prospects.
a. If she calls on 10 people, what is the mean and the variance of the number of sales she
makes?(2)
b. If she calls on a large number of people, what is the probability that her first sale is among her
first three calls? (2)
c. What is the probability that she makes at least 7 sales in 10 calls? (Don’t give me exactly seven
– or exactly any other number. )(2)
d. What is the probability that she makes at least 63 sales in 90 calls? (3)
e. Assume that she calls on 10 people every day, so that the mean and variance of her daily sales
are as you stated in a). Let x be the sample mean number of sales she makes a day over 100
days. What does the central limit theorem say the expected value and standard error of x will be
over 100 days? (2)
e. What is the probability that x will exceed 6.2? (3) 14 - 97
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5. As everyone knows, a jorcillator has two components, a phillinx and a flubberall. It seems that the
particular jorcillator we have works as long as either component works (This has changed!).
The probability of the phillinx failing is given by a continuous uniform distribution with a range between
c  0.5 and d  2.0, so , if x represents the life of the phillinx, the probability of the phillinx failing in the
first year is P0  x  1 , the probability of it failing in the second year is P1  x  2 and the probability
of it lasting beyond the 2nd year is Px  2 .
The probability of the flubberall failing is given by a Normal distribution with a mean of 1.5 and a standard
deviation of 0.25, so , if y represents the life of the phillinx, the probability of the phillinx failing in the
first year is P0  y  1 , etc.
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 in the first year is .7, the probability of both
components failing in the first year is (.1) (.7) =.07 (This is not necessarily the probability that the
jorcillator will fail in the first year!)
a) What is the probability the Phillinx will fail in the first year? The second year? After the
second year? (1.5)
b) What is the probability the Flubberall will fail in the first year? The second year? After the
second year? (1.5)
c) What is the probability that the jorcillator will fail in the first year? (2)
d) What is the probability that the jorcillator will fail in the second year? (2)
e) What is the probability that the jorcillator will last beyond 2 years? (2)
f) If my firm owns two Jorcillators, what is the probability that at least one lasts beyond two
years? (3)
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6. The weights of players on my university football team have a Normal distribution with a mean of 115
pounds, with a standard deviation of 24 pounds. (We don’t win many games.)
a. The opposing team has weights that also have a Normal distribution with a much higher mean.
If Cadwallader is on my team and he is tackled by someone who is on the opposing team, what is
the probability that the tackler is above the median weight for his team? (1)
b. If Spike, the guy who tackled Cadwallader, weighs 210 pounds and Spike tackles someone on
my team, what is the chance that Spike tackles a person who weighs half of what he weighs.? (2)
c. Lets say that I take a sample of 36 from my team ( consider the team to have lots of players on
it) what is the expected value and standard deviation of the sample mean?(1.5)
d. What is the probability that that the sample mean of this sample exceeds 120 pounds?(2)
e. If I put 36 members of the team on a bus and the bus has a capacity of 4320 pounds, what is
the probability that the bus will be overloaded? (1) 8.5
f. If I randomly pick 36 members of the team and weigh the entire group, the expected value of
the sum of their weights is 4140 pounds. What is the standard deviation of the sum of their
weights? (2)
g. If you did not do e) using the mean and variance in f), do it now. (2) (If you did do e) this way,
you got 2 points for it.) 12.5
h. If you learned that this sample of 36 was taken from a team with 80 members, how ould this
affect your answer to c, d, e and f? Do it! (Some of this involves hard thinking.) (6.5) 19
i. If we implement mandatory testing for steroids, even though we believe that 95% of our players
do not use them ( PU   .05 ), we will use a drug testing procedure that has a 93% chance of
testing positive for a user. (Use Po for the event that someone tests positive and U for the event
that someone is a user.) The procedure has a 3% chance of testing positive for a non - user, a 7%
chance of testing negative for a user and a 97% chance of testing negative for a non - user.
Cadwallader tests negative. If we use the 95% as a prior probability, what is the chance that
Cadwallader is a steroid user? (6) 25
j. Actually, my university football team gets an average of 1.2 goals per game.
(i) If we assume that the Poisson distribution applies, what is the probability that we get
at least one goal in a game? (2)
(ii) What is the probability of at least one goal in the first quarter? (2)
(iii) Why might the Poisson distribution not apply in this case? (1) 30 - 139
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(blank page)
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