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ECO 251 QBA1
FINAL EXAM
MAY 2, 2005
Name
Class ________________
Part I. Do all the Following (14 Points) Make Diagrams! Show your work! Exam is normed on 75 points.
There are actually 134 possible points. If you haven’t done it lately, take a fast look at ECO 251 - Things
That You Should Never Do on a Statistics Exam (or Anywhere Else).
x ~ N 12, 4.6
1. P17.26  x  28 
2. F 17 .26  (Cumulative Probability)
3. Px  19 .30 
4. P12  x  16 
5. P0  x  16 
6. x.13 (Find
z .13
first)
7. A symmetrical region around the mean with a probability of 33%. [14, 14]
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II. (10 points+, 2 point penalty for not trying part a.) Show your work!
x
1
2
3
4
5
6
7
8
9
10
11
x2
y
y2
xy
4.0 16.00 10.2
104.04
*****
4.5 20.25 10.4
108.16
*****
5.0 25.00 10.5
110.25
*****
5.5 30.25 21.8
475.24 ******
6.0 36.00 36.8 1354.24 ******
6.5 42.25 51.6 2662.56 ******
7.0 49.00 66.2 4382.44 ******
7.5 56.25 68.7 4719.69 ______
8.0 64.00 68.0 4624.00 ______
8.5 72.25 69.4 4816.36 ______
9.5 90.25 75.0 5625.00 ******
72.0 501.50 488.6 28981.98 3641.25
The data above represent a random sample of the intensity of advertising x , measured in number of
exposure on evenings in prime TV, and y , the intensity of awareness of the product according to a
consumer survey taken after the advertising campaign for 11 products. Calculate the following:.
a. The sample standard deviation s y of awareness. (The standard deviation of x is 1.51383.) (2)
b. The sample covariance s xy between x and y after computing the 3 numbers not replaced by
asterisks. (3)
c. The sample correlation rxy between x and y . (2)
d. Given the size and sign of the correlation, what conclusion might you draw on the relation
between x and y ? (1)
e. Assume that the intensity of awareness was 15% higher ( v  1.15 y ). Find
v , s v2 , s xv and rxv . 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 x1 and x 2 and
recompute the results, you will receive no credit. (4).
f. Do a 90% confidence interval for the mean number of exposures. (2)
[14, 28]
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III. Do at least 5 of the following 7 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!) This is not an opinion
questionnaire. Answers without reasons or supporting calculations or table references will not be
accepted!!!! Note that some problems extend over 2 pages.
1.
Find P4  x  12  for the following distributions:
a) Normal with mean of 5.5 and standard deviation of 3 (1)
b) Continuous Uniform with c  2 and d  10 (2)
c)
Binomial with n  25 and p  .05 (2)
d) Binomial with n  25 and p  .55 (2)
e)
Geometric with p  .05 (2)
f)
Poisson with a parameter of 20. (2)
g) Find Px  2 for a Hypergeometric distribution with N  25, M  6 and n  4 . (3)
[14, 42]
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2.
Find the following:
The mean and standard deviation for
a) Continuous Uniform with c  2 and d  10 (1)
b) Binomial with n  25 and p  .55 (1)
c)
Geometric with p  .05 (1)
d) Poisson with a parameter of 20. (1)
e)
f)
Hypergeometric distribution with N  25, M  6 and n  4 . (1)
You wish to find P4  x  12  for a binomial distribution with n  90 and p  .02 . You
do not have the appropriate binomial table, so show that you can use an approximation
and do the problem using that approximation. (4)
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g) You are taking a sample of 90 from a population of 2000 that is 10% defective and want
the probability of more than 25 defective items in the sample.
(i)
Show that you can use the Binomial distribution in this result. (1)
(ii)
Show that you can use the Normal or Poisson distribution to replace the
Binomial distribution and do the problem (4+)
(iii)
Show the effect of using a finite population correction on this result. (2)
h) You are taking a sample of n  100 from a population that is 60% in favor of your
candidate.You will declare your candidate the victor if p  .55 . Find P p  .55  when
the true population proportion is p  .60 .(3)
i)
(Extra Credit) The Negative Binomial distribution gives the probability that there will be
x failures before success n . We have the following formulas for it:
nq
nq
and  2  2 . Assume that p  .3 .
Px   C nn1x 1 p n q x ,  
p
p
(i)
(ii)
(iii)
What is the chance that our 5th success occurs after the 8th failure? (2)
What is the average number of failures that will occur before the 5 th
success? (2)
Show that the Geometric distribution is a special case of the Negative
Binomial distribution by showing under what conditions Px ,  and
 2 are the same for both distributions. (5)
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3.
A random sample is taken of the time necessary for a firm to approve 49 life insurance policies.
From the sample we get a sample mean of 43 days and a standard deviation of 24 days.
a) Find a 99% confidence interval for the mean processing time assuming that the sample
standard deviation is correct and that the sample of 49 policies was taken from only 400
policies. (4)
b) On the basis of long experience, we know that the population standard deviation for the
policies was 20. Find a 99% confidence interval assuming that this population standard
deviation is correct and that the sample comes from a large number of policies. (4)
c)
Find a 99% confidence interval for the mean processing time assuming that the population
standard deviation of 20 is correct and that the sample of 49 policies was taken from only
400 policies. (4)
d) Repeat b) with a 74% confidence level. You cannot use the t table to answer this
question correctly. (3)
[12, 73]
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4.
A supermarket clerk is believed to take a mean time of 2.6 minutes with a standard deviation of 2.1
minutes to check out a customer. Assume that the underlying distribution is Normal.
a) What is the probability that the clerk will take more than 2.8 minutes to check out an
individual? (2)
b) If 3 people come to this clerk, what is the probability that at least one of the people takes
more than 2.8 minutes to check out? (2)
c)
If 3 people come to this clerk, what is the probability that their average checkout time will
be more than 2.8 minutes? (3)
d) If 7 people are in line, what is the probability that the clerk will be able to check them all
out before the clerk goes on break in 18 minutes? (3)
e)
Do not guess on this question. Show what formula you should use! Let x represent the
supermarket’s revenues and y represent their costs, so that profits are a random variable
w  x  y. Will a smaller correlation between x and y
(i)
decrease or increase the expected value of profit? (1)
(ii)
decrease or increase the variability of profit? (2)
[13, 86]
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5.
(Example modified from Hildebrand, Ott, and Gray) A bank officer estimates the joint
probabilities for percent return on two utility bonds as shown below. For your convenience, most
row and column sums are given in the joint probability table, and most column sums, which you do
not need but which may speed your computations, are given in the xyPx, y  table.
xyPx, y 
Original Data
x
4.0

4.0
.03

4.5 .04
y 5.0 .02

5.5 .00
6.0 .00
sum .09
4.5 5.0 5.5 6.0 sum
.04 .03 .00 .00  .10

.06 .06 .04 .00  .20
.08 .20 .08 .02  .40

.04 .06 .06 .03  .19
.00 .03 .03 __ 
.22 .38 .21 __
0.480
0.720

0.400

0.000
0.000
0.720
1.215
1.800
0.990
0.000
0.060
1.350
5.000
1.650
0.900
0.000
0.990
2.200
1.815
0.990
0.000 
0.000 
0.600 

0.990 
___
Column sums are 1.600, 4.725, 9.500, 5.995 and ____
Note that E x   5.005 and  x2  0.297475.
a) If x and y were independent, what would the number be in the blank space on the
original joint probability table? (1)
b) What should the number be in the blank space in the joint probability table? Fill in all the
blank spots in the tables. (2)
c)
Compute  y and the population standard deviation of y (2)
x
4.0
.03

4.5 .04
y 5.0 .02

5.5 .00
6.0 .00
sum .09
4.0
4.5 5.0 5.5 6.0 sum
.04 .03 .00 .00  .10

.06 .06 .04 .00  .20
.08 .20 .08 .02  .40

.04 .06 .06 .03  .19
.00 .03 .03 __ 
.22 .38 .21 __
d) Compute  xy  Covx, y  and  xy  Corrx, y  (4)
e)
f)
Compute Ex  y  and Var x  y  (2)
If the officer buys one unit of bond x and 5 units of bond y , so that the total return is
R  x  4 y , what are the mean, standard deviation and coefficient of variation of R ? (4)
[15, 101]
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6. Consider the following joint probability table.
B1

A1
.10

A2 .00
A3 .18

sum .28
B2
.15
.05
.02
.22
B3
.50 

.00 
.00 

.50
Find P A1  B2  (2)
sum
a)
.75
.05
20
b) Find P A1 B2 (2)

c)

Find P A1  B2  (1)
B1
B2
B3
sum
 __ __ __  __


d) Fill in the blanks if A1 and B1 are mutually exclusive. A2  __ __ .05  .25 (2)
A3  __ .15 .10  .45


sum __ .45 .30
B1 B 2 B3
sum
A1  __ __ __  __


e) Fill in the blanks if A1 and B1 are independent. A2  __ __ .06  .20 (2)
A3  __ .16 .12  .40


sum __ .40 .30
B1 B 2 B3
sum


A1
__ __ __
__


f) Fill in the blanks if A1 and B1 are collectively exhaustive. A2  __ __ .00  .05 (2)
A3  __ .00 .00  .25


sum __ .15 .50
g) (Ben Horim and Levy) The senate is about to vote on a tax bill. There is a 45% chance
that it will pass. If it passes there is a 15% chance that unemployment will increase in the
next six months. If it does not pass there is a 60% chance that unemployment will increase
in the next six months. Use the following notation: TP is the event that the bill passes and
UI is the event that unemployment increases.
(i)
What is the probability that unemployment increases in the next six
months? (2)
(ii)
You were out hunting minces (You make mince pie from them) in BanjiWanjiland for the last six months and never found out if the tax bill
passed. When you get back you find out that unemployment has increases.
What is the (posterior) probability that the tax bill passed? (3) [16,
117]
A1
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7.
Do the following.
a) Assume that the average number of students logging onto a system every hour is 900.
(i) What is the chance that none will log on in 30 seconds? (2)
(ii) What is the chance that none will log on in one minute? (1)
(iii) What is the chance that none will log on in ten minutes? (2)
b) A shipment of 20 items is 30% defective. To decide whether to accept the shipment, you
will take a sample of 4 items and reject the shipment if more than one item in the sample
is defective. What is the probability that you will reject the shipment? (3)
c)
Repeat b) assuming that there are many items in the shipment. (3)
d) A large shipment 30% defective. To decide whether to accept the shipment, you will take
a sample of 30 items and reject the shipment if more than three items in the sample are
defective. What is the probability that you will reject the shipment? (3)
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e)
A large shipment 2% defective. To decide whether to accept the shipment, you will take a
sample of 30 items and reject the shipment if more than three items in the sample are
defective. What is the probability that you will reject the shipment? (You will find that
because of the small probability, this problem is done very differently from d)). (3)
f)
(Extra Credit) If a teller at the Grover’s Corner bank can serve 30 customers an hour, the
average time to serve a customer is 2 minutes. Assume that the exponential distribution
applies and find the following:
(i) The probability that it takes more than 5 minutes to serve a customer. (2)
(ii) The probability that it takes no more than 2 minutes to serve a customer. (2)
(iii) The probability that it takes between 2 and 5 minutes to serve a customer. (1)
[17, 134]
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(Blank)
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