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ECO 251 QBA1
FINAL EXAM, Version 2
MAY 6, 2008
Name
Class ________________
Part I. Do all the Following (14 Points) Make Diagrams! Show your work! Illegible and poorly
presented sections will be penalized. Exam is normed on 75 points. There are actually 123+ 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 8.40, 6.00 
1. P6.88  x  6.88 
2. P0  x  8.00 
3. F 6.85  (Cumulative Probability)
4. Px  6.92 
5. P10.30  x  26.50 
6. x.055
7. A symmetrical region around the mean with a probability of 13%
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Name:_________________
You may find it convenient to tear out exhibit 1. It must be handed in with the exam. You can do
calculations on this page if you wish.
Exhibit 1:
I have looked up the price and number of pages for books published by Gaugem textbook publishers.
Row
Pages
1
2
3
4
5
6
7
Price
x
y
490
690
740
580
530
640
450
120
130
160
110
120
120
70
Minitab has kindly given me the following descriptive sample statistics
Descriptive Statistics: Pages, Price
Variable
Pages
Price
N
7
7
Mean
588.6
118.6
SE Mean StDev
40.3 106.7
………… ……………
Minimum Q1
450.0 490.0
70.0 110.0
Median Q3
580.0 690.0
120.0 130.0
Maximum
740.0
160.0
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II. (10 points+, 2 point penalty for not trying parts a) and b) Show your work! Mark individual sections
clearly. Answers without work and/or reasons cannot be accepted!
a) To show me that you know how to compute a sum of squares, compute
y
2
, the sum of squares for
the book prices. (1) (Hint: You can save a bit of time by dividing the numbers in y by 10 by dropping the
last zero in the number and then multiplying the resulting sum of squares by 100.) Show your work!
b) Compute the sample standard deviation of the book prices using the 7 prices given above.(2)
c) Compute a 99% confidence interval for the mean contents (3)
d) The company representative claims that their average book price is ‘only $85. On the basis of your
calculations is the mean significantly different from $85 ? Why? (1)
e) Actually the book company only publishes 60 titles. Assuming that the sample of 7 was taken from these
60 titles, recompute the confidence interval is c). (2) [9]
f) The costs of producing a book involve a fixed cost per book of $15 and a cost of 15 cents per page.
Express this as a formula C  ax  b . Use this and the mean and standard deviation given by Minitab to
find the mean and standard deviation of the producer’s cost per book. There will be no credit in this section
unless you use the mean and standard deviation given by Minitab. (2.5) [11.5}
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III. Do at least 5 of the following 6 sections (at least 12 each) (or do items 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 (except in multiple choice questions)!!!! Answers that are hard to follow will be penalized.
Note that some sections extend over more than one page.
A. Remember the problem in Exhibit 1. x is the number of pages in the text and y is the cost to the student.
The publisher claims that the book price is a direct reflection of the cost to produce the book which is base
on the number of pages.
1) Compute
 xy
from the data in Exhibit 1. (Hint: You can save a bit of time by dividing the numbers
in x and those in y both by 10 by dropping the last zero in the numbers and then multiplying the resulting
xy 100.) Show your work (2)

2). Compute the sample covariance s xy between the number of pages x and the price y . (2) Explain what
this covariance tells us. (0.5)
In the following any actual or implied reference to the variance or standard deviation refers to the value of
the variance or standard deviation that you computed in section c) on the previous page.
3) Compute the sample correlation rxy between x and y . (2) Explain what this correlation tells us about
the relationship between the number of pages and the price of the book. (1) [7.5]
4) Using the correlation and covariance you computed in b) and c) and the equation relating the number of
pages to the cost of production that you used on the last page, find the covariance and correlation between
the cost of production and the consumer price of the book. (4) [11.5]
5) The profit on each book is price per book less cost per book. Using the mean and variance that you have
computed for the price, the mean and variance that you have computed for cost per book and the covariance
between these two quantities. (3) [14.5]
6) (Extra credit) The true test of a correlation is a significance test which not only looks at the size of the
r
r
correlation, but the size of the sample. To do such a test compute t 
. On the assumption

sr
1 r 2
n2
that there is no significant correlation and using a 99% confidence level, 99% of the values of this t-ratio
n  2)
n  2)
will fall between  t .(025
and t .(025
. If your value of the t ratio does not fall between these two values, you
can reject the hypothesis of no correlation and say that there is significant correlation. Find the appropriate
values of t compute the t—ratio and tell me if the correlation is significant. (4)
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(Blank)
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B. (Multiple Choice – 2.5 with a .5 penalty for wrong answers each unless marked otherwise)
1. If y tends to decrease as x increases, what can we say about the population correlation  ?
a)   0 .
b)   0 .
c)   1 .
d)   1 .
e) Not enough information
2.
3.
One hundred items are randomly selected from a pilot production run of N items to check their
quality. x is the number of defective items in the sample. The distribution of x can be considered
approximately binomial if N is at least
a) 10000
b) 2000
c) 1000
d) 500
e) 100
f) 5 2  25
g) 5
h) All of the above
i) None of the above.
A student calculates that P9  x  35   F 35   F 9 . The student could be working with the
a) Binomial Distribution
b) Continuous Uniform Distribution
c) Poisson Distribution.
d) Geometric distribution
e) Any of the above
f) None of the above.
4.
What type of probability distribution will most likely be used to analyze the number of cars with
defective radios in the following problem?
From an inventory of 48 new cars being shipped to local dealerships, corporate reports
indicate that 12 have defective radios installed. The sales manager of one dealership wants to
predict the probability that out of the 8 new cars it just received, when each is tested, no more
than 2 of the cars have defective radios.
a) Hypergeometric distribution.
b) Binomial distribution.
c) Poisson distribution.
d) Any of the above could be used
e) None of the above are appropriate
5.
For some value of z , the probability that a standardized normal variable is below z is 0.2090.
The value of z is
a) 1.96.
b) 0.31.
c) – 0.31.
d) – 0.81.
e) None of the above. Write in the correct solution.
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6.
If we know that the length of time it takes a college student to find a parking spot in the library
parking lot follows a Normal distribution with a mean of 3.5 minutes and a standard deviation of 1
minute, find the probability that a randomly selected college student will find a parking spot in the
library parking lot in less than 4 minutes.
[12 + 2, 26.5]
a) 0.1915
b) 0.2674
c) 0.3085
d) 0.3551
e) 0.6915
f) Almost 1.0000
g) None of the above is within .001 of being correct. Here is the correct answer.
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C. Exhibit 2: The following table contains the joint probability distribution for x and y . Note: This is
not a sample! All of your answers depend on the given probabilities.
Show your work neatly in questions 6-9.
x
y
0
1
2
0
.10
.20
.15
0
0
0
6
0
0
.20
.15
.10
.10
Total
.10
.20
.35
.15
.10
.10
3
4
5
Questions are 2 each except as marked.
1. Referring to Exhibit 2 the probability that x is at least 2 is ________.
In Exhibit 2 define two events: let A  x  2 and B  y  6
2. P A  Px  2  ________. (1)
 
3. P B A ________.
4. P A  B  ________.
5. P A  B  ________. [9]
6. In Exhibit 2, the events A and B are
a) Both independent and mutually exclusive
b) Mutually exclusive, collectively exhaustive and complements
c) Complements
d) Collectively exhaustive
e) Mutually exclusive
f) Independent
g) None of the above
h) All of the above.
7. Referring to Exhibit 2, the mean or expected value of y is ________.
8. Referring to Exhibit 2, the standard deviation of y is ________.
9. Referring to Exhibit 2, the covariance of x and y is ________. (4) [19]
10. Referring to Exhibit 2, the correlation between x and y is ________. [21, 47.5]
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D. Mostly 2 points each.
1. A multiple-choice test has 30 questions. There are 4 choices for each question. A student who
has not studied for the test decides to answer all questions randomly. What type of probability
distribution can be used to figure out his chance of getting at least 20 questions right? (1)
a) Poisson distribution.
b) Hypergeometric distribution.
c) Binomial distribution.
d) Geometric Distribution
e) Continuous Uniform Distribution
f) None of the above.
2. The number of 911 calls in Butte, Montana, has a Poisson distribution with a mean of 15
calls a day. The probability of between six and ten calls to 911 in a day is ____________.
3 A debate team of 4 members for a high school will be chosen randomly from a potential group
of 16 students. Three quarters of the 16 students have no prior competition experience while the
others have some degree of experience. What is the probability that exactly half of the members
chosen for the team have some prior competition experience? (3)
V2-4 Assume instead that there are 84 students to pick our 4 from Show that you can find the
probability that half the students have prior experience by an easier method than you used
previously and do the problem by that method.
V2 -5 Can you do problem 4 using the Poisson distribution? Explain why or why not.(1) [9]
V2-6 In 2000 estimates suggested that only 25% of the home-based computers have access to
on-line services. Suppose people with home-based computers were randomly and independently
sampled. Find the probability that the first person that has access to on-line services is between
the 5th and the tenth person sampled.
[11]
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D.(ctd.) 8. A factory has three veeblefetzers. Veeblefetzer A produces defective items 4 percent of
the time and thus is only used for 8% of the output. Veeblefetzer B produces 24% of the output,
and 3% of its output is defective. Because only 2% of the output of Veeblefetzer C is defective,
68% of the output is produced on Veeblefetzer C. If a defective item is found, what is the
probability that it comes from Veeblefetzer C .
Before you do this problem let event A be that an output unit was produced by Veeblefetzer A.
(Define events B and C similarly.) Let event D be the event that a unit is defective, so that the
proportion of the total output that is both produced by Veeblefetzer A and defective is P A  D  .
List the numbers given by the problem and identify them by probabilities like P A  D  , P A or


P A  D . (I am nor implying that any of these particular probabilities was actually given. Then
identify what you are looking for the same way and do the problem. (3)
9. In problem 8, the proportion of output that comes from Veeblefetzer C and is not defective is
____________.
[16, 63.5]
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E. Short answer. Show your work!
A jorcillator’s lifespan (failure time) can be represented by a continuous uniform distribution
between 75 and 350 years
1. What is the probability that it lasts between 100 and 200 years? (1)
2. What is the probability that it lasts between 200 and 1000 years? (1)
3. What is the mean life of such a jorcillator ? (1)
4. What is the standard deviation of the life of such a jorcillator ? (1)
5. If I have two such jorcillators, what is the probability that both fail before the 100 th year?
6. If I have six such jorcillators, what is the probability that at least one lasts between 200 and
1000 years? (2)
? (2) [7}
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E(ctd)
The average weekly starting salary of dogcatchers is $1600 with a standard deviation of $88.
We take a random sample of 64 dogcatchers.
7. What are the mean and standard deviation of the average salaries of the 64 dogcatchers? (1)
8. What is the probability that the mean from our sample will be above $1578 (2)
9. If the sample of 64 is taken from a population of 320 dogcatchers, what is the chance that the
sample mean will be above $1578. (2)
10. Faced with a plague of abandoned dogs, West Chester Borough hires 64 randomly selected
dogcatchers from a nationwide pool of 320. It budgets $100,992.00 to pay them for the next week.
What is the probability that this is not enough money? (2) [14]
11. If I compute a sample mean from four successive samples of 64 dogcatchers, what is the
probability that all three will be below $1600? (2)
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F. A local health club is trying to put together plans for expansion.
1. A number of free-lance trainers use the club facilities and the club takes a survey of 12 of them
with the following results.
Hourly Wages 8.17 9.23 10.50 7.55 9.45 8.65 8.20 10.35 13.50 9.60 6.10 7.40
Minitab has kindly given us the following sample statistics.
Variable
x
n
12
Mean
9.058
SE Mean StDev
0.546 1.893
Minimum
Q1
6.100 7.705
Median
8.940
Q3
Maximum
10.163 13.500
The club has also taken a random sample of 200 members and found that 160 members favor the
construction of improved (and more expensive) facilities.
Construct the following 98% confidence intervals. (2 each)
1) Assume that the population standard deviation is 1.00 and construct a confidence interval for the trainers’
mean hourly wage.
2) Change the confidence interval in 1 to reflect the fact that the sample of 12 was chosen randomly from
the 40 trainers who actually work regularly at the club.
3) Change the confidence interval in 2 to reflect the fact that we really aren’t too sure about the population
standard deviation and prefer to use the results of our sample survey.
4) Construct a confidence interval for the proportion of the club members that favor the construction of
improved facilities.
5) Modify your confidence interval in 4 to reflect the fact that the sample of 200 people was taken in a club
that only has 500 members.
6) Change the confidence level for the interval in 5 to 87%. Hint: You probably have the number you need
already and should not waste your time trying to find it on the t-table.
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