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ECO 251 QBA1
FINAL EXAM, Version 1
MAY 7 and 9, 2007
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 10, 7.9
1. Px  14 
2. P0  x  9
3. F 2.00  (Cumulative Probability)
4. P2  x  36 
5. P10  x  10 
6. x.34
7. A symmetrical region around the mean with a probability of 32%.
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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!
(Webster) A local bar sells ’16 oz’ glasses of beer. A group of students buys 22 glasses of beer and, using
their own measuring cup, tries to estimate the mean contents. The measurements are below.
13.5
15.3
13.8
15.4
14.2
15.4
14.3
15.6
14.6
15.6
14.6
15.8
14.6
15.8
22
Using Minitab, I have computed the following

i 1
14.9
15.9
15.0
16.1
15.1
16.7
15.3
16.9
20
x  334 .4 and
x
2
 4533 .84 .
i 1
a) Note that the sum of squares is the sum of only the first 20 numbers. To show me that you know how to
x 2 using the total that I have given to you. (1)
compute a sum of squares, compute

b) Compute the sample variance of the contents of the 22 glasses given above. (2)
c) Compute a 99% confidence interval for the mean contents (3)
d) Is the mean significantly different from 16 oz.? Why? (1)
e) Because of all the hoopla around the students’ experiment only 100 glasses of beer were sold that night.
Assuming that 100 is the population size from which the sample of 22 was taken, recompute the confidence
interval in c). (2) [9]
f) Assume that the costs to the bar owner in drawing a glass of beer are $0.15 per glass plus $0.015 per oz,
use the mean and variance that you found in b) to compute the mean and variance of the costs to the bar
owner. (The formula relating cost to ounces is w  0.015 x  0.15 .) There will be no credit in this section
unless you use the mean and variance from b). (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 on the previous page. The data is repeated here. x is volume in ounces. y is
simply the order in which the beers were drawn. The students believe that the bartender got more generous
as the evening passed. To test this they computed the correlation of the volume with the order. You have
 x and  x
2
from the previous problem. It is quite easy to compute
 y  253
and
y
2
 3795
20
and Minitab says
 xy  3237 .3
i 1
1) Note that the sum xy is the sum of only the first 20 numbers. To show me that you know how to
compute a sum of this type, compute
 xy
using the total that I have given to you. (1)
2). Compute the sample covariance s xy between x and y . (2) Explain what this covariance tells us. (0.5)
3) Compute the sample correlation rxy between x and y . (2) Explain what this correlation tells us about
the relationship between the size of the beer and the order in which it was drawn. (1) [6.5]
4) Using the correlation and covariance you computed in b) and c) and the equation relating the size of the
beer and its cost that you used on the last page, find the covariance and correlation between cost of the beer
to the bar owner and the order in which it was drawn. (4) [10.5]
5) The true test of a correlation is a significance test which not only looks at the size of the correlation, but
r
r
the size of the sample. To do such a test compute t 
. On the assumption that there is no

sr
1 r 2
n2
significant correlation and using a 99% confidence level, 99% of the values of this t-ratio will fall between
n  2)
n  2)
 t .(005
and t .(005
. If your value of the t ratio does not fall between these two values, you can say that there
is significant correlation. Find the appropriate values of t, compute the t-ratio and tell me if the correlation
is significant. (5) [15.5]
Row
1
2
3
4
5
6
7
8
9
10
11
x
13.5
13.8
14.2
14.3
14.6
14.6
14.6
14.9
15.0
15.1
15.3
y
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
15.3
15.4
15.4
15.6
15.6
15.8
15.8
15.9
16.1
16.7
16.9
12
13
14
15
16
17
18
19
20
21
22
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B. Let us define the following events. A1 : y  3, A2 : y  5 , B1 : x  1 B 2 : x  2 .
B1
A1
A2
B2
1) Create a joint probability table like the one at the left on the assumption that the
four events are the only ones that can occur, P A1   .4 , PB1   .3 and that A1 and B1 are independent.
(3)
2) Compute the variance of y and the covariance and correlation between x and y in 1). (3)
3) Repeat 1) on the assumption that A1 and B1 are independent are mutually exclusive. (2)
4) Compute the variance of y and the covariance and correlation between x and y in 3). (3)
5) Use the addition rule to show that if P A1   .4 and PB1   .3 , these two events cannot be collectively
exhaustive no matter what the relationship between these two events. (1) [12]
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C. Answer the following 6 multiple choice questions. (These should be 2 each, but to discourage guessing,
how about 2.5 each for right answers and 0.5 penalty for wrong answers.)
1) Which of the following is a major difference between the binomial and the hypergeometric
distributions?
a). The sum of the outcomes can be greater than 1 for the hypergeometric
b) The probability of a success changes in the hypergeometric distribution.
c) The number of trials changes in the hypergeometric distribution
d) The outcomes cannot be whole numbers in the hypergeometric distribution
e) None of the above is correct.
2) The continuity correction factor is used when
a) The sample size is at least 5.
b) Both np and nq are at least 30.
c) A continuous distribution is used to approximate a discrete distribution
d) A discrete distribution is used to approximate a continuous distribution
e) A binomial distribution is used to approximate the hypergeometric distribution
f) None of the above.
3) Suppose a population consisted of 20 items. How many different samples of n = 3 are possible?
a) 20
b) 40
c) 120
d) 20 3  8000
e) 1140
f) 6840
g) None of the above is correct.
4) The finite population correction factor is used when
a) n/N is more than .05.
b) N is more than 1000
c) np is greater than 5.
d) n is more than 30.
e) None of the above
5) In the formula   x  z  x , the 
2
2
is
a) The confidence level
b) The area of one tail of the sampling distribution of the mean
c) The probability that the interval would not include the mean
d) The proportion of confidence intervals that will contain the mean
e) None of the above.
6) All Normal distributions have
a) A finite range
b) The same coefficient of variation
c) The same probability density function f x 
d) The same area between    and   2
e) All of the above must be true.
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D.
1) We are baking chocolate chip cookies again. Assume that all cookies are discarded that have
less than 4 chips, but the mean is only 4.5, what proportion of cookies will be discarded. (1)
2) In view of the results of problem 1 and using only the distributions for which you have tables,
what is the lowest value the mean can take if we want to be sure that no more than 1% of the cookies are
discarded?
3) Find P1  x  15  for the following distributions. If you must substitute one distribution for
another, show that the substitution criterion is satisfied.
a) Poisson with parameter of 2.5 (1)
b) Poisson with parameter of 36 (2)
c) Binomial with n  35 and p  .02 . (2)
d) Binomial with n  35 and p  .2 (2)
e) Binomial with n  50 and p  .2 (1)
f) Binomial with n  50 and p  .55 (2)
[13]
g) Continuous uniform c  8 and d  13 (1)
4) Find Px  1 for the following.
a) Hypergeometric N  16 , p  .25 , n  5 (2)
b) Hypergeometric N  600 , p  .25 , n  5 (2)
c) Geometric p  .25 (1)
[19 – 58.5]
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E.
1) Do the following confidence intervals.
(Webster) Assume that a real estate developer wants to estimate the mean family income in an area
where a mall is proposed. The developer assumes that the population standard deviation for
income is $7200
a) If a survey of 100 families yields a sample mean of $35500, create a 90% confidence
interval for the mean. (3)
b) Find a confidence interval using the data in a) with a 32% confidence level. (2)
c) Assume a 90% confidence level again and that the sample of 100 families comes from
a neighborhood of only 1000 families. Repeat a) (3)
d) (Extra Credit) Assume that your results in a) are correct and that there are 30000
families in the target area. Can you make the 90% confidence interval in a) into a
confidence interval for total income? (2)
d) Is the income found in a) significantly different from $36000? Why? (1) [9]
e) A Business Week article claims that 25% of CEOs are ‘outsiders.’ A survey of 350
corporations is taken to check this ‘fact’ and the survey finds that 77 of the 350 firms
have outsider CEOs. Create a 90% confidence interval for the proportion of firms that
have outsider CEOs. Does this conflict with the statement in Business Week? Why? (4)
2) The amount of time a bank teller spends with each customer has a population mean   3.10
minutes and a population standard deviation   0.40 minute.
a) If a random sample of 16 customers is selected from a large normally distributed
population, what is the probability that the average (mean) time spent per customer is less
than 3 minutes? (2.5)
b) How would this change if the sample was taken from a population of only 144
customers? Give a specific answer! (2)
c) If the teller puts in a 420 minute day, and the conditions in 2a) prevail, what is the
probability of serving 150 customers or more? (2.5) [20]
d) Under the circumstances in a), what is the probability that the time spent with 1 of the
16 customers is less than 3 minutes? (2)
e) Under the circumstances in a), what is the probability that the time spent with each of
the 16 customers in 2a) is less than 3 minutes apiece? (2)
[26 – 84.5]
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F.
1) (Dunleavy) Assume that for every 100,000 people treated in an emergency room, 5,000 are
misdiagnosed, and 1,000 misdiagnosed patients die. Overall, 90,000 of the 100,000 live through
the experience.
Define the following events MD misdiagnosed ; S survives ; MD not misdiagnosed and
S dies.
Don’t just guess, make a table. Distinguish clearly between conditional and joint probabilities.
a) What is the probability that a patient will be diagnosed correctly? (1)
b) What is the probability that a patient will be diagnosed correctly and live? (1)
c) What is the probability that a patient will die given that they were diagnosed correctly?
(2)
d) Are the diagnosis and survival independent? (1)
e) If a patient dies, what is the probability that the patient was not misdiagnosed? (2) [7]
f) What is the value of P MD  S ?
2) OK. I ran out of ideas, so here is a mini-Jorcillator problem.
Here is the joint probability table and the table relating joint events to failure times.


Failure of the phillinx in period 1, 2, 3, 4 are events A1, A2 , A3 , and A4 .
Failure of the flubberall in period 1, 2, 3, 4 are events B1, B2 , B3 , and B4 .
Failure of the jorcillator in period 1, 2, 3, 4 are events C1, C2 , C3 , and C 4 .
a) Assuming that the Jorcillator lives as long as one of its components lives, fill in the
second table with failure times of 1, 2, 3 and 4. (1)
A1
A2
A3
A4
B1
B2
B3
B4
.20
.12
.04
.04
.40
.15
.09
.03
.03
.30
.10
.06
.02
.02
.20
.05
.03
.01
.01
.10
b) Find PC1  (1)
B1
.5
.3
.1
.1
1.0
B2
B3
B4
A1
A2
A3
A4
c) Find PC 2  (1)
d) Find PC3  (1)
e) Find PC 4  (1)


f) Find P C3 A3 (1) [13 – 97.5]
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Blank page for calculations.
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