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NA387(3)
Exam #1 - SOLUTIONS
February 13, 2007
60 minutes (but you have the full 80 mins to do it — closed book and notes — 105 points ( 5 are
bonus)
One formula sheet allowed, 8.5 x 11", one-sided
Before you start, please
1.
Be sure you have written and signed the honor pledge on the cover of your bluebook. I am not allowed
to grade it otherwise.
2.
Take some time and read each question carefully, checking if all questions are clear to you.
3.
Solve, do not just answer, the non-multiple choice, non-fill-in-the-blanks type problems; show all work
clearly; partial credit will be given whenever appropriate.
4.
Clearly state your final answer to each question and cross out any material you don't want graded.
GOOD LUCK!
Problem 1 (29%) : Fill in the blanks (1 point per blank)
1.
The __________ of an experiment is the set of all possible outcomes of that experiment.
ANSWER: sample space
2.
An __________ is any collection (subset) of outcomes contained in the sample space.
ANSWER: event
3.
A pictorial representation of events and manipulations with events is obtained by using
__________. (two words)
ANSWER: Venn diagrams
4.
If A = {3, 5, 7} and B = {2, 4, 6}, then A and B are said to be __________ events.
ANSWER: mutually exclusive
5.
For any event A, P(A) should be larger than or equal to __________ , but smaller than or equal to
__________.
(insert numbers)
ANSWER: 0,1
6.
If P(A) = .70, P(B) = .60, and P ( A  B) = .90, then P ( A  B) = __________.
ANSWER: .40
7.
If the first element of an ordered pair can be selected in 6 ways, and for each of these 6 ways the
second element of the pair can be selected in 4 ways, then the number of pairs is __________.
1
ANSWER: 24
8.
If a six-sided die is tossed three times in succession, then there are __________ possible different
outcomes (3-tuples).
ANSWER: 216
9.
If P(A) = .35, P(B) = .50 and P ( A  B) = .20, then P(A/B) is __________.
ANSWER: .40
10.
If P ( A  B) = .123 and P(A/B) = .60, then P(B) is __________.
ANSWER: .205
11.
Two events A and B are __________ if P(A/B) = P(A).
ANSWER: independent
12.
If P(A) = .25, P(B) = .40, and events A and B are independent, then P ( A  B) is __________.
ANSWER: .10
13.
Two events A and B , where P ( A  B)  P( A)  P( B) , are _____________.
ANSWER: dependent
14.
If P(A) = .30, P(B) = .20, and events A and B are independent, then P ( A  B) is __________.
ANSWER: .14
15.
“__________ statistics” involves summarizing and describing important features of the data.
ANSWER: Descriptive
16.
Techniques for using sample information to draw dependable conclusions about the population are
gathered within the branch of statistics called “__________ statistics”.
ANSWER: inferential
17.
The number of traffic citations a person will receive during 2008 is not an example of a
__________ variable.
ANSWER: continuous
18.
The median of the data set 8, 20, 18, 8, 9, 16, 10, and 12 is __________.
ANSWER: 11
19.
A 5% trimmed mean would be computed by eliminating the smallest__________% and the largest
2
__________% of the sample values, and then averaging the remaining data values.
ANSWER: 5,5
20.
The __________, __________, and __________ are the three most common measures of location.
ANSWER: mean, median, mode
21.
The simplest measure of variability in a sample is the __________, which is the difference
between the largest and smallest sample values.
ANSWER: range
n
22.
The sum of the deviations from the mean; namely
 ( x  x ), is always __________.
i i
i
ANSWER: zero
23.
The sample variance of the data set 5,5,5,5, and 5 is __________.
ANSWER: zero
24.
For a sample of size 4, if
x1  x  10, x2  x  8, and x3  x  6, then the sample variance
is equal to __________.
ANSWER: 72
25.
If sx  16 and yi  3 xi for all i  1, 2,...., n, then the sample variance of the y’s is
2
__________.
ANSWER: 144
3
Problem 2: (44%) Multiple-Choice Questions (2 points each)
1.
Which of the following statements is always true?
A. The complement of an event A, denoted by A , is the set of all outcomes in the sample space that
are not contained in A.
B. The union of two events A and B, denoted by A  B , is the event consisting of all outcomes that
are in both events.
C.
The intersection of two events A and B, denoted by A  B , is the event consisting of all
outcomes that are either in A or in B.
D. All of the above
ANSWER: A
2.
A.
B.
C.
D.
The sample space for the experiment in which a six-sided die is thrown twice consists of how
many outcomes?
6
36
12
None of the above
ANSWER: B
3.
Which of the following is true for events M and N?
( M  N )  M   N 
B. ( M  N )  M   N 
C. Neither A nor B is true
D. Both A and B are true
A.
ANSWER: D
4.
For any two events A and B,
A. P ( A  B) = P(A) + P(B)
B. P ( A  B) = P( A)  P( B)
C. P ( A  B) = P(A) + P(B) - P ( A  B)
D. P ( A  B) = P( A)  P( B)
ANSWER: C
5.
If A and B are mutually exclusive events, then
P ( A  B) = 0
P ( A  B) = 0
P ( A  B) = P ( A  B)
All of the above
ANSWER: B
4
If P(A) =.35 and P(B) = .45, then P ( A  B)
6.
A. is .10
B. is .80
C. is .20
D. Cannot be determined from the given information
ANSWER: D
7.
A.
B.
C.
D.
A homeowner doing some remodeling requires the services of both a plumbing contractor and an
electrical contractor. If there are 15 plumbing contractors, and 10 electrical contractors available
in the area, in how many ways can the contractors be chosen?
15
10
25
150
ANSWER: D
8.
A.
B.
C.
D.
How many permutations of size 3 can be constructed from the set (A, B, C, D, E)?
60
20
15
8
ANSWER: A
9.
A.
B.
C.
D.
How many combinations of size 4 can be formed from a set of 6 distinct objects?
36
24
15
10
ANSWER: C
10.
Which of the following statements is true?
A. The number of combinations of size k from a particular set is larger than the number of permutations.
B. The number of circular permutations of size k from a particular set is smaller than the number of
permutations.
C. The number of combinations of size k from a particular set is equal to the number of permutations.
D. The number of circular permutations of size k is larger than the number of permutations.
ANSWER: B
11.
Let A1 , A2 ,.... Ak be mutually exclusive and exhaustive events. Then for any other event B, P(B) =
P( B / A1 ) P( A1 ) 
 P( B / Ak ) P( Ak ) is well known as
5
A.
B.
C.
D.
De Morgan’s laws of probability
The multiplication rule
Bayes’ theorem
The law of total probability
ANSWER: D
12.
A.
B.
C.
D.
If P(B) = .40 and P(A/B) = .75, then
P(A) cannot be determined from the given information
P(A) = .35
P(A) = .30
P(A) = .25
ANSWER: A
13.
Let A1 and A2 be mutually exclusive and exhaustive events, with P ( A1 ) = .10 and P ( A2 ) = .25.
Let B be any event such that P ( B / A1 ) = .50 and P ( B / A2 ) = .80, then P ( A1 / B) is
A.
B.
C.
D.
.40
.25
.20
.05
ANSWER: C
14.
A.
B.
C.
D.
For events A and B with P(B) > 0, which of the following is always true?
P ( A / B)  P( B / A)  P( A  B)
P (A/B) = P(B/A)
P (A/B) + P(B/A) = 1
P (A/B) + P ( A / B) = 1
ANSWER: D
15.
A.
B.
C.
D.
E.
Which of the following is (are) true if events A and B are mutually exclusive with P(B) > 0?
P ( A  B) = 0
P(A/B) = 0
A and B cannot be independent
None of the above is true
All of the above are true
ANSWER: E
16.
If P(A) = .30, P(B) = .10 and events A and B are independent, then P ( A  B) is
A. .63
B. .70
C. .90
6
D. .40
ANSWER: A
17.
Which of the following statements are correct?
A. Constructing a histogram for continuous data (measurements) entails subdividing the
measurement axis into a suitable number of class intervals or classes, such that each
observation is contained in exactly one class.
B. The reaction time to a particular stimulus is an example of a discrete variable.
C. Constructing a histogram for discrete data is generally not different from constructing a
histogram for continuous data.
D. The total area of all rectangles in a histogram is 100.
ANSWER: A
18.
Which of the following statements are false?
A. A physical interpretation of the sample mean x demonstrates how it measures the location
(center) of a sample.
B. The sample median, represented by x, is the middle value when the observations are ordered
from smallest to largest.
C. The sample median is very sensitive to extremely small or extremely large data values
(outliers).
D. The sample mean is very sensitive to extremely small or extremely large data values
(outliers).
ANSWER: C
19.
Which of the following is not a measure of location?
A.
B.
C.
D.
E.
The mean
The median
The mode
The variance
All of the above
ANSWER: D
20.
Which of the following statements are true?
A. The population mean  and population median  will not generally be equal to one
another.
B. There are four quartiles; they divide the data set into four equal parts.
C. A 10% trimmed mean would be computed by eliminating the smallest 5% and the largest 5%
of the sample values and then averaging what is leftover.
D. All of the above.
ANSWER: A
7
21.
Which of the following is not a measure of variability?
A.
B.
C.
D.
E.
Variance
Standard deviation
Mean absolute deviation
Mode
all of the above
ANSWER: D
22.
Boxplots have been used successfully to describe
A.
B.
C.
D.
E.
center of a data set
spread of a data set
the extent and nature of any departure from symmetry
identification of “outliers”
All of the above
ANSWER: E
8
Problem 3
(12%) (Chapter 1 material:)
Consider the following data on bearing load-life (million revs.) for bearings of a certain type when
subjected to a 9.56 kN load appeared:
14.5
a.
b.
c.
66.3
69.3
25.6
52.4
69.8
73.1
Calculate the values of the sample mean and median. (4%)
What do their values relative to one another tell you about the sample? (3%)
Compute the values of the sample variance and standard deviation. (5%)
ANSWER:
371
= 53.0, whereas x  66.3
7
The fact that x < x suggests that observations on the lower end of the sample are more
stretched out than those on the upper end (a longer lower tail than upper tail).
[23,025.2  (371)2 / 7] 3362.2
b.  xi2  23,025.2, so s 2 

 560.367 and
6
6
a.
n =7, n  7,  x  371, so x 
s  560.367  23.672
Problem 4 (8%) Chapter 1 material
A sample of 20 glass bottles of a particular type was selected, and the internal pressure strength of each
bottle was determined. Consider the following partial sample information:
Median = 202.2
lower fourth = 196.0
Upper fourth = 216.8
Three smallest observations
Three largest observations
125.8
221.3
188.1
230.5
193.7
250.2
Are there any outliers in the sample? Any extreme outliers? Which ones are they?
ANSWER:
1.5(IQR) = 1.5(216.8-196.0) = 31.2 and 3(IQR) = 3(216.8-196.0) = 62.4.
Mild outliers: observations below 196-31.2 = 164.6 or above 216.8+31.2=248.
Extreme outliers: observations below 196-62.4 = 133.6 or above 216.8+62.4 = 279.2.
Of the observations given, 125.8 is an extreme outlier and 250.2 is a mild outlier.
Problem 5:
(Combinations, permutations etc) (12%)
A real estate agent is showing homes to a prospective buyer. There are ten homes in the desired price
range listed in the area. The buyer has time to visit only four of them.
a. (3%)
In how many ways could the four homes be visited if the order of visiting is considered?
b. (3%)
In how many ways could the four homes be visited if the order is disregarded?
c. (6%)
If four of the homes are new and six have previously been occupied and if the four homes
to visit are randomly chosen, what is the probability that all four are new? ((calculate twice, once for the
case in a. and once for b. above)
9
ANSWER:
a. (10)(9)(8)(7) = 5040
 10 
10!
 210
b.   =
4!6!
4 
c.
P(all 4 are new) =
4!
1

5040 210
4
 
4
1
P(all 4 are new) = (# of ways of visiting all new)/(# of ways of visiting) =   =
210
 10 
 
4 
10