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Subject
Mathematics
Date
Apr 3, 2024
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MATH 3342-022 Test 1
Fall 2022
Test 1
Key
Name (Last, First):
INSTRUCTIONS
This exam consists of two parts.
Part I is comprised of 10 multiple-choice questions (no partial
credit) and is worth 50 points.
Part II consists of partial-credit-type questions and is worth 50
points.
Write the answer of your choice, A-E, to the Part I questions in the
Answer:
space provided.
Use CAPITALS, e.g. B insteab of b.
Give complete answers to the Part II questions in order to get maximum credit. Even if you do not
know the answer try to write something down; remember that blank space equals zero points.
You are allowed to use a formula sheet consisting of no more than
1 sheet (2 pages)
of 8
.
5
×
11
paper. A standard scientific calculator is permitted (and necessary). Usage of a smartphone is per-
mitted provided it is ONLY used as a scientific calculator (you may NOT use it as a communication
device, to access the internet, stored text, etc.).
This
Technometrics Review Times
will be used in some Questions.
The time (in months)
taken to review articles submitted to the journal
Technometrics
on a specific year is given below as
frequency table (the last column gives a few of the corresponding relative frequencies):
Time (months)
Number of articles
(Relative Frequency)
0 to
<
1
45
1 to
<
2
17
2 to
<
3
18
0.105
3 to
<
4
19
4 to
<
5
12
0.070
5 to
<
6
14
6 to
<
7
13
0.076
7 to
<
8
22
8 to
<
9
11
0.064
PART I
1. Refer to the
Technometrics Review Times
dataset. What percentage of this sample of articles
had review times under 7 months?
(a) 19.3%
(b) 13%
(c) 77.8%
(d) 80.7%
***
(e) None of the above
Answer:
1
MATH 3342-022 Test 1
Fall 2022
2. Refer to the
Technometrics Review Times
dataset. We can say that the median review time
for this sample of articles is:
(a) somewhere between 2 and 3 months
(b) somewhere between 3 and 4 months
***
(c) somewhere between 4 and 5 months
(d) exactly 4.5 months
(e) None of the above
Answer:
3. An engineer reaches into a large box containing a shipment of 10,000 bolts just received from a
vendor, and selects the first 10 bolts that are most easily accessible on top for inspection. Which of
the following statements most accurately describes the type of population and sampling mechanism.
(a) The population of 10,000 bolts is
concrete
, and the 10 selected bolts constitute a
convenience
sample.
***
(b) The population of 10,000 bolts is
hypothetical
, and the 10 selected bolts constitute a
convenience
sample.
(c) The population of 10,000 bolts is
concrete
, and the 10 selected bolts constitute a
simple random
sample.
(d) The population of 10,000 bolts is
hypothetical
, and the 10 selected bolts constitute a
stratified
sample.
(e) It would be more appropriate to view the 10 bolts as the population, and the 10,000 bolts as
a sample.
Answer:
4. Consider the following 11 observations on shear strength (MPa) of a bonded joint:
4
.
4
,
16
.
4
,
21
.
8
,
30
.
0
,
33
.
1
,
36
.
6
,
40
.
4
,
66
.
7
,
72
.
8
,
81
.
5
,
108
.
6
Compute the lower fourth (or first quartile).
(a) 21.8
(b) 25.9
***
(c) 30
(d) 36.6
(e) None of the above
Answer:
5. Refer to Question 4. Compute the fourth spread,
f
S
(or interquartile range). (Choose the closest
value.)
(a) 36.6
(b) 43.9
***
(c) 51.0
(d) 51.5
(e) None of the above
Answer:
2
MATH 3342-022 Test 1
Fall 2022
6. Which of the following statements is always
true for any two events
A
and
B
?
(a)
P
(
A
∪
B
) =
P
(
A
) +
P
(
B
)
(b)
P
(
A
∪
B
) =
P
(
A
) +
P
(
B
)
-
P
(
A
∩
B
)
***
(c)
P
(
A
∪
B
) =
P
(
A
)
·
P
(
B
)
(d)
P
(
A
∩
B
) =
P
(
A
)
·
P
(
B
)
(e) None of the above.
Answer:
7. How many permutations of size 3 can be constructed from the elements of the set
{
A, B, C, D, E
}
?
(a) 10
(b) 15
(c) 20
(d) 60
***
(e) None of the above
Answer:
8. A pool of 70 people is available for selection as jurors in a court case. From this pool of 70, the trial
lawyers will randomly select 12 people to form the jury panel. How many different 12-member jury
panels can be formed in this way?
(a) 12!
(b)
70!
12!
(c)
70!
58!
(d)
70!
(12!)(58!)
***
(e) None of the above
Answer:
9. A production facility employs 20 workers on the day shift, 15 workers on the swing shift, and 10
workers on the graveyard shift. Suppose 8 of these workers are to be randomly selected. Compute
the probability that all 8 selected workers are from the day shift. .
(a)
(
20
8
)
/
(
45
8
)
***
(b)
(
15
8
)
/
(
45
8
)
(c)
(
10
8
)
/
(
45
8
)
(d) 20
/
45
(e) None of the above
Answer:
10. Refer to Question 9. What is the probability that when randomly selecting the 8 workers, we end
up with
at least one
worker from each shift?
(a) [
(
20
5
)
+
(
15
5
)
+
(
10
5
)
]
/
(
45
8
)
(b) [
(
20
8
)
+
(
15
8
)
+
(
10
8
)
]
/
(
45
8
)
(c) [
(
19
8
)
+
(
14
8
)
+
(
9
8
)
]
/
(
45
8
)
(d) [
(
19
5
)
+
(
14
5
)
+
(
9
5
)
]
/
(
45
8
)
(e) None of the above
***
Answer:
3
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MATH 3342-022 Test 1
Fall 2022
PART II
1. (10 points) Consider the
Technometrics Review Times
dataset in Part I.
(a) Compute the five relative frequency values that are missing from the table.
Time (months)
Number of articles
(Relative Frequency)
0 to
<
1
45
0.263
1 to
<
2
17
0.099
3 to
<
4
19
0.111
5 to
<
6
14
0.082
7 to
<
8
22
0.129
(b) Construct a relative frequency histogram for these data (using the entire set of relative fre-
quencies), and comment on the shape of the data (skewed or symmetric).
0 to 1
1 to 2
2 to 3
3 to 4
4 to 5
5 to 6
6 to 7
7 to 8
8 to 9
Histogram of Technometrics
Review Times
0.00
0.05
0.10
0.15
0.20
0.25
Approximately symmetric, but the first class makes it a little right-skewed.
2. (10 points) Construct a boxplot for
n
observations where you are given only the following:
x
(1)
=
-
3
,
x
(2)
= 4
,
Q
1
= 15
,
Q
2
= 20
,
Q
3
= 25
,
x
(
n
-
2)
= 39
,
x
(
n
-
1)
= 43
,
x
(
n
)
= 50
[Note that
Q
i
, i
= 1
,
2
,
3 denotes the
i
-th quartile or fourth. Recall that the whiskers only extend
up to points that are not outliers; and the latter should be individually marked.]
Since 1
.
5(
Q
3
-
Q
1
) = 1
.
5(10) = 15, any point below 0 or above 40 is an outlier. Thus, the box goes
from
Q
1
to
Q
3
, left whisker from 4 to
Q
1
, and right whisker from
Q
3
to 39.
Put a line through box at the median value of 20.
The min value of -3 and the two largest values of 43 and 50 are all outliers.
4
MATH 3342-022 Test 1
Fall 2022
3. (15 points) A large company offers its employees two different health insurance plans and two
different dental insurance plans. Plan A of each type is relatively inexpensive, whereas plan B is
more expensive. The following table gives the percentages of employees who have chosen the various
plans. (Note that each employee chooses exactly one health and one dental plan.)
Health
Dental Plan
Plan
A
B
Total
A
27%
14%
41%
B
24%
35%
59%
Total
51%
49%
100%
Compute the following probabilities for a randomly selected employee.
(a) The probability that the employee chooses the cheaper of each plan (Plan A).
The joint probability is:
P
(Health A and Dental A) = 0
.
27
(b) The probability that the employee chooses dental Plan B.
The marginal probability is:
P
(Dental B) = 0
.
14 + 0
.
35 = 0
.
49
(c) If we randomly select the employee from among those who chose dental Plan B, what is the
probability that they will also have chosen health Plan B.
The conditional probability is:
P
(Health B
|
Dental B) =
P
(Health B
∩
Dental B)
P
(Dental B)
=
0
.
35
0
.
49
= 0
.
71
5
MATH 3342-022 Test 1
Fall 2022
4. (15 points) An executive on a business trip must rent a car in each of two different cities. Let A
denote the event that the executive is offered a free upgrade in the first city, and B the event he’s
offered a free upgrade in the second city. Suppose that
P
(
A
) = 0
.
3,
P
(
B
) = 0
.
4, and that
A
and
B
are independent events.
(a) If the executive is not offered a free upgrade in the first city, what is the probability of not
getting a free upgrade in the second city? (Explain the reasoning behind your computation.)
Want
P
(
B
0
|
A
0
), but since
A
and
B
are independent, the same follows for the compliments, so
that:
P
(
B
0
|
A
0
) =
P
(
B
0
) = 0
.
6
(b) Show that the probability that the executive is offered a free upgrade in at least one of the two
cities is 0
.
58.
Want
P
(
A
∪
B
), which we obtain from the Multiplicaion Rule, and using the fact that
A
and
B
are independent:
P
(
A
∪
B
)
=
P
(
A
) +
P
(
B
)
-
P
(
A
∩
B
)
=
P
(
A
) +
P
(
B
)
-
P
(
A
)
P
(
B
)
=
0
.
3 + 0
.
4
-
(0
.
3)(0
.
4)
=
0
.
58
(c) If the executive is offered a free upgrade in at least one of the two cities, what is the probability
that such an offer was made only in the first city?
Want
P
(
A
∩
B
0
), but conditional on the event
{
A
∪
B
}
. From the def. of conditional probability,
and using the indep. of A & B:
P
(
A
∩
B
0
|
A
∪
B
)
=
P
((
A
∩
B
0
)
∩
(
A
∪
B
))
P
(
A
∪
B
)
=
P
(
A
∩
B
0
)
P
(
A
∪
B
)
=
P
(
A
)
P
(
B
0
)
P
(
A
∪
B
)
=
(0
.
3)(0
.
6)
0
.
58
= 0
.
31
6
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