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Probability and Statistics final exam Fall semester review
Name___________________________________
Is the following the empty set?
1) {x|x is a day of the week whose name begins with thte letter Y}
2) {x|x is a number less than 2 or greater than 6}
Determine whether the statement is true or false.
3) 9 ∈ {2, 4, 6, ..., 20}
4) 16 ∉ {1, 2, 3, ..., 10}
Fill in the blank with either ∈ or ∉ to make the statement true.
5) 49,872 _____ the set of even natural numbers
Express the set using the roster method.
6) the set of natural numbers less than or equal to 8
Find the cardinal number for the set.
7) Determine the cardinal number of the set {x | x is a letter of the alphabet}
8) {27, 29, 31, 33, 35}
Are the sets equivalent?
9) A = {31, 33, 35, 37, 39}
B = {32, 34, 36, 38, 40}
Are the sets equal?
10) {4, 4, 11, 11, 17} = {4, 11, 17}
Write ⊆ or ⊈ in the blank so that the resulting statement is true.
11) {red, blue, green} _____ {blue, green, yellow, black}
Determine whether the statement is true or false.
12) {Ted} ⊆ {Bob, Carol, Ted, Alice}
1
Place the various elements in the proper regions of the following Venn diagram.
13) Let U = {g, h, j, k, m, n} and A = {g, h, n}. Find Aʹ. Then use a Venn diagram to illustrate the relationship among
the sets U, A, and Aʹ.
Use the Venn diagram to list the elements of the set in roster form.
14) The set of students who studied Saturday
15) The set of students who studied Saturday and Sunday
Let U = {1, 2, 4, 5, a, b, c, d, e}. Use the roster method to write the complement of the set.
16) T = {2, 4, b, d}
Let U = {q, r, s, t, u, v, w, x, y, z}
A = {q, s, u, w, y}
B = {q, s, y, z}
C = {v, w, x, y, z}. List the elements in the set.
17) A ∩ Bʹ
18) (A ∩ B)ʹ
19) Aʹ ∪ B
20) Cʹ ∩ Aʹ
2
Use the Venn diagram to list the elements of the set in roster form.
21)
A ∪ B
Use the formula for the cardinal number of the union of two sets to solve the problem.
22) Set A contains 5 elements, set B contains 11 elements, and 3 elements are common to sets A and B. How many
elements are in A ∪ B?
Solve the problem by applying the Fundamental Counting Principle with two groups of items.
23) There are 5 roads leading from Bluffton to Hardeeville, 7 roads leading from Hardeeville to Savannah, and 3
roads leading from Savannah to Macon. How many ways are there to get from Bluffton to Macon?
24) You are taking a multiple-choice test that has 6 questions. Each of the questions has 4 choices, with one correct
choice per question. If you select one of these options per question and leave nothing blank, in how many
ways can you answer the questions?
Use the Fundamental Counting Principle to solve the problem.
25) You want to arrange 10 of your favorite CDʹs along a shelf. How many different ways can you arrange the
CDʹs assuming that the order of the CDʹs makes a difference to you?
Use the formula for nPr to evaluate the expression.
26) 7P7
Use the formula for nPr to solve.
27) In a contest in which 8 contestants are entered, in how many ways can the 5 distinct prizes be awarded?
Solve the problem.
28) In how many distinct ways can the letters in ENGINEERING be arranged?
In the following exercises, does the problem involve permutations or combinations? Explain your answer. It is not
necessary to solve the problem.
29) How many different user IDʹs can be formed from the letters W, X, Y, Z if no repetition of letters is allowed?
30) Five of a sample of 100 computers will be selected and tested. How many ways are there to make this
selection?
Use the formula for nCr to evaluate the expression.
31) 6C5
3
32) From 8 names on a ballot, a committee of 3 will be elected to attend a political national convention. How many
different committees are possible?
Use the theoretical probability formula to solve the problem. Express the probability as a fraction reduced to lowest
terms.
33) Use the spinner below to answer the question. Assume that it is equally probable
that the pointer will land on any one of the five numbered spaces. If the pointer lands
on a borderline, spin again.
Find the probability that the arrow will land on an odd number.
Use the empirical probability formula to solve the exercise. Express the answer as a fraction. Then express the
probability as a decimal, rounded to the nearest thousandth, if necessary.
34) In 1999 the stock market took big swings up and down. A survey of 996 adult investors asked how often they
tracked their portfolio. The table shows the investor responses. What is the probability that an adult investor
tracks his or her portfolio daily?
How frequently? Response
Daily 222
Weekly 281
Monthly 292
Couple times a year 140
Donʹt track 61
Solve the problem.
35) Amy, Jean, Keith, Tom, Susan, and Dave have all been invited to a birthday party. They arrive randomly and
each person arrives at a different time. In how many ways can they arrive? In how many ways can Jean arrive
first and Keith last? Find the probability that Jean will arrive first and Keith will arrive last.
The chart shows the probability of a certain disease for men by age. Use the information to solve the problem. Express
all probabilities as decimals, estimated to two decimal places.
Age Probability of Disease X
20-24 less than 0.008
25-34 0.009
35-44 0.14
45-54 0.39
55-64 0.42
65-74 0.67
75+ 0.79
36) What is the probability that a randomly selected man between the ages of 55 and 64 does not have this disease?
4
The chart shows the probability of dying from four conditions in the U.S. Express all probabilities as decimals to three
decimal places. Assume all events are mutually exclusive.
Causes of Death Percentage of all Deaths
Disease A 30.3%
Disease B 23.0%
Disease C 5.8%
Disease D 4.7%
37) What is the probability of dying from disease A or B?
Solve the problem that involves probabilities with events that are not mutually exclusive.
38) In a class of 50 students, 31 are Democrats, 13 are business majors, and 3 of the business majors are Democrats.
If one student is randomly selected from the class, find the probability of choosing a Democrat or a business
major.
Solve the problem involving probabilities with independent events.
39) A spinner is used for which it is equally probable that the pointer will land on any one of six regions. Three of
the regions are colored red, two are colored green, and one is colored yellow. If the pointer is spun once, find
the probability it will land on green and then yellow.
Solve the problem that involves probabilities with events that are not mutually exclusive.
40) There are 37 chocolates in a box, all identically shaped. There are 10 filled with nuts, 12 with caramel, and 15
are solid chocolate. You randomly select one piece, eat it, and then select a second piece. Find the probability
of selecting 2 solid chocolates in a row.
The table shows claims and their probabilities for an insurance company.
Amount of Claim Probability
$0 0.60
$50,000 0.25
$100,000 0.09
$150,000 0.04
$200,000 0.01
$250,000 0.01
41) (a) Calculate the expected value.
(b) How much should the company charge as an average premium so that it breaks even on its claim costs?
(c) How much should the company charge to make a profit of $140 per policy?
Solve the problem that involves computing expected values in a game of chance.
42) A game is played using one die. If the die is rolled and shows a 2, the player wins $8. If the die shows any
number other than 2, the player wins nothing. If there is a charge of $1 to play the game, what is the gameʹs
expected value?
Determine whether the data are qualitative or quantitative.
43) the number of seats in a movie theater
44) the numbers on the shirts of a girlʹs soccer team
5
Decide which method of data collection you would use to collect data for the study. Specify either observational
study, experiment, simulation, or survey
45) A study where you would like to determine the chance getting three girls in a family of three children
Identify the sampling technique used.
46) A researcher randomly selects and interviews fifty male and fifty female teachers.
47) Based on 12,500 responses from 42,000 questionnaires sent to its alumni, a major university estimated that the
annual salary of its alumni was $96,500 per year.
Use the given frequency distribution to find the
(a) class width.
(b) class midpoints of the first class.
(c) class boundaries of the first class.
48) Height (in inches)
Class Frequency, f
50 - 52 5
53 - 55 8
56 - 58 12
59 - 61 13
62 - 64 11
Provide an appropriate response.
49) Use the ogive below to approximate the number in the sample.
50) For the stem-and-leaf plot below, what is the maximum and what is the minimum entry?
Key : 11 2 = 11.2
11
12
13
14
15
16
17
0 2
4 6 6 7 8 9
0 1 1 2 3 6 6 7 8 8
3 4 6 6 8 9 9 9
0 1 1 2 3 7 7 8 9
2 2 5 7 8 8 9 9
0 5
6
For the given data , construct a frequency distribution and frequency histogram of the data using five classes. Describe
the shape of the histogram as symmetric, uniform, negatively skewed, or positively skewed.
51) Data set: ages of 20 cars randomly selected in a student parking lot
12 6 4 9 11 1 7 8 9 8
9 13 5 15 7 6 8 8 2 1
Provide an appropriate response.
52) Use the histogram below to approximate the median heart rate of adults in the gym.
53) The top 14 speeds, in miles per hour, for Pro-Stock drag racing over the past two decades are listed below.
Find the median speed.
181.1 202.2 190.1 201.4 191.3 201.4 192.2
201.2 193.2 201.2 194.5 199.2 196.0 196.2
54) Find the range of the data set represented by the graph.
7
55) Find the sample standard deviation.
22 29 21 24 27 28 25 36
56) Without performing any calculations, use the stem-and-leaf plots to determine which statement is accurate.
(i)
0
1
2
3
4
9
5 8
3 3 7 7
2 5
1
(ii)
10
11
12
13
14
9
5 8
3 3 7 7
2 5
1
(iii)
0
1
2
3
4
5
3 3 3 3 7 7 7 7
5
A) Data set (ii) has the greatest standard deviation.
B) Data sets (i) and (ii) have the same standard deviation.
C) Data set (i) has the smallest standard deviation.
D) Data sets (i) and (iii) have the same range.
57) Adult IQ scores have a bell-shaped distribution with a mean of 100 and a standard deviation of 15. Use the
Empirical Rule to find the percentage of adults with scores between 70 and 130.
58) A placement exam for entrance into a math class yields a mean of 80 and a standard deviation of 10. The
distribution of the scores is roughly bell-shaped. Use the Empirical Rule to find the percentage of scores that lie
between 60 and 80.
59) SAT verbal scores are normally distributed with a mean of 489 and a standard deviation of 93. Use the
Empirical Rule to determine what percent of the scores lie between 303 and 582.
60) The test scores of 30 students are listed below. Find Q3.
31 41 45 48 52 55 56 56 63 65
67 67 69 70 70 74 75 78 79 79
80 81 83 85 85 87 90 92 95 99
61) The weights (in pounds) of 30 preschool children are listed below. Find the interquartile range of the 30
weights listed below. What can you conclude from the result?
25 25 26 26.5 27 27 27.5 28 28 28.5
29 29 30 30 30.5 31 31 32 32.5 32.5
33 33 34 34.5 35 35 37 37 38 38
8
62) Use the box-and-whisker plot below to determine which statement is accurate.
A) About 75% of the adults have cholesterol levels less than 180.
B) One half of the cholesterol levels are between 180 and 197.5.
C) One half of the cholesterol levels are between 180 and 211.
D) About 25% of the adults have cholesterol levels of at most 211.
63) Find the z-score for the value 88, when the mean is 95 and the standard deviation is 7.
64) Many firms use on-the-job training to teach their employees computer programming. Suppose you work in
the personnel department of a firm that just finished training a group of its employees to program, and you
have been requested to review the performance of one of the trainees on the final test that was given to all
trainees. The mean and standard deviation of the test scores are 74 and 2, respectively, and the distribution of
scores is bell-shaped and symmetric. Suppose the trainee in question received a score of 69. Compute the
traineeʹs z-score.
65) State whether the variable is discrete or continuous.
The number of pills in a container of vitamins
66) State whether the variable is discrete or continuous.
The age of the oldest student in a statistics class
67) The random variable x represents the number of cars per household in a town of 1000 households. Find the
probability of randomly selecting a household that has less than two cars.
Cars Households
0 125
1 428
2 256
3 108
4 83
68) Determine the probability distributionʹs missing value.
The probability that a tutor will see 0, 1, 2, 3, or 4 students
x 0 1 2 3 4
P(x) 0.01 0.04 0.37 0.34 ?
9
69) The random variable x represents the number of credit cards that adults have along with the corresponding
probabilities. Find the mean and standard deviation.
x P(x)
0 0.07
1 0.68
2 0.21
3 0.03
4 0.01
70) In a recent survey, 80% of the community favored building a police substation in their neighborhood. If 15
citizens are chosen, what is the mean number favoring the substation?
71) A test consists of 10 multiple choice questions, each with five possible answers, one of which is correct. To pass
the test a student must get 60% or better on the test. If a student randomly guesses, what is the probability that
the student will pass the test?
72) Find the area under the standard normal curve between z = 0 and z = 3.
73) Find the area under the standard normal curve between z = -1.5 and z = 2.5.
74) Find the area of the indicated region under the standard normal curve.
75) Use the standard normal distribution to find P(0 < z < 2.25).
76) Find the area of the indicated region under the standard normal curve.
77) For the standard normal curve, find the z-score that corresponds to the third quartile.
Provide an appropriate response. Use the Standard Normal Table to find the probability.
78) IQ test scores are normally distributed with a mean of 100 and a standard deviation of 15. An individualʹs IQ
score is found to be 110. Find the z-score corresponding to this value.
10
79) IQ test scores are normally distributed with a mean of 100 and a standard deviation of 15. An individualʹs IQ
score is found to be 120. Find the z-score corresponding to this value.
80) The distribution of cholesterol levels in teenage boys is approximately normal with μ = 170 and σ = 30 (Source:
U.S. National Center for Health Statistics). Levels above 200 warrant attention. Find the probability that a
teenage boy has a cholesterol level greater than 200.
Provide an appropriate response.
81) Find the z-score that is greater than the mean and for which 70% of the distributionʹs area lies to its left.
82) For the standard normal curve, find the z-score that corresponds to the 90th percentile.
11
Answer KeyTestname: FALL FINALREV14
1) Yes
2) No
3) False
4) True
5) ∈
6) {1, 2, 3, . . . , 8}
7) 26
8) 5
9) Yes
10) Yes
11) ⊈
12) True
13) Aʹ = {j, k, m}
14) {Karen, Charly, Sam, Sophia}
15) {Sam, Sophia}
16) {1, 5, a, c, e}
17) {u, w}
18) {r, t, u, v, w, x, z}
19) {q, r, s, t, v, x, y, z}
20) {r, t}
21) {11, 12, 13, 14, 15, 16, 17}
22) 13
23) 105
24) 4096
25) 3,628,800
26) 5040
27) 6720
28) 277,200
29) Permutations, because the order of the letters matters.
30) Combinations, because the order of the computers selected does not matter.
31) 6
32) 56
33)3
5
34)222
996; 0.223
35) 720; 24; 1
30
36) 0.58
12
Answer KeyTestname: FALL FINALREV14
37) 0.533
38)41
50
39)1
18
40)35
222
41) (a) $32,000 (b) $32,000 (c) 32,140
42) $0.33
43) quantitative
44) qualitative
45) simulation
46) stratified
47) random
48) (a) 3
(b) 51
(c) 49.5-52.5
49) 80
50) max: 17.5; min: 11.0
51) symmetric
52) 70
53) 196.1
54) 6
55) 4.8
56) B
57) 95%
58) 47.5%
59) 81.5%
60) 83
61) IQR = Q3 - Q1 = 34 - 28 = 6. This means that the weights of the middle half of the data set vary by 6 pounds.
62) C
63) z = -1.00
64) z = -2.50
65) discrete
66) continuous
67) 0.553
68) 0.24
69) mean: 1.23; standard deviation: 0.66
70) 12
71) 0.006
72) 0.4987
73) 0.9270
74) 0.9032
75) 0.4878
76) 0.1504
77) 0.67
78) 0.67
79) 1.33
80) 0.1587
13
Answer KeyTestname: FALL FINALREV14
81) 0.53
82) 1.28
14