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Chapter 5 Practice Problems
1) Find the area under the standard normal curve between z = 0 and z = 3. 1
A) 0.9987 B) 0.4987 C) 0.0010 D) 0.4641
Normalcdf(0,3.1) = 0.4990 B
2) Find the area under the standard normal curve to the left of z = 1.25.
A) 0.1056 B) 0.8944 C) 0.2318 D) 0.7682
Normalcdf(-10000,1.25) = 0.8944 B
Chapter 5 Practice Problems3) Find the area of the indicated region under the standard normal curve.
A) 0.0968 B) 0.0823 C) 0.9032 D) 0.9177
Normalcdf(-1.30,1.0000) = 0.9032 C
4) For the standard normal curve, find the z-score that corresponds to the first quartile. A) 0.67 B) -0.23 C) 0.77 D) -0.67
The first quartile means that P = 0.2500. The z-score that corresponds to this percentage from the Normal Distribution chart is – 0.67. The answer is D.
Chapter 5 Practice Problems
5) IQ test scores are normally distributed with a mean of 99 and a standard deviation of 11. An individual's IQ score is found to be 109. Find the z-score corresponding to this value.
A) 0.91 B) 1.10 C) -1.10 D) -0.91
Given that the value of = 99, = 11, and x = 109, we can solve for the value of z using the equation z = (x - )/ . Z = 0.91 A
6) An airline knows from experience that the distribution of the number of suitcases that get lost each week on a certain route is approximately normal with μ = 15.5 and = 3.6. What is the probability that during a given week the airline will lose between 10 and 20 suitcases?
A) 0.3944 B) 0.8314 C) 0.1056 D) 0.4040
Normalcdf(10,20,15.5,3.6) = 0.8311 B
Chapter 5 Practice Problems
7.
With 5% of the data not included on either side, we look up the percent value of .0500 on the chart and we get that z = -1.645. The z value for the other side of the chart will be +1.645. The solution is B.
8.
Using the equation x = z + , we see that x = (15)(2.33) + 100 = 134.95. The correct choice will be D.
Chapter 5 Practice Problems9.
Solve for the value of z for each of the situations and then compare the results; the larger of the two z scores will have performed better. The first z-value will be z = (75 – 65)/8 which equals 1.25. The second z-value will be z = (75 – 70)/4 which equals 1.25. Since both z scores are the same the correct choice will be C.
Chapter 4 Practice Problems10. State whether the variable is discrete or continuous. The number of cups of coffee sold in a cafeteria during lunch
A) discrete B) continuous A) discrete
11. State whether the variable is discrete or continuous. The blood pressures of a group of students the day before their final exam
A) continuous B) discrete A) continuous
Chapter 4 Practice Problems
12) 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.
A) 0.809 B) 0.125 C) 0.553 D) 0.428
Cars Households0 1251 4282 2563 1084 83
P(x) = (428 + 125)/1000 = 0.553 C) 0.553
Chapter 4 Practice Problems
x P(x) xP(x) (x - ) (x - )2 (x - )2P(x)
0 0.07 0.00 -1.23 1.513 0.106
1 0.68 0.68 -0.23 .053 0.036
2 0.21 0.42 0.77 .593 0.125
3 0.03 0.09 1.77 3.133 0.094
4 0.01 0.04 2.77 7.673 0.077
Total 1.00 1.23 0.4.38
13) The random variable x represents the number of credit cards that adults have along with the corresponding probabilities. Find the mean and standard deviation.
A) mean: 1.30; standard deviation: 0.44 B) mean: 1.23; standard deviation: 0.66
C) mean: 1.23; standard deviation: 0.44 D) mean: 1.30; standard deviation: 0.32
x P(x)0 0.071 0.682 0.213 0.034 0.01
2( ) ( ) 0.438 0.66x P x B) mean: 1.23; standard deviation: 0.66
Chapter 4 Practice Problems14) At a raffle, 10,000 tickets are sold at $5 each for three prizes valued at $4,800, $1,200, and $400. What is the expected value of one ticket?
A) $4.36 B) -$4.36 C) $0.64 D) -$0.64
1 1 1( ) ( $4,800) ( $1,200) ( $400)
10,000 9,999 9,998
999( $5) $0.48 $0.12 $0.04 ( $5.00) $4.3610000
xP x
B) -$4.36
15) A sports analyst records the winners of NASCAR Winston Cup races for a recent season. The random variable x represents the races won by a driver in one season. Use the frequency distribution to construct a probability distribution.
Wins 1 2 3 4 5 6 7Drivers 12 2 0 2 0 0 1 x 1 2 3 4 5 6 7
P(x) 0.71 0.12 0 0.12 0 0 0.06
Chapter 4 Practice Problems16) Determine whether the distribution represents a probability distribution. If not, identify any requirements that are not satisfied.
x P(x)1 0.22 0.23 0.24 0.25 0.2
It is a probability distribution, because each value is between 0 and 1 and the sum of P(x) = 1.
17. According to government data, the probability that a woman between the ages of 25 and 29 was never married is 40%. In a random survey of 10 women in this age group, what is the probability that at least eight were married?
B) 0.167P(8) + P(9) +P(10) = 0.1209 + .0403 + .0060 = .0.1672
Binompdf(10,.60,8) = 0.1209; Binompdf(10,.60,9) = 0.0403; Binompdf(10,.60,10) = 0.0060
Chapter 4 Practice Problems17) The probability that a tennis set will go to a tiebreaker is 18%. In 60 randomly
selected tennis sets, what is the mean and the standard deviation of the number of tiebreakers?
A) mean: 10.8; standard deviation: 2.98 B) mean: 10.2; standard deviation: 2.98B) C) mean: 10.8; standard deviation: 3.29 D) mean: 10.2; standard deviation: 3.29
Mean: μ = np = 60 x 0.18 = 10.8
Standard Deviation: =npq = 60x.18x.82 = 2.98
A) mean: 10.8; standard deviation: 2.98
Chapter 4 Practice Problems
18) Fifty-seven percent of families say that their children have an influence on their vacation plans. Consider a sample of eight families who are asked if their children influence their vacation plans. Identify the values of n, p, and q, and list the possible values of the random variable x.
n = 8; p = 0.57; q = 0.43; x = 0, 1, 2, 3, 4, 5, 6, 7, 8
19) You observe the gender of the next 100 babies born at a local hospital. You count the number of girls born. Identify the values of n, p, and q, and list the possible values of the random variable x
n = 100; p = 0.5; q = 0.5; x = 0, 1, 2, . . ., 99, 100
Chapter 4 Practice Problems20. A statistics professor finds that when he schedules an office hour at the 10:30 a.m. time slot, an average of three students arrive. Use the Poisson distribution to find the probability that in a randomly selected office hour in the 10:30 a.m. time slot exactly five students will arrive.
A) 0.0137 B) 0.0519 C) 0.0070 D) 0.10083) D
= 3; x = 5;
5 33( ) 0.1008
! 5!
xe eP x
x
Poisson(3,5) = 0.1008
Chapter 4 Practice Problems21. A mail-order company receives an average of five orders per 500 solicitations. If it sends out 100 advertisements, find the probability of receiving at least two orders. Use the Poisson distribution.
A) 0.1839 B) 0.9596 C) 0.9048 D) 0.2642
5) D0 1
1 1
1(0) 0.3679
! 0!
1(1) 0.3679
! 1!( 2) 1 (0.3679 .03679) 0.2642
x
x
e eP
x
e eP
xP x
Poisson(1,0) = 0.3679; Poisson(1,1) = 0.3679; P(0) +P(1)= 2(0.3679)= 0.7358
P( x 2) = 1 – 0.7358 = 0.2642
Chapter 4 Practice Problems22. Given: The probability that a federal income tax return is filled out incorrectly with an error in favor of the taxpayer is 20%. Question: What is the probability type that of the ten tax returns randomly selected for an audit, three returns will contain only errors favoring the taxpayer?
A) Poisson B) geometric C) binomial
7) C1. The experiment is repeated for a fixed number of trials, where
each trial is independent of other trials.
2. There are only two possible outcomes of interest for each trial. The outcomes can be classified as a success (S) or as a failure (F).
3. The probability of a success P(S) is the same for each trial.
4. The random variable x counts the number of successful trials.
Chapter 4 Practice Problems23 Given: The probability that a federal income tax return is filled out incorrectly with an error in favor of the taxpayer is 20%. Question: What is the probability type that when the ten tax returns are randomly selected for an audit, the sixth return will contain only errors favoring the taxpayer?
A) Poisson B) binomial C) geometric
8) C
Geometric distribution A discrete probability distribution. Satisfies the following conditions
a) A trial is repeated until a success occurs.b) The repeated trials are independent of each other.c) The probability of success p is constant for each trial.
Chapter 4 Practice Problems Binomial
repeated for a fix # of trials Random variable x, counts the # of successes out of n trials
Geometric repeated until a successful outcome occurs Random variable x, is the # of where the first success occurs
Poisson Counting the # times an event occurs for a specified interval The number of occurrences of an event in a specified interval
is independent of the # of occurrences of the event in other specified intervals
Chapter 4 Practice Problems
Binomial
Geometric
Poisson
2
np
npq
npq
Mean
Variance
Standard Deviation
Mean
Variance
Standard Deviation
Mean
Variance
Standard Deviation
22
2
1
p
q
p
q
p
2
#mean occurrences
Chapter 4 Practice Problems24. Assume the probability that you will make a sale on any given telephone call is 0.19. Find the probability that you (a) make your first sale on the fifth call, (b) make your first sale on the first, second, or third call, and (c) do not make a sale on the first three calls.
x = # of trials for 1st success
p = probability of success
q = probability of failure
P(x) = p(q) x-1
(a) geometpdf(.19,5) = 0.082
(b) geometpdf(.19,1) + geometpdf(.19,2) + geometpdf(.19,3) = 0.19 + 0.154 + 0.125 = 0.469
(c) No sales on the first 3 calls = Q(1) + Q(2) + Q(3) = 1 – 0.469 = 0.531
Chapter 4 Practice Problems24. During a 36-year period, lightning killed 2457 people in the United States. Assume that this rate holds true today and is constant throughout the year. Find the probability that tomorrow (a) no one in the United States will be struck and killed by lightning, (b) one person will be struck and killed, (c) more than one person will be struck and killed.
x = # occurrences in a given time or space
= average # of occurrences in a given time or space
( )!
xeP x
x
Poissoinpdf(,x)
= 2457/13140 = 0.1870
(a) x = 0; poissonpdf(0.1870,0) = 0.829
(b) x = 1; poissonpdf(0.1870,1) = 0.155
(c) x > 1 = 1 – (P(0) + P(1)) = 0.016
36 yrs = 13140 days
Chapter 4 Practice Problems
x = # occurrences in a given time or space
= average # of occurrences in a given time or space
( )!
xeP x
x
Poissoinpdf(,x)
(a) = 4, x = 3; P(3) poissonpdf(4,3) = 0.195
(b) P(x 3) = P(0) + P(1) + P(2) + P(3) = .0183 + .0733 + 0.1465 + 0.1954 = 0.433
(c) P(x > 3) = 1 – P( x 3) = 1 – 0.433 = 0.567
25. A newspaper finds that the mean number of typographical errors per page is four. Find the probability that (a) exactly three typographical errors will be found on a page, (b) at most three typographical errors will be found on a page, and (c) more than three typographical errors will be found on a page.
Chapter 3 Practice Problems
26. A single six-sided die is rolled. Find the probability of rolling a number less than 3.
A) 0.25 B) 0.333 C) 0.1 D) 0.5
1) B 2 correct answers/6 possible answers =0.33
27. In a survey of college students, 880 said that they have cheated on an exam and 1721 said that they have not. If one college student is selected at random, find the probability that the student has cheated on an exam.
A) 1721/2601 B) 2601/1721 C) 2601/880 D) 880/2601
4) D 880 correct answers/(1721+880) possible answers = 880/2601
Chapter 3 Practice Problems28. The distribution of blood types for 100 Americans is listed in the table. If one donor is selected at random, find the probability of selecting a person with blood type A+ or A-. Blood Type O+ O- A+ A- B+ B- AB+ AB-
Number 37 6 34 6 10 2 4 1
A) 0.34 B) 0.06 C) 0.4 D) 0.02
5) C (34 +6) correct answers/100 possible answers = 40/100 = 0.4
29. A card is picked at random from a standard deck of 52 playing cards. Find the odds that it is not a heart.
A) 1:3 B) 1:4 C) 4:1 D) 3:1
10) D 1 suit of correct answers/4 suits of possible answers = ¼. To not be a heart means that we have 4/4 – ¼ = ¾ . There are a 3:1 odds that a heart will not be chosen randomly.
Chapter 3 Practice Problems29. Classify the events as dependent or independent. Events A and B whereP(A) = 0.6, P(B) = 0.3, and P(A and B) = 0.17
A) independent B) dependent
12) B The results of P(A and B) = 0.17 indicates that the event A and the event B are interrelated and so they are dependent.
30. Find the probability of answering two true or false questions correctly if random guesses are made. Only one of the choices is correct.
A) 0.25 B) 0.75 C) 0.1 D) 0.5
15) A 1 correct answer/4 possible answers =0.25
Chapter 3 Practice Problems30. Find the probability of getting four consecutive aces when four cards are drawn without replacement from a standard deck of 52 playing cards.
31. A multiple-choice test has five questions, each with five choices for the answer. Only one of the choices is correct. You randomly guess the answer to each question. What is the probability that you do not answer any of the questions correctly?
32. The probability it will rain is 40% each day over a three-day period. What is theprobability it will rain at least one of the three days?
30. P(4-Aces) = (4/52)(3/51)(2/50)(1/49 )= 0.00000369
31. P(all five questions answers incorrect) = (4/5)(4/5)(4/5)(4/5)(4/5) = 0.32768
32. P(rain at least one day) = 1 - P(no rain all three days)= 1 - (0.60)(0.60)(0.60) = 0.7845
Chapter 3 Practice Problems32. A card is selected at random from a standard deck. Find each
probability.
(a) Randomly selecting a diamond or a 7._________________________
(b) Randomly selecting a red suit or a queen._________________________
(c) Randomly selecting a 3 or a face card_________________________33. You roll one die. Find each probability.
(a) Rolling a 6 or a number greater than 4. _________________________
(b) Rolling a number less than 5 or an odd number. _________________________
(c) Rolling a 3 or an even number. _________________________
13 4 1 160.308
52 52 52 52
26 4 2 280.538
52 52 52 52
4 12 160.308
52 52 52
1 2 1 20.333
6 6 6 6
4 3 2 50.833
6 6 6 6
1 3 40.667
6 6 6
Chapter 3 Practice Problems
34. Determine the probability that an 8-sided die, numbered 1-8 is rolled and the results of the roll is an even number or a number that is greater than 6.
___________________________________
4 2 1 50.625
8 8 8 8
Chapter 3 Practice Problems
Gender
Male Female Total
Level Associates 260 405 665
Of Bachelor’s 595 804 1399
Degree Master’s 230 329 559
Doctorate 25 23 48
Total 1110 1561 2671
(a) earned a master’s degree or is a female _________________
(b) earned an associate degree and is a male _________________
(c) is a female given that the person earned a bachelor’s degree_________________
8040.575
1399
2600.097
2671
559 1561 3290.671
2671 2671 2671
35.
Chapter 3 Practice Problems
1 13 140.269
52 52 52
1 12 120.005
52 51 2652
13 13 1690.064
52 51 2652
36. Using a standard 52-card deck for each situation, determine the probability.
(a) Drawing a five and a spade.
(b) Drawing a five of spades.
(c) Drawing a spade and then without replacement, drawing club.
(d) Drawing a 3 of clubs and then without replacement, drawing another club.
(e) Drawing a 4 of diamonds or a heart on the first draw.
4 1 40.019
52 4 208
10.019
52
Chapter 3 Practice Problems37. How many ways can a jury of five men and three women be selected from twelve men and ten women?
(12C5)(10C3) = 95,040
38. How many different permutations of the letters in the word PROBABILITY are there?
11!/(2!2!) = 9,979,200
39. A student must answer six questions on an exam that contains twelve questions.a) How many ways can the student do this?b) How many ways are there if the student must answer the first and last question?
(a) 12C6 = 924; (b) 10C4 = 210
Chapter 3 Practice Problems
40. In the California State lottery, you must select six numbers from fifty-two numbers to win the big prize. The numbers do not have to be in a particular order. What is the probability that you will win the big prize if you buy one ticket?
52C6 = 1/20,358,520 = 0.00000004911
41. From a group of 40 people, a jury of 12 people is selected. In how many different ways can a jury of 12 people be selected?
5,586,853,48040 12
40!5,586,853,480
(40 12)!(12!)C
Chapter 3 Practice Problems42. An area code consists of three digits. How many area codes are possible if (a) there are no restrictions and (b) if the first digit can not be 1 or 0. (c) What isthe probability of selecting an area code at random that ends in an odd number if the first digit cannot be a 1 or a 0?
42. (a) 10x10x10= 1000 (b) 10x10x8= 800 (c) 0.5
43. A shipment of 10 microwave ovens contains two defective units. In how many ways can a restaurant buy three of these units and receive (a) no defective units, (b) one defective unit, and (c) at least two non-defective units? (d) What is the probability of the restaurant buying at least two defective units?
43. (a) 56 (b) 56 (c) 112 (d) 0.067
(a) (8C3 )(2C0 ) = (56 )(1 )= 56
(b) (8C2 )(2C1 ) = (28 )(2 )= 56
(c) At least two good units one or fewer defective units. 56 + 56 = 112
(d) Pat least 2 defective units =8 1 2 2
10 3
8 10.067
120
C C
C