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Introduction To Probability And Statistics For Science Engineering And Finance 1st Edition Walter A. Rosenkrantz - Solutions

Find the errors in each of the following statements:1. The probabilities that an automobile salesperson will sell 0, 1, 2, or 3 cars on any given day in February are, respectively, 0.19, 0.38, 0.29, and 0.15.2. The probability that it will rain tomorrow is 0.40, and the probability that it will not
How many ways are there that no two students will have the same birth date in a class of size 100?
How many ways are there to select 4 candidates from 8 equally qualified recent graduates for openings in an accounting firm?
In how many ways can 2 oaks, 4 pines, and 4 maples be arranged along a property line if one does not distinguish among trees of the same kind?
How many distinct permutations can be made from the letters of the word INFINITY?
In how many ways can a caravan of 8 covered wagons from Arizona be arranged in a circle?
In how many ways can 7 different trees be planted in a circle?
Three lottery tickets for first, second, and third prizes are drawn from a group of 40 tickets. Find the number of sample points in S for awarding the 3 prizes if each contestant holds only 1 ticket.
Find the number of ways that 6 teachers can be assigned to 3 sections of an introductory psychology course if no teacher is assigned to more than one section.
In how many ways can 5 starting positions on a basketball team be filled with 9 men who can play any of the positions?
In a regional spelling bee, the 8 finalists consist of 3 boys and 5 girls. Find the number of sample points in the sample space S for the number of possible orders at the conclusion of the contest for 1. all 8 finalists;2. the first 3 positions.
Four married couples have bought 8 seats in the same row for a concert. In how many different ways can they be seated 1. with no restrictions?2. if each couple is to sit together?3. if all the men sit together to the right of all the women?4. if all the men sit together to the right of all the
In how many ways can 4 boys and 5 girls sit in a row if the boys and girls must alternate?
1. How many three-digit numbers can be formed from the digits 0, 1, 2, 3, 4, 5, and 6 if each digit can be used only once?2. How many of these are odd numbers?3. How many are greater than 330?
A contractor wishes to build 7 houses, each different in design. In how many ways can he place these houses on a street if 5 lots are on one side of the street and 2 lots are on the opposite side?
1. How many distinct permutations can be made from the letters of the word COLUMNS?2. How many of these permutations start with the letter M?
If a multiple-choice test consists of 5 Questions, each with 4 possible answers of which only 1 is correct, 1. in how many different ways can a student check off one answer to each Question?2. in how many ways can a student check off one answer to each Question and get all the answers wrong?
1. In how many ways can 6 people be lined up to get on a bus?2. If 3 specific persons, among 6, insist on following each other, how many ways are possible?3. If 2 specific persons, among 6, refuse to follow each other, how many ways are possible?
A witness to a hit-and-run accident told the police that the license number contained the letters RLH followed by 4 digits, the first and second of which were a 5. If the witness cannot recall the last 2 digits, but is certain that all 4 digits are different, find the maximum number of automobile
In how many different ways can a true-false test consisting of 5 Questions be answered?
In a fuel economy study, each of 3 race cars is tested using 5 different brands of gasoline at 7 test sites located in different regions of the country. If 2 drivers are used in the study, and test runs are made once under each distinct set of conditions, how many test runs are needed?
A drug for the relief of asthma can be purchased from 7 different manufacturers in liquid, tablet, or capsule form, all of which come in regular and extra strength. How many different ways can a doctor prescribe the drug for a patient suffering from asthma?
A developer of a new subdivision offers a prospective home buyer a choice of 5 designs, 2 different heating systems, a garage or carport, and a patio or screened porch. How many different plans are available to this buyer?
A California study concluded that following 7 simple health rules can extend a man’s life by 11 years on the average and a woman’s life by 7 years. These 7 rules are as follows: no smoking, get regular exercise, use alcohol only in moderation, get 7 to 8 hours of sleep, maintain proper weight,
A certain brand of shoes comes in 5 different styles, with each style available in 4 distinct colors. If the store wishes to display pairs of these shoes showing all of its various styles and colors, how many different pairs will the store have on display?+ – + – +– + –
Students at a private liberal arts college are classified as being freshmen, sophomores, juniors, or seniors, and also according to whether they are male or female. Find the total number of possible classifications for the students of that college.
If an experiment consists of tossing a coin and then drawing a number within 0 to 9 at random, how many points are there in the sample space?
In a medical study, patients are classified in 8 ways according to whether they have blood type AB , AB , A , A , B , B , O , or O , and also according to whether their blood pressure is low, normal, or high. Find the number of ways in which a patient can be classified.
Registrants at a large convention are offered 4 sightseeing tours on each of 5 days. In how many ways can a person arrange to go on a sightseeing tour planned by this convention?
Referring to Exercise 2.19 and the Venn diagram of Figure 2.5, list the numbers of the regions that represent the following events:1. The family will experience no mechanical problems and will not receive a ticket for a traffic violation but will arrive at a campsite with no vacancies.2. The family
Suppose that a family is leaving on a summer vacation in their camper and that M is the event that they will experience mechanical problems, T is the event that they will receive a ticket for committing a traffic violation, and V is the event that they will arrive at a campsite with no vacancies.
Which of the following pairs of events are mutually exclusive?1. A golfer scoring the lowest 18-hole round in a 72-hole tournament and losing the tournament.2. A poker player getting a flush (all cards in the same suit) and 3 of a kind on the same 5-card hand.3. A mother giving birth to a baby girl
Let A, B, and C be events relative to the sample space S.Using Venn diagrams, shade the areas representing the following events:1. (A ∩ B)′;2. (A ∪ B)′;3. (A ∩ C) ∪ B.
If S = {x | 0 < x < 10}, M = {x | 2 < x < 9}, and N = {x | 0 < x
Consider the sample space S = {copper, sodium, nitrogen, potassium, uranium, oxygen, zinc} and the events List the elements of the sets corresponding to the following events:1. A′;2. B ∪ A;3. (A ∪ B′) ∩ C;4. A′ ∪ C;5. A ∪ B ∪ C;6. (A′ ∩ B′) ∪ (A′ ∪ C).
If S ={0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {2, 4, 6, 8}, B = {1, 3, 5, 7, 9}, C = {0, 2, 3, 4, 5}, and D = {1, 6, 7}, list the elements of the sets corresponding to the following events:1. A ∪ D;2. A ∩ B;3. B′;4. (B′ ∩ D) ∪ A;5. (S ∩ D)′;6. A ∩ C ∩ D′.
Construct a Venn diagram to illustrate the possible intersections and unions for the following events relative to the sample space consisting of all automobiles made in the United States.
Exercise and diet are being studied as possible substitutes for medication to lower blood pressure. Three groups of 2 1 subjects will be used to study the effect of exercise. Group 1 is sedentary, while group 2 walks and group 3 swims for 1 hour a day. Half of each of the three exercise groups will
The resumés of two male applicants for a college teaching position in chemistry are placed in the same file as the resumés of two female applicants. Two positions become available, and the first, at the rank of assistant professor, is filled by selecting one of the four applicants at random. The
An engineering firm is hired to determine if certain waterways in Virginia are safe for fishing. Samples are taken from three rivers.1. List the elements of a sample space S, using the letters F for safe to fish and N for not safe to fish.2. List the elements of S corresponding to event E that at
For the sample space of Exercise 2.5, 1. list the elements corresponding to the event A that a number less than 3 occurs on the die;2. list the elements corresponding to the event B that two tails occur;3. list the elements corresponding to the event A′;4. list the elements corresponding to the
For the sample space of Exercise 2.4, 1. list the elements corresponding to the event A that the sum is greater than 8;2. list the elements corresponding to the event B that a 2 occurs on either die;3. list the elements corresponding to the event C that a number greater than 4 comes up on the green
Four students are selected at random from a chemistry class and classified as male or female. List the elements of the sample space S , using the letter M for male and F for female. Define a second sample space S where the elements represent the number of females selected.
Two jurors are selected from 4 alternates to serve at a murder trial. Using the notation A A , for example, to denote the simple event that alternates 1 and 3 are selected, list the 6 elements of the sample space S.
An experiment consists of tossing a die and then flipping a coin once if the number on the die is even. If the number on the die is odd, the coin is flipped twice. Using the notation 4H, for example, to denote the outcome that the die comes up 4 and then the coin comes up heads, and 3HT to denote
An experiment involves tossing a pair of dice, one green and one red, and recording the numbers that come up. If x equals the outcome on the green die and y the outcome on the red die, describe the sample space S 1. by listing the elements (x, y);2. by using the rule method.
Which of the following events are equal?1. A = {1, 3};2. B = {x | x is a number on a die};3. C = {x | x – 4x + 3 = 0};4. D = {x | x is the number of heads when six coins are tossed}.
Use the rule method to describe the sample space S consisting of all points in the first quadrant inside a circle of radius 3 with center at the origin.
List the elements of each of the following sample spaces:1. the set of integers between 1 and 50 divisible by 8;2. the set S = {x | x + 4x – 5 = 0};3. the set of outcomes when a coin is tossed until a tail or three heads appear;4. the set S = {x | x is a continent};5. the set S = {x | 2x – 4
Group Project: Collect the data for weight of everyone in the class. Use the sample means and variances and the types of plots presented in this chapter to summarize any features that draw a distinction between the distributions of weights for males and females. Do the same for the shoe sizes of
Use the results of Exercises 1.28 and 1.31 to create a plot that illustrates the interaction evident from the data. Use the plot in Figure 1.3 in Example 1.3 as a guide. Could the type of information found in Exercises 1.28 and 1.31 have been found in an observational study in which there was no
Consider the situation of Exercise 1.28. But now use the following data set, in which shrinkage is measured once again at low injection velocity and high injection velocity. However, this time the mold temperature is raised to a high level and held constant.Shrinkage values at low injection
Below are the lifetimes, in hours, of fifty 40-watt, 110-volt internally frosted incandescent lamps, taken from forced life tests:Construct a box plot for these data.
Use the data in Exercise 1.24 to construct a box plot.
Many manufacturing companies in the United States and abroad use molded parts as components of a process. Shrinkage is often a major problem. Thus, a molded die for a part is built larger than nominal size to allow for part shrinkage. In an injection molding study it is known that the shrinkage is
A study is done to determine the influence of the wear, y, of a bearing as a function of the load, x, on the bearing. A designed experiment is used for this study. Three levels of load were used, 700 lb, 1000 lb, and 1300 lb. Four specimens were used at each level, and the sample means were,
Suppose it is of interest to use the data sets in Exercises 1.24 and 1.25 to derive a model that would predict staff salaries as a function of percentage of families in a high income level for current school systems. Comment on any disadvantage in carrying out this type of analysis.
The following data set is related to that in Exercise 1.24. It gives the percentages of the families that are in the upper income level, for the same individual schools in the same order as in Exercise 1.24.1. Calculate the sample mean.2. Calculate the sample median.3. Construct a relative
The following are historical data on staff salaries (dollars per pupil) for 30 schools sampled in the eastern part of the United States in the early 1970s.1. Compute the sample mean and sample standard deviation.2. Construct a relative frequency histogram of the data.3. Construct a stem-and-leaf
The hydrocarbon emissions at idling speed in parts per million (ppm) for automobiles of 1980 and 1990 model years are given for 20 randomly selected cars.1980 models:1990 models:1. Construct a dot plot as in Figure 1.1.2. Compute the sample means for the two years and superimpose the two means on
The following data are the measures of the diameters of 36 rivet heads in 1/100 of an inch.1. Compute the sample mean and sample standard deviation.2. Construct a relative frequency histogram of the data.3. Comment on whether or not there is any clear indication that the sample came from a
The lengths of power failures, recorded in the following table.1. Find the sample mean and sample median of the powerfailure times.2. Find the sample standard deviation of the power-failure times.
The following data represent the length of life, in seconds, of 50 fruit flies subject to a new spray in a controlled laboratory experiment:1. Construct a double-stem-and-leaf plot for the life span of the fruit flies using the stems , and such that stems coded by the symbols and · are associated,
The following data represent the length of life in years, measured to the nearest tenth, of 30 similar fuel pumps:1. Construct a stem-and-leaf plot for the life in years of the fuel pumps, using the digit to the left of the decimal point as the stem for each observation.2. Set up a relative
The following scores represent the final examination grades for an elementary statistics course:1. Construct a stem-and-leaf plot for the examination grades in which the stems are 1, 2, 3, …, 9.2. Construct a relative frequency histogram, draw an estimate of the graph of the distribution, and
A study of the effects of smoking on sleep patterns is conducted. The measure observed is the time, in minutes, that it takes to fall asleep. These data are obtained:8 1. Find the sample mean for each group.2. Find the sample standard deviation for each group.3. Make a dot plot of the data sets A
Show that the n pieces of information in are not independent; that is, show that
Eight independent coin tosses result in HHHHHHHH. It turns out that if the coin is fair, the probability of this outcome is (1/2) = 0.00390625. Does this produce strong evidence that the coin is not fair? Comment and use the concept of P-value discussed in Section 1.1.
A tire manufacturer wants to determine the inner diameter of a certain grade of tire. Ideally, the diameter would be 570 mm. The data are as follows:1 2 3 41. Find the sample mean and median.2. Find the sample variance, standard deviation, and range.3. Using the calculated statistics in parts (a)
A manufacturer of electronic components is interested in determining the lifetime of a certain type of battery. A sample, in hours of life, is as follows:1. Find the sample mean and median.2. What feature in this data set is responsible for the substantial difference between the two?
For Exercise 1.6 on page 33, compute the sample standard deviation in tensile strength for the samples separately for the two temperatures. Does it appear as if an increase in temperature influences the variability in tensile strength?Explain.
Consider the data in Exercise 1.5 on page 33. Compute the sample variance and the sample standard deviation for both control and treatment groups.
For the data of Exercise 1.4 on page 33, compute both the mean and the variance in “flexibility” for both company A and company B. Does there appear to be a difference in flexibility between company A and company B?
Exercise 1.3 on page 33 showed tensile strength data for two samples, one in which specimens were exposed to an aging process and one in which there was no aging of the specimens.1. Calculate the sample variance as well as standard deviation in tensile strength for both samples.2. Does there appear
Compute the sample variance and standard deviation for the water absorbency data of Exercise 1.2 on page 33.
Consider the drying time data for Exercise 1.1 on page 33.Compute the sample variance and sample standard deviation.
The tensile strength of silicone rubber is thought to be a function of curing temperature. A study was carried out in which samples of 12 specimens of the rubber were prepared using curing temperatures of 20°C and 45°C. The data below show the tensile strength values in megapascals.1. Show a dot
Twenty adult males between the ages of 30 and 40 participated in a study to evaluate the effect of a specific health regimen involving diet and exercise on the blood cholesterol.Ten were randomly selected to be a control group, and ten others were assigned to take part in the regimen as the
In a study conducted by the Department of Mechanical Engineering at Virginia Tech, the steel rods supplied by two different companies were compared. Ten sample springs were made out of the steel rods supplied by each company, and a measure of flexibility was recorded for each. The data are as
A certain polymer is used for evacuation systems for aircraft. It is important that the polymer be resistant to the aging process. Twenty specimens of the polymer were used in an experiment. Ten were assigned randomly to be exposed to an accelerated batch aging process that involved exposure to
According to the journal Chemical Engineering, an important property of a fiber is its water absorbency. A random sample of 20 pieces of cotton fiber was taken and the absorbency on each piece was measured. The following are the absorbency values:1. Calculate the sample mean and median for the
The following measurements were recorded for the drying time, in hours, of a certain brand of latex paint.Assume that the measurements are a simple random sample.1. What is the sample size for the above sample?2. Calculate the sample mean for these data.3. Calculate the sample median.4. Plot the
Problem 2.39 From a batch of 20 radios a sample of size 3 is randomly selected for inspection. If there are six defective radios in the batch what is the probability that the sample contains: (a) only defectives? (b) one defective and two non-defectives? (c) only non-defectives? (d) two defectives
Problem 2.38 A bus starts with 5 people and makes 10 stops. Assume that passengers are equally likely to get off at any stop. (a) Describe the sample space and calculate the number of different outcomes. (b) Compute the probability that no two passengers get off at the same stop.
Problem 2.37 In how many ways can 12 people be divided into 3 groups of 4 persons (in each group) for an evening of bridge?
Problem 2.36 A group of 10 components contains 3 that are defective. (a) What is the probability that one component drawn at random is defective? (b) Suppose a random sample of size 4 is drawn without replacement. What is the probability that exactly one of these components is defective? (c)
Problem 2.34 How many distinguishable arrangements of the letters of the word MISSISSIPPI are there?Problem 2.35 Let S denote the sample space consisting of the 24 distinguishable permutations of the symbols 1,2,3,4 and assign to each permutation the probability 1 24 . Let Ai =event that the
Problem 2.33 To win megabucks of the Massachusetts state lottery one must select 6 different digits out of {1,2,. . . ,36} that match the winning combination. What is the probability of winning?
Problem 2.32 The format of a Virginia license plate is 3 letters followed by 4 numbers as in ABC1234. How many such license plates are there?
Problem 2.31 The format of a Massachusetts license plate is 3 digits followed by 3 letters as in 736FSC. How many such license plates are there?
Problem 2.30 Poker dice is played by simultaneously rolling 5 dice. (a) Describe the sample space S; how many sample points are there? (b) Show that P(no two dice show the same number)=0.0926. (c) What is the probability that at least two of the dice show the same number?
Problem 2.29 Consider the four-step binomial tree displayed next. Starting at the root node (point A) you can go one step up (u) or one step down (d) at each node. After four steps you arrive at one of the end points of the tree {0, 1, 2, 3, 4}. (a) How many different paths are there from A to 3?
Problem 2.28 Consider the following variation on Feller’s birthday problem (Example 2.24). The martian year is 669 days long. (a) There are k martians in a room; derive a formula for the probability that at least two martians in the room have a common birthday. (b) Use your answer to part (a) to
Problem 2.27 (a) In how many ways can two distinct numbers be chosen from the set {1, 2,..., 100} so that their sum is an even number? (b) In how many ways can three distinct numbers be chosen from the set {1, 2,..., 100} so that their sum is an even number?
Problem 2.26 Find the value of the fourth term in the binomial expansion of (2x + 1 x )10 when x = 1/2.
Problem 2.25 (a) Find the fifth term of the binomial expansion of (1 + √x)9. (b) Find the seventh term of the binomial expansion of (1 + √x)9.
Problem 2.24 (a) Find the second term of the binomial expansion of (2x + 3y)4. (b) Find the third term of the binomial expansion of (2x + 3y)4.
Problem 2.23 Using the recurrence Equation 2.20 compute: 6 k , for k = 1,..., 6.
Problem 2.22 Verify the binomial identity (*)+(-1)=(+1) k in the following cases: (a) n = 8, k=3 (d) n = 9, k (b) n = 5, k=2 4 (e) n = 12, k=3 (c) n=10, k=4 (f) n=11, k=7
Problem 2.21 Each permutation of the integers {1, 2, 3, 4, 5} determines a five digit number. If the numbers corresponding to all possible permutations are listed in increasing order of magnitude, i.e., {12345, 12354,... 54321}, show that the 73rd number on the list is 41235. Hint: It is too

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