Question: Compute the sample variance and standard deviation

> According to USA Today (March 18, 1997), of 4 million workers in the general workforce, 5.8% tested positive for drugs. Of those testing positive, 22.5% were cocaine users and 54.4% marijuana users. (a) What is the probability that of 10 workers testing

> Twelve people are given two identical speakers, which they are asked to listen to for differences, if any. Suppose that these people answer simply by guessing. Find the probability that three people claim to have heard a difference between the two speake

> How many ways are there to select 3 candidates from 8 equally qualified recent graduates for openings in an accounting firm?

> In how many ways can 3 oaks, 4 pines, and 2 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 5 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 4 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 8 men who can play any of the positions?

> 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 (a) by listing the elements (x, y); (b) b

> 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 (a) all 8 finalists; (b) the first 3 positions.

> As a student drives to school, he encounters a traffic signal. This traffic signal stays green for 35 seconds, yellow for 5 seconds, and red for 60 seconds. Assume that the student goes to school each weekday between 8:00 and 8:30 a.m. Let X1 be the numb

> Four married couples have bought 8 seats in the same row for a concert. In how many different ways can they be seated (a) with no restrictions? (b) if each couple is to sit together? (c) if all the men sit together to the right of all the women?

> In how many ways can 4 boys and 5 girls sit in a row if the boys and girls must alternate?

> (a) 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? (b) How many of these are odd numbers? (c) How many are greater than 330?

> A contractor wishes to build 9 houses, each different in design. In how many ways can he place these houses on a street if 6 lots are on one side of the street and 3 lots are on the opposite side?

> (a) How many distinct permutations can be made from the letters of the word COLUMNS? (b) 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, (a) in how many different ways can a student check off one answer to each question? (b) in how many ways can a student check off one answer to ea

> (a) In how many ways can 6 people be lined up to get on a bus? (b) If 3 specific persons, among 6, insist on following each other, how many ways are possible? (c) 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 3 digits, the first of which was a 5. If the witness cannot recall the last 2 digits, but is certain that all 3 digits are different, find t

> In how many different ways can a true-false test consisting of 9 questions be answered?

> Which of the following events are equal? (a) A = {1, 3}; (b) B = {x | x is a number on a die}; (c) C = {x | x2 − 4x+3 = 0}; (d) D = {x | x is the number of heads when six coins are tossed}.

> (a) In Exercise 5.9, how many of the 15 trucks would you expect to have blowouts? (b) What is the variance of the number of blowouts experienced by the 15 trucks? What does that mean? Data from Exercise 5.9: In testing a certain kind of truck tire over

> 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

> A drug for the relief of asthma can be purchased from 5 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 asth

> A developer of a new subdivision offers a prospective home buyer a choice of 4 designs, 3 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

> 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 displa

> 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 throwing a die and then drawing a letter at random from the English alphabet, 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

> Registrants at a large convention are offered 6 sightseeing tours on each of 3 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: (a) The family will experience no mechanical problems and will not receive a ticket for a traffic violation but will arriv

> If X represents the number of people in Exercise 5.13 who believe that antidepressants do not cure but only cover up the real problem, find the mean and variance of X when 5 people are selected at random. Exercise 5.13: A national study that examined at

> 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: (a)

> 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 w

> Which of the following pairs of events are mutually exclusive? (a) A golfer scoring the lowest 18-hole round in a 72- hole tournament and losing the tournament. (b) A poker player getting a flush (all cards in the same suit) and 3 of a kind on the same 5

> Let A, B, and C be events relative to the sample space S. Using Venn diagrams, shade the areas representing the following events: (a) (A ∩ B)’; (b) (A ∪ B)’; (c) (A ∩ C) ∪ B.

> If S = {x | 0 < x < 12}, M = {x | 1 < x < 9}, and N = {x | 0 < x < 5}, find (a) M ∪ N; (b) M ∩ N; (c) M’ ∩ N’.

> Consider the sample space S = {copper, sodium, nitrogen, potassium, uranium, oxygen, zinc} and the events A = {copper, sodium, zinc}, B = {sodium, nitrogen, potassium}, C = {oxygen}. List the elements of the sets corresponding to the following events: (a

> If S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {0, 2, 4, 6, 8}, B = {1, 3, 5, 7, 9}, C = {2, 3, 4, 5}, and D = {1, 6, 7}, list the elements of the sets corresponding to the following events: (a) A ∪ C; (b) A ∩ B; (c) C’; (d) (C’ ∩ D) ∪ B; (e) (S ∩ C)’; (f

> 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. F: Four door, S: Sun roof, P: Power steering.

> Exercise and diet are being studied as possible substitutes for medication to lower blood pressure. Three groups of 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

> The resumes of two male applicants for a college teaching position in chemistry are placed in the same file as the resumes of two female applicants. Two positions become available, and the first, at the rank of assistant professor, is filled by selecting

> Suppose that airplane engines operate independently and fail with probability equal to 0.4. Assuming that a plane makes a safe flight if at least one-half of its engines run, determine whether a 4-engine plane or a 2- engine plane has the higher probabil

> An engineering firm is hired to determine if certain waterways in Virginia are safe for fishing. Samples are taken from three rivers. (a) List the elements of a sample space S, using the letters F for safe to fish and N for not safe to fish. (b) List the

> List the elements of each of the following sample spaces: (a) the set of integers between 1 and 50 divisible by 8; (b) the set S = {x | x2 + 4x − 5 = 0}; (c) the set of outcomes when a coin is tossed until a tail or three heads appear; (d) the set S = {x

> Exercise 1.3 on page 13 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. (a) Calculate the sample variance as well as standard deviation in tensile

> Consider the drying time data for Exercise 1.1 on page 13. Compute the sample variance and sample standard deviation. Data from Exercise 1.1: The following measurements were recorded for the drying time, in hours, of a certain brand of latex paint.

> 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 &acirc;&#151;&brvbar; C and 45 &acirc;&#151;&b

> 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 assig

> 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

> 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

> 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 const

> It is known that 60% of mice inoculated with a serum are protected from a certain disease. If 5 mice are inoculated, find the probability that (a) none contracts the disease; (b) fewer than 2 contract the disease; (c) more than 3 contract the disease.

> 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. 919 1196 785 1126 936 918 1156 920 948 1067 1092 1162 1170 929 950 905 972 1035

> 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 batc

> Use the data in Exercise 1.24 to construct a box plot. Data in Exercise 1.24: 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. 3.79 2.99 2.77

> 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 mol

> 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

> 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 carryi

> 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. (a) Calculate the sample mean. (b) Calculate

> 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. (a) Compute the sample mean and sample standard deviation. (b) Construct a relative frequency hi

> 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. (a) Construct a dot plot as in Figure 1.1. (b) Compute the sample means for the two years and sup

> The following data are the measures of the diameters of 36 rivet heads in 1/100 of an inch. (a) Compute the sample mean and sample standard deviation. (b) Construct a relative frequency histogram of the data. (c) Comment on whether or not there is any

> The percentage of wins for the Chicago Bulls basketball team going into the playoffs for the 1996–97 season was 87.7. Round the 87.7 to 90 in order to use Table A.1. (a) What is the probability that the Bulls sweep (4-0) the initial best-of-7 playoff ser

> The lengths of power failures, in minutes, are recorded in the following table. (a) Find the sample mean and sample median of the power-failure times. (b) Find the sample standard deviation of the power failure times. 22 18 135 15 90 78 69 98 102 8

> The following data represent the length of life, in seconds, of 50 fruit flies subject to a new spray in a controlled laboratory experiment: (a) Construct a double-stem-and-leaf plot for the life span of the fruit flies using the stems 0, 0&Acirc;&midd

> The following data represent the length of life in years, measured to the nearest tenth, of 30 similar fuel pumps: (a) 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

> The following scores represent the final examination grades for an elementary statistics course: (a) Construct a stem-and-leaf plot for the examination grades in which the stems are 1, 2, 3, . . . , 9. (b) Construct a relative frequency histogram, draw

> 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: (a) Find the sample mean for each group. (b) Find the sample standard deviation for e

> Show that the n pieces of information in E(ri – )2 i=1 n Σα 2(ri – x) = 0. i=1

> Five independent coin tosses result in HHHHH. It turns out that if the coin is fair the probability of this outcome is (1/2)5 = 0.03125. Does this produce strong evidence that the coin is not fair? Comment and use the concept of P-value discussed in Sec

> 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: 572, 572, 573, 568, 569, 575, 565, 570. (a) Find the sample mean and median. (b) Find the sample varianc

> 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: 123, 116, 122, 110, 175, 126, 125, 111, 118, 117. (a) Find the sample mean and median. (b) What fea

> For Exercise 1.6 on page 13, 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. Data fro

> A national study that examined attitudes about antidepressants revealed that approximately 70% of respondents believe “antidepressants do not really cure anything; they just cover up the real trouble.” According to this study, what is the probability tha

> Consider the data in Exercise 1.5 on page 13. Compute the sample variance and the sample standard deviation for both control and treatment groups. Data in Exercise 1.5: Twenty adult males between the ages of 30 and 40 participated in a study to evaluate

> For the data of Exercise 1.4 on page 13, compute both the mean and the variance in &acirc;&#128;&#156;flexibility&acirc;&#128;&#157; for both company A and company B. Does there appear to be a difference in flexibility between company A and company B? D

> There is a 50-50 chance that the queen carries the gene of hemophilia. If she is a carrier, then each prince has a 50-50 chance of having hemophilia independently. If the queen is not a carrier, the prince will not have the disease. Suppose the queen has

> During bad economic times, industrial workers are dismissed and are often replaced by machines. The history of 100 workers whose loss of employment is attributable to technological advances is reviewed. For each of these individuals, it is determined if

> A survey of those using a particular statistical software system indicated that 10% were dissatisfied. Half of those dissatisfied purchased the system from vendor A. It is also known that 20% of those surveyed purchased from vendor A. Given that the soft

> A firm is accustomed to training operators who do certain tasks on a production line. Those operators who attend the training course are known to be able to meet their production quotas 90% of the time. New operators who do not take the training course o

> An industrial plant is conducting a study to determine how quickly injured workers are back on the job following injury. Records show that 10% of all injured workers are admitted to the hospital for treatment and 15% are back on the job the next day. In

> In the field of quality control, the science of statistics is often used to determine if a process is “out of control.” Suppose the process is, indeed, out of control and 20% of items produced are defective. (a) If three items arrive off the process line

> A construction company employs two sales engineers. Engineer 1 does the work of estimating cost for 70% of jobs bid by the company. Engineer 2 does the work for 30% of jobs bid by the company. It is known that the error rate for engineer 1 is such that 0

> A rare disease exists with which only 1 in 500 is affected. A test for the disease exists, but of course it is not infallible. A correct positive result (patient actually has the disease) occurs 95% of the time, while a false positive result (patient doe

> A traffic control engineer reports that 75% of the vehicles passing through a checkpoint are from within the state. What is the probability that fewer than 4 of the next 9 vehicles are from out of state?

> A producer of a certain type of electronic component ships to suppliers in lots of twenty. Suppose that 60% of all such lots contain no defective components, 30% contain one defective component, and 10% contain two defective components. A lot is picked,

> A certain form of cancer is known to be found in women over 60 with probability 0.07. A blood test exists for the detection of the disease, but the test is not infallible. In fact, it is known that 10% of the time the test gives a false negative (i.e., t

> Consider the situation in Exercise 2.116, and suppose that the manufacturer can try only two combinations in a day. (a) What is the probability that any given set of two runs is chosen? (b) What is the probability that the highest temperature is used in

> A manufacturer is studying the effects of cooking temperature, cooking time, and type of cooking oil for making potato chips. Three different temperatures, 4 different cooking times, and 3 different oils are to be used. (a) What is the total number of co

> A certain federal agency employs three consulting firms (A, B, and C) with probabilities 0.40, 0.35, and 0.25, respectively. From past experience it is known that the probability of cost overruns for the firms are 0.05, 0.03, and 0.15, respectively. Supp

> A shipment of 12 television sets contains 3 defective sets. In how many ways can a hotel purchase 5 of these sets and receive at least 2 of the defective sets?

> From a box containing 6 black balls and 4 green balls, 3 balls are drawn in succession, each ball being replaced in the box before the next draw is made. What is the probability that (a) all 3 are the same color? (b) each color is represented?

> From 4 red, 5 green, and 6 yellow apples, how many selections of 9 apples are possible if 3 of each color are to be selected?

> In a certain federal prison, it is known that 2/3 of the inmates are under 25 years of age. It is also known that 3/5 of the inmates are male and that 5/8 of the inmates are female or 25 years of age or older. What is the probability that a prisoner sele

> The probability that a patient recovers from a delicate heart operation is 0.8. What is the probability that (a) exactly 2 of the next 3 patients who have this operation survive? (b) all of the next 3 patients who have this operation survive?

> The probability that a patient recovers from a delicate heart operation is 0.9. What is the probability that exactly 5 of the next 7 patients having this operation survive?