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statistics for engineers and scientists

Introduction To Probability And Statistics For Engineers And Scientists 4th Edition Sheldon M. Ross - Solutions

The probability of the closing of the ith relay in the circuits shown is given by pi , i =1, 2, 3, 4, 5. If all relays function independently, what is the probability that a current flows between A and B for the respective circuits?(a) If the ith component functions with probability Pi , i =1, 2,
A pair of fair dice is rolled. Let E denote the event that the sum of the dice is equal to 7.(a) Show that E is independent of the event that the first die lands on 4.(b) Show that E is independent of the event that the second die lands on 3.
Suppose that an insurance company classifies people into one of three classes —good risks, average risks, and bad risks. Their records indicate that the probabilities that good, average, and bad risk persons will be involved in an accident over a 1-year span are, respectively, .05, .15, and .30.
Prostate cancer is the most common type of cancer found in males. As an indicator of whether a male has prostate cancer, doctors often perform a test that measures the level of the PSA protein (prostate specific antigen) that is produced only by the prostate gland. Although higher PSA levels are
Each of 2 cabinets identical in appearance has 2 drawers. Cabinet A contains a silver coin in each drawer, and cabinet B contains a silver coin in one of its drawers and a gold coin in the other. A cabinet is randomly selected, one of its drawers is opened, and a silver coin is found. What is the
Each of 2 balls is painted black or gold and then placed in an urn. Suppose that each ball is colored black with probability 12, and that these events are independent.(a) Suppose that you obtain information that the gold paint has been used (and thus at least one of the balls is painted gold).
A total of 600 of the 1,000 people in a retirement community classify themselves as Republicans, while the others classify themselves as Democrats. In a local election in which everyone voted, 60 Republicans voted for the Democratic candidate, and 50 Democrats voted for the Republican candidate. If
Two balls, each equally likely to be colored either red or blue, are put in an urn.At each stage one of the balls is randomly chosen, its color is noted, and it is then returned to the urn. If the first two balls chosen are colored red, what is the probability that(a) both balls in the urn are
You ask your neighbor to water a sickly plant while you are on vacation.Without water it will die with probability .8; with water it will die with probability .15. You are 90 percent certain that your neighbor will remember to water the plant.(a) What is the probability that the plant will be alive
A red die, a blue die, and a yellow die (all six-sided) are rolled. We are interested in the probability that the number appearing on the blue die is less than that appearing on the yellow die which is less than that appearing on the red die. (That is, if B (R) [Y ] is the number appearing on the
There are two local factories that produce radios. Each radio produced at factory A is defective with probability .05, whereas each one produced at factory B is defective with probability .01. Suppose you purchase two radios that were produced at the same factory, which is equally likely to have
A total of 500 married working couples were polled about their annual salaries, with the following information resulting.Thus, for instance, in 36 of the couples the wife earned more and the husband earned less than $25,000. If one of the couples is randomly chosen, what is (a) the probability that
Fifty-two percent of the students at a certain college are females. Five percent of the students in this college are majoring in computer science. Two percent of the students are women majoring in computer science. If a student is selected at random, find the conditional probability that(a) this
A couple has 2 children. What is the probability that both are girls if the eldest is a girl?
Of three cards, one is painted red on both sides; one is painted black on both sides;and one is painted red on one side and black on the other. A card is randomly chosen and placed on a table. If the side facing up is red, what is the probability that the other side is also red?
The sample mean of the annual salaries of a group of 100 accountants who work at a large accounting firm is $130,000 with a sample standard deviation of $20,000.If a member of this group is randomly chosen, what can we say about(a) the probability that his or her salary is between $90,000 and
There is a 60 percent chance that the event A will occur. If A does not occur, then there is a 10 percent chance that B will occur.(a) What is the probability that at least one of the events A or B occurs?(b) If A is the event that the democratic candidate wins the presidential election in 2012 and
Suppose that distinct integer values are written on each of 3 cards. These cards are then randomly given the designations A, B, and C. The values on cards A and B are then compared. If the smaller of these values is then compared with the value on card C, what is the probability that it is also
Consider a set of 23 unrelated people. Because each pair of people shares the same birthday with probability 1/365, and there are23 2=253 pairs, why isn’t the probability that at least two people have the same birthday equal to 253/365?
A group of 5 boys and 10 girls is lined up in random order — that is, each of the 15! permutations is assumed to be equally likely.(a) What is the probability that the person in the 4th position is a boy?(b) What about the person in the 12th position?(c) What is the probability that a particular
Show thatFor a combinatorial argument, consider a set of n items and fix attention on one of these items. How many different sets of size r contain this item, and how many do not? ()=(-1)+("") r
Show thatNow present a combinatorial argument for the foregoing by explaining why a choice of r items from a set of size n is equivalent to a choice of n−r items from that set. n -r ("-")=(1)
Calculate93,96,7 2,7 5,10 7.
Show that the probability that exactly one of the events E or F occurs is equal to P(E)+P(F )−2P(EF ).
Prove that(a) P(EF c ) = P(E)−P(EF )(b) P(EcF c ) = 1−P(E)−P(F )+P(EF )
If P(E)=.9 and P(F )=.9, show that P(EF )≥.8. In general, prove Bonferroni’s inequality, namely that P(EF)≥P(E)+P(F )−1
Prove Boole’s inequality, namely that P =1 ) i=1
Show that if E⊂F then P(E)≤P(F ). (Hint:Write F as the union of two mutually exclusive events, one of them being E.)
For the following Venn diagram, describe in terms of E, F, and G the events denoted in the diagram by the Roman numerals I through VII. E F S I II IV III VII V VI G
Use Venn diagrams (or any other method) to show that(a) EF⊂E, E ⊂E ∪ F ;(b) if E ⊂F then F c ⊂Ec ;(c) the commutative laws are valid;(d) the associative laws are valid;(e) F =FE ∪ FEc ;(f ) E ∪ F =E ∪ EcF ;(g) DeMorgan’s laws are valid.
Find simple expressions for the events(a) E ∪ Ec ;(b) EE c ;(c) (E ∪ F )(E ∪ F c );(d) (E ∪ F )(Ec ∪ F )(E ∪ F c );(e) (E ∪ F )(F ∪G).
Let E, F, G be three events. Find expressions for the events that of E, F, G(a) only E occurs;(b) both E and G but not F occur;(c) at least one of the events occurs;(d) at least two of the events occur;(e) all three occur;(f ) none of the events occurs;(g) at most one of them occurs;(h) at most two
Asystem is composed of four components, each of which is either working or failed.Consider an experiment that consists of observing the status of each component, and let the outcome of the experiment be given by the vector (x1, x2, x3, x4) where xi is equal to 1 if component i is working and is
Two dice are thrown. Let E be the event that the sum of the dice is odd, let F be the event that the first die lands on 1, and let G be the event that the sum is 5.Describe the events EF, E ∪ F, FG, EF c , EFG.
Let S = {1, 2, 3, 4, 5, 6, 7}, E = {1, 3, 5, 7}, F = {7, 4, 6}, G = {1, 4}. Find(a) EF ; (c) EG c ; (e) Ec (F ∪ G);(b) E ∪ FG ; (d) EF c ∪G; (f ) EG ∪ FG.
An experiment consists of tossing a coin three times. What is the sample space of this experiment? Which event corresponds to the experiment resulting in more heads than tails?
A box contains three marbles — one red, one green, and one blue. Consider an experiment that consists of taking one marble from the box, then replacing it in the box and drawing a second marble from the box. Describe the sample space.Repeat for the case in which the second marble is drawn without
A set of k coupons, each of which is independently a type j coupon with probability pj ,nj=1 pj =1, is collected. Find the probability that the set contains a type j coupon given that it contains a type i, i = j.
A system composed of n separate components is said to be a parallel system if it functions when at least one of the components functions. (See Figure 3.7.) For such a system, if component i, independent of other components, functions with probability pi , i = 1, . . . , n, what is the probability
A plane is missing and it is presumed that it was equally likely to have gone down in any of three possible regions. Let 1−αi denote the probability the plane will be found upon a search of the ith region when the plane is, in fact, in that region, i =1, 2, 3. (The constants αi are called
Let us now suppose that the new evidence is subject to different possible interpretations, and in fact only shows that it is 90 percent likely that the criminal possesses this certain characteristic. In this case, how likely would it be that the suspect is guilty (assuming, as before, that he has
At a certain stage of a criminal investigation, the inspector in charge is 60 percent convinced of the guilt of a certain suspect. Suppose now that a new piece of evidence that shows that the criminal has a certain characteristic (such as left-handedness, baldness, brown hair, etc.) is uncovered.
A laboratory blood test is 99 percent effective in detecting a certain disease when it is, in fact, present. However, the test also yields a “false positive” result for 1 percent of the healthy persons tested. (That is, if a healthy person is tested, then, with probability .01, the test result
In answering a question on a multiple-choice test, a student either knows the answer or she guesses. Let p be the probability that she knows the answer and 1−p the probability that she guesses. Assume that a student who guesses at the answer will be correct with probability 1/m, where m is the
An insurance company believes that people can be divided into two classes — those that are accident prone and those that are not. Their statistics show that an accident-prone person will have an accident at some time within a fixed 1-year period with probability .4, whereas this probability
Ms. Perez figures that there is a 30 percent chance that her company will set up a branch office in Phoenix. If it does, she is 60 percent certain that she will be made manager of this new operation. What is the probability that Perez will be a Phoenix branch office manager?
The organization that Jones works for is running a father–son dinner for those employees having at least one son. Each of these employees is invited to attend along with his youngest son. If Jones is known to have two children, what is the conditional probability that they are both boys given
A bin contains 5 defective (that immediately fail when put in use), 10 partially defective (that fail after a couple of hours of use), and 25 acceptable transistors.A transistor is chosen at random from the bin and put into use. If it does not immediately fail, what is the probability it is
If n people are present in a room, what is the probability that no two of them celebrate their birthday on the same day of the year? How large need n be so that this probability is less than 1/2?
A basketball team consists of 6 black and 6 white players. The players are to be paired in groups of two for the purpose of determining roommates. If the pairings are done at random, what is the probability that none of the black players will have a white roommate?
A committee of size 5 is to be selected from a group of 6 men and 9 women.If the selection is made randomly, what is the probability that the committee consists of 3 men and 2 women?
A class in probability theory consists of 6 men and 4 women. An exam is given and the students are ranked according to their performance. Assuming that no two students obtain the same score, (a) how many different rankings are possible? (b) If all rankings are considered equally likely, what is the
Mr. Jones has 10 books that he is going to put on his bookshelf. Of these, 4 are mathematics books, 3 are chemistry books, 2 are history books, and 1 is a language book. Jones wants to arrange his books so that all the books dealing with the same subject are together on the shelf. How many
A recent study (reported in the May 5, 2008, LA Times) yielded a positive correlation between breast-fed babies and scores on a vocabulary test taken at age 6.Discuss the potential difficulties in interpreting the results of this study.
In a study of children in grades 2 through 4, a researcher gave each student a reading test. When looking at the resulting data the researcher noted a positive correlation between a student’s reading test score and height. The researcher concluded that taller children read better because they can
Verify property 4 of the sample correlation coefficient.
Verify property 3 of the sample correlation coefficient.
Using data on the first 10 cities listed inTable 2.5, draw a scatter diagram and find the sample correlation coefficient between the January and July temperatures.
If for each of the fifty states we plot the paired data consisting of the average income of residents of the state and the number of foreign-born immigrants who reside in the state, then the data pairs will have a positive correlation. Can we conclude that immigrants tend to have higher incomes
A random sample of individuals were rated as to their standing posture. In addition, the numbers of days of back pain each had experienced during the past year were also recorded. Surprisingly to the researcher these data indicated a positive correlation between good posture and number of days of
Use the data concerning the first 10 states listed in the table given in Problem 15.(a) Draw a scatter diagram relating the 1992 and 1993 salaries.(b) Determine the sample correlation coefficient.
Use the data of Problem 16.(a) Compute the sample mean and sample median.(b) Are the data approximately normal?(c) Compute the sample standard deviation s.(d) What percentage of the data fall within ¯x ±1.5s?(e) Compare your answer in part (d) to that given by the empirical rule.(f ) Compare your
Would you expect that a histogram of the weights of all the members of a health club would be approximately normal?
Do the data in Problem 26 appear to be approximately normal? For parts (c) and(d) of this problem, compare the approximate proportions given by the empirical rule with the actual proportions.
The following are the grade point averages of 30 students recently admitted to the graduate program in the Department of Industrial Engineering and Operations Research at the University of California at Berkeley.3.46, 3.72, 3.95, 3.55, 3.62, 3.80, 3.86, 3.71, 3.56, 3.49, 3.96, 3.90, 3.70, 3.61,
A chemical engineer desiring to study the evaporation rate of water from brine evaporation beds obtained data on the number of inches of evaporation in each of 55 July days spread over 4 years. The data are given in the following stem and leaf plot, which shows that the smallest data value was .02
The average particulate concentration, in micrograms per cubic meter, was measured in a petrochemical complex at 36 randomly chosen times, with the following concentrations resulting:5, 18, 15, 7, 23, 220, 130, 85, 103, 25, 80, 7, 24, 6, 13, 65, 37, 25, 24, 65, 82, 95, 77, 15, 70, 110, 44, 28, 33,
The table on page 51 gives the numbers of dogs, categorized by breed, registered in the American Kennel Club in 2006. Represent these numbers in a box plot.
Use the part of the table given in Problem 4 that gives the percentage of workers in each state that use public transportation to get to work to draw a box plot of these 50 percentages.
Use the following figure (see next page), which gives the amounts of federal research money given to 15 universities in 1992, to answer this problem.(a) Which universities were given more than $225 million?(b) Approximate the sample mean of the amounts given to these universities.(c) Approximate
Use the following table to find the quartiles of the average annual pay in the specified areas. Average Annual Pay by New York State Metropolitan Areas, 1999 Rank Amt. Rank Amt. Albany-Sch'dy-Troy $31,901 Nassau-Suffolk $36,944 Binghamton 29,167 New York City 52,351 Buffalo-Niagara Falls 30,487
Use the data of Problem 8 to find the(a) 10 percentile of the winning scores;(b) 90 percentile of the winning scores.
A computationally efficient way to compute the sample mean and sample variance of the data set x1, x2, . . . , xn is as follows. Letbe the sample mean of the first j data values, and letbe the sample variance of the first j, j ≥ 2, values. Then, with s2 1 =0, it can be shown thatand(a) Use the
An experiment measuring the percent shrinkage on drying of 50 clay specimens produced the following data:18.2 21.2 23.1 18.5 15.6 20.8 19.4 15.4 21.2 13.4 16.4 18.7 18.2 19.6 14.3 16.6 24.0 17.6 17.8 20.2 17.4 23.6 17.5 20.3 16.6 19.3 18.5 19.3 21.2 13.9 20.5 19.0 17.6 22.3 18.4 21.2 20.4 21.4 20.3
The following data represent the lifetimes (in hours) of a sample of 40 transistors:112, 121, 126, 108, 141, 104, 136, 134 121, 118, 143, 116, 108, 122, 127, 140 113, 117, 126, 130, 134, 120, 131, 133 118, 125, 151, 147, 137, 140, 132, 119 110, 124, 132, 152, 135, 130, 136, 128(a) Determine the
The following table gives the average annual pay, per state, in the years 1992 and 1993.(a) Do you think that the sample mean of the averages for the 50 states will equal the value given for the entire United States?(b) If the answer to part (a) is no, explain what other information aside from just
The sample mean and sample variance of five data values are, respectively, ¯x =104 and s2 =4. If three of the data values are 102, 100, 105, what are the other two data values?
Using the table given in Problem 4, find the sample variance of the average travel time for those states in the(a) south Atlantic;(b) mountain region.
The following are the percentages of ash content in 12 samples of coal found in close proximity:9.2, 14.1, 9.8, 12.4, 16.0, 12.6, 22.7, 18.9, 21.0, 14.5, 20.4, 16.9 Find the(a) sample mean, and(b) sample standard deviation of these percentages.
The following table gives the number of pedestrians, classified according to age group and sex, killed in fatal road accidents in England in 1922.(a) Approximate the sample means of the ages of the males.(b) Approximate the sample means of the ages of the females.(c) Approximate the quartiles of
A total of 100 people work at company A, whereas a total of 110 work at company B. Suppose the total employee payroll is larger at company A than at company B.(a) What does this imply about the median of the salaries at company A with regard to the median of the salaries at company B?(b) What does
Using the table given in Problem 4, find the sample mean and sample median of the average travel time for those states in the(a) northeast;(b) midwest;(c) south;(d) west.
The following table gives the winning scores in the Masters golf tournament for the years from 1973 to 2008. Use it(a) to construct a stem and leaf plot, and(b) to find the sample median of the winning scores in these years. Year Player Score 1973 Tommy Aaron 283 1974 Gary Player 278 1975 1976 Jack
(Use the table from Problem 6.)(a) Represent the number of yearly airline fatalities in a histogram.(b) Represent the number of yearly airline fatalities in a stem and leaf plot.(c) Find the sample mean of the number of yearly airline fatalities.(d) Find the sample median of the number of yearly
The following table gives the number of commercial airline accidents and fatalities in the United States in the years from 1985 to 2006.(a) Represent the number of yearly airline accidents in a frequency table.(b) Give a frequency polygon graph of the number of yearly airline accidents.(c) Give a
Choose a book or article and count the number of words in each of the first 100 sentences. Present the data in a stem and leaf plot. Now choose another book or article, by a different author, and do the same. Do the two stem and leaf plots look similar? Do you think this could be a viable method
The following table gives the average travel time to work for workers in each of the 50 states as well as the percentage of those workers who use public transportation.(a) Represent the data relating to the average travel times in a histogram.(b) Represent the data relating to the percentage of
The following are the estimated oil reserves, in billions of barrels, for four regions in theWestern Hemisphere.United States 38.7 South America 22.6 Canada 8.8 Mexico 60.0 Represent these data in a pie chart.
Explain how a pie chart can be constructed. If a data value had relative frequency r, at what angle would the lines defining its sector meet?
The following is a sample of prices, rounded to the nearest cent, charged per gallon of standard unleaded gasoline in the San Francisco Bay area in June 1997.3.88, 3.90, 3.93, 3.90, 3.93, 3.96, 3.88, 3.94, 3.96, 3.88, 3.94, 3.99, 3.98 Represent these data in(a) a frequency table;(b) a relative
The following stem and leaf plot gives the scores on a statistics exam taken by industrial engineering students.By standing the stem and leaf plot on its side we can see that the corresponding histogram is approximately normal. Use it to assess the empirical rule. 9 0, 1,4 8 7 3,5,5,7,8
Find the sample variances of the data sets A and B given below.A : 3, 4, 6, 7, 10 B :−20, 5, 15, 24
In a study reported in Hoel, D. G., “A representation of mortality data by competing risks,” Biometrics, 28, pp. 475–488, 1972, a group of 5-week-old mice were each given a radiation dose of 300 rad. The mice were then divided into two groups;the first group was kept in a germ-free
The winning scores in the U.S. Masters golf tournament in the years from 1999 to 2008 were as follows:280, 278, 272, 276, 281, 279, 276, 281, 289, 280 Find the sample mean of these scores.
An article reported that a survey of clothing worn by pedestrians killed at night in traffic accidents revealed that about 80 percent of the victims were wearing darkcolored clothing and 20 percent were wearing light-colored clothing. The conclusion drawn in the article was that it is safer to wear
A university plans on conducting a survey of its recent graduates to determine information on their yearly salaries. It randomly selected 200 recent graduates and sent them questionnaires dealing with their present jobs. Of these 200, however, only 86 were returned. Suppose that the average of the
To determine the proportion of people in your town who are smokers, it has been decided to poll people at one of the following local spots:(a) the pool hall;(b) the bowling alley;(c) the shopping mall;(d) the library.Which of these potential polling places would most likely result in a reasonable
A researcher is trying to discover the average age at death for people in the United States today. To obtain data, the obituary columns of the New York Times are read for 30 days, and the ages at death of people in the United States are noted. Do you think this approach will lead to a
The approach used in Problem 1(e) led to a disastrous prediction in the 1936 presidential election, in which Franklin Roosevelt defeated Alfred Landon by a landslide. A Landon victory had been predicted by the Literary Digest. The magazine based its prediction on the preferences of a sample of
An election will be held next week and, by polling a sample of the voting population, we are trying to predict whether the Republican or Democratic candidate will prevail. Which of the following methods of selection is likely to yield a representative sample?(a) Poll all people of voting age

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