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Statistics For Business And Economics 4th Edition Paul Newbold - Solutions

Scores on an achievement test are known to be normally distributed, with mean 420 and standard deviation 80.(a) For a randomly chosen person taking this test, what is the probability of a score be- tween 400 and 480? (b) What is the minimum score needed in order to be in the top 10% of all people
A company services copiers. A review of its records shows that the time taken for a ser- vice call can be represented by a normal random variable with mean 75 minutes and stan- dard deviation 20 minutes. (a) What proportion of service calls take less than one hour? (b) What proportion of service
An instructor has found that times spent by students on a particular homework assignment follow a normal distribution with mean 150 minutes and standard deviation 40 minutes. (a) The probability is .9 that a randomly chosen student spends more than how many min- utes on this assignment? (b) The
I am considering two alternative investments. In both cases, I am unsure about the per- centage return but believe that my uncertainty can be represented by normal distributions with the means and standard deviations shown in the accompanying table. I want to make the investment that is more likely
Scores on an examination taken by a very large group of students are normally distributed with mean 700 and standard deviation 120.(a) An A is awarded for a score higher than 820.What proportion of all students obtain an A? (b) A B is awarded for scores between 730 and 820.An instructor has a
A new television series is to be shown. A broadcasting executive feels that his uncertainty. about the rating which the show will receive in its first month can be represented by a nor- mal distribution with mean 18.2 and standard deviation 1.6. According to this executive, the probability is that
Scores on a test follow a normal distribution. What is the probability that a randomly se- lected student will achieve a score that exceeds the mean score by more than 1.5 standard deviations?
A contractor regards the cost of fulfilling a particular contract as a normally distributed random variable with mean $500,000 and standard deviation $50,000. (a) What is the probability that the cost of fulfilling the contract will be between $460,000 and $540,000? (b) The probability is .2 that
A company produces bags of a chemical, and it is concerned about impurity content. It is believed that the weights of impurities per bug are normally distributed, with mean 12.2 grams, and standard deviation 2.8 grams. A bag is chosen at random. (a) What is the probability that it contains less
Let the random variable Z follow a standard normal distribution. (a) Find P(Z 1.33) (c) Find P(Z-1.70) (d) Find P(Z-1.00) (e) Find P(1.20
Continuing Example 5.4, assume the same specifications as that example, except that now we no longer assume that the random variables X and Y are independent of one another. Denote by C the covariance between these random variables. Show now that the variance of total return is Var(R)-(2a -2,000+
A homeowner has installed a new energy-efficient furnace. It is estimated that, over a year, the new furnace will reduce energy costs by an amount that can be regarded as a ran- dom variable with mean $200 and standard deviation $60. Stating any assumptions you need to make, find the mean and
An investor plans to divide $200,000 between two investments. The first yields a certain profit of 10%, while the second yields a profit with expected value 18% and standard devi ation 6%. If the investor divides the money equally between these two investments, find the mean and standard deviation
A coin is thrown three times, and interest is in the number, X, of heads resulting. (a) Find the probability function of the random variable X. (b) Find the mean and standard deviation of the random variable X. (c) Consider a game in which you gain $2 if either one or three heads result, you lose
A recent estimate suggested that of all individuals and couples reporting income in excess of $200,000, 6.5% either paid no federal tax or paid tax at an effective rate of less than 15%. A random sample of 100 of those reporting income in excess of $200,000 was taken. What is the probability that
A team of five analysts is about to examine the earnings prospects of twenty corporations. Each of the five analysts will study four of the corporations. These analysts are not equally competent. In fact, one of them is a star, having an excellent record of anticipating chang ing trends. Ideally,
Using detailed cash flow information, a financial analyst claims to be able to spot compa- nies that are likely candidates for bankruptcy. The analyst is presented with information on the past records of fifteen companies and told that in fact five of these have failed. He selects as candidates for
It is estimated that 55% of the freshmen entering a particular college will graduate from that college in four years. (a) For a random sample of five entering freshmen, what is the probability that exactly three will graduate in four years? (b) For a random sample of five entering freshmen, what is
A basketball team's star 3-point shooter takes six 3-point shots in a game. Historically, he makes 40% of all 3-point shots attempted. Answer the following questions about the out- come of the six 3-point shots taken in this game, stating at the outset what assumptions you have made. (a) Find the
A long-distance taxi service owns four vehicles. These are of different ages and have dif- ferent repair records. The probabilities that on any given day, each vehicle will be avail- able for use are 95, 90, .90, and .80. Whether one vehicle is available is independent of whether any other vehicle
Develop realistic examples of pairs of random variables for which you would expect to find: (a) Positive covariance (b) Negative covariance (c) Zero covariance
A multiple-choice test has nine questions. For each question, there are four possible an swers from which to select. One point is awarded for each correct answer, and points are not subtracted for incorrect answers. The instructor awards a bonus point if the student spells his or her name
A car salesman estimates the following probabilities for the number of cars that he will sell in the next week. NUMBER OF CARS 0 2 3 4 PROBABILITY .10 .20 35 .16 12 07 (a) Find the expected number of cars that will be sold in the week. (b) Find the standard deviation of the number of cars that will
A contractor estimates the probabilities for the number of days required to complete a cer- tain type of construction project as follows:(b) Find the expected time to complete a project. (c) Find the standard deviation of time required to complete a project. (d) The contractor's project cost is
Explain what can be learned from each of the following: (a) A graph of the probability function of a random variable (b) A graph of the cumulative probability function of a random variable (c) The standard deviation of a random variable (d) The covariance between a pair of random variables
Develop a realistic business example (other than those in the text and in other exercises) in which each of the following probability distributions would be appropriate. (a) The binomial distribution (b) The hypergeometric distribution (c) The Poisson distribution
Explain carefully, with an illustrative example, what is meant by the expected value of a random variable. Why is this concept important?
A state has a law requiring motorists to carry insurance. It was estimated that, despite this law, 7.5% of all motorists in the state are uninsured. A random sample of 60 motorists was taken. Use the Poisson approximation to the binomial distribution to estimate the probabil- ity that at least
An insurance company holds fraud insurance policies on 6,000 firms. In any given year, the probability that any single policy will result in a claim is 001.Find the probability that at least three claims are made in a given year. Use the Poisson approximation to the bino- mial distribution.
A corporation has 250 personal computers. The probability that any one of them will re- quire repair in a given week is .01. Find the probability that fewer than four of the personal computers will require repair in a particular week. Use the Poisson approximation to the binomial distribution.
The Internal Revenue Service reported that 5.5% of all taxpayers filling out the 1040 short form make mistakes. If 100 of these forms are chosen at random, what is the probability that fewer than three of them contain errors? Use the Poisson approximation to the bino- mial distribution.
Records indicate that on average, 3.2 breakdowns per day occur on an urban highway dur- ing the moming rush hour. Assume that the distribution is Poisson. (a) Find the probability that on any given day, there will be fewer than two breakdowns on this highway during the early morning rush hour. (b)
A professor receives, on average, 4.2 telephone calls from students the day before a final examination. If the distribution of calls is Poisson, what is the probability of receiving at least three of these calls on such a day?
The number of accidents in a production facility has a Poisson distribution with mean 2.6 per month. (a) For a given month, what is the probability there will be fewer than two accidents? (b) For a given month, what is the probability there will be more than three accidents?
Customers arrive at a busy check-out counter at an average rate of three per minute. If the distribution of arrivals is Poisson, find the probability that in any given minute there will be two or fewer arrivals.
A bank executive is presented with loan applications from ten people. The profiles of the applicants are similar, except that five are minorities and five are nonminorities. In the end, the executive approved six of the applications. If these six approvals had been chosen at random from the ten
A bond analyst was given a list of twelve corporate bonds. From that list, she selected three whose ratings she felt were in danger of being downgraded in the next year. In actu- ality, a total of four of the twelve bonds on the list had their ratings downgraded in the next year. Suppose that the
A committee of eight members is to be formed from a group of eight men and eight women. If the choice of committee members is made randomly, what is the probability that precisely half of these members will be women?
A company receives a shipment of sixteen items. A random sample of four items is se- lected, and the shipment is rejected if any of these items proves to be defective. (a) What is the probability of accepting a shipment containing four defective items? (b) What is the probability of accepting a
A company receives large shipments of parts from two sources. Seventy percent of the shipments come from a supplier whose shipments typically contain 10% defectives, while the remainder are from a supplier whose shipments typically contain 20% defectives. A manager receives a shipment but does not
The following two acceptance rules are being considered for determining whether to take delivery of a large shipment of components: (i) A random sample of ten components is checked, and the shipment is accepted only if none of them is defective. (i) A random sample of twenty components is checked,
A company receives a very large shipment of components. A random sample of sixteen of these components are checked, and the shipment is accepted if fewer than two of these components are defective. What is the probability of accepting a shipment containing:(a) 5% defectives? (b) 15% defectives? (c)
A campus finance officer finds that for all parking tickets issued, fines on 78% are paid. The fine is $2. In the most recent week, 620 parking tickets have been issued. (a) Find the mean and standard deviation of the number of these tickets for which the fines will be paid. (b) Find the mean and
We have seen that, for a binomial distribution with a trials, each with probability of suc- cess p, the mean is Verify this result for the data of Example 4.9 by calculating the mean direct from () showing that, for the binomial distribution, the two formulas produce the same answer.
A family of mutual funds maintains a service that allows clients to switch money among accounts through a telephone call. It was estimated that 3.2% of callers either got a busy signal or were kept on hold so long that they hung up. Fund management assesses any fail- ure of this sort as a $10
An automobile dealer mounts a new promotional campaign, in which it is promised that purchasers of new automobiles may, if dissatisfied for any reason, return them within two days of purchase and receive a full refund. It is estimated that the cost to the dealer of such a refund is $250. The dealer
Following a touchdown, a college football coach has the option to elect to attempt a "2-point conversion", that is, 2 additional points are scored if the attempt is successful. and none if it is unsuccessful. The coach believes that the probability is 4 that his team will be successful in any
A small commuter airline flies planes that can seat up to eight passengers. The airline has determined that the probability that a ticketed passenger will not show up for a flight is .2. For each flight, the airline sells tickets to the first ten people placing orders. The probabil- ity
The Cubs are to play a series of five games in St. Louis against the Cardinals. For any one game, it is estimated that the probability of a Cubs win is 4.The outcomes of the five games are independent of one another. (a) What is the probability that the Cubs will win all five games? (b) What is the
A company installs new central heating furnaces, and has found that for 15% of all instal- lations a return visit is needed to make some modifications. Six installations were made in a particular week. Assume independence of outcomes for these installations. (a) What is the probability that a
Suppose that the probability is .5 that the value of the U.S. dollar will rise against the Japanese yen over any given week, and that the outcome in one week is independent of that in any other week. What is the probability that the value of the U.S. dollar will rise against the Japanese yen in a
A public interest group hires students to solicit donations by telephone. After a brief train- ing period, students make calls to potential donors and are paid on a commission basis. Experience indicates that early on, these students tend to have only modest success, and that 70% of them give up
A politician believes that 25% of all macroeconomists in senior positions would strongly support a proposal he wishes to advance. Suppose that this belief is correct and that five senior macroeconomists are approached at random. (a) What is the probability that at least one of the five would
From the results given in Section 4.4, it follows that for any pair of random variables X and Y, the variance of (XY) is the same as the variance of (YX). Explain why you would expect this to be so. [Note: (Y-X)=-(XY).J
A company has five representatives covering large territories and ten representatives cov- ering smaller territories. The probability distributions for the numbers of orders received by each of these types of representatives in a day are shown in the accompanying table. Assuming that the number of
Refer to the information in Exercise 24.Find the mean and standard deviation of the total number of complaints received in a week. Having reached this point, you are concerned that numbers of food and service complaints may not be independent of each other. However, you have no information about
A restaurant manager receives occasional complaints about the quality of both the food and the service. The marginal probability functions for the number of weekly complaints in each category are shown in the accompanying table. If complaints about food and ser- vice are independent of each other,
A college bookseller makes calls at the offices of professors and forms the impression that professors are more likely to be away from their offices on Friday than any other working day. A review of the records of calls, one-fifth of which are on Fridays, indicates that for 16% of Friday calls, the
A market researcher wants to determine whether a new model of a personal computer, which had been advertised on a late-night talk show, had achieved more brand name recognition among people who watched the show regularly than among people who did. not. After conducting a survey, it was found that
The accompanying table shows, for credit card holders with one to three cards, the joint probabilities for number of cards owned (X) and number of credit purchases made in a week (Y). NUMBER OF CARDS (X) 0 NUMBER OF PURCHASES IN WEEK (Y) 1 2 3 4 .08 .13 .09 .03 08 .08 .01 03 .06 888 .06 .09 .08 .08
A real estate agent is interested in the relationship between the number of lines in a news- paper advertisement for an apartment and the volume of enquiries from potential renters. Let volume of enquiries be denoted by the random variable X, with the value 0 for little in- terest, 1 for moderate
A researcher suspected that the number of between-meals snacks eaten by students in a day during final examinations week might depend on the number of tests a student had to take on that day. The accompanying table shows joint probabilities, estimated from a survey. NUMBER OF TESTS (N) NUMBER OF
An investor is considering three strategies for a $1,000 investment. The probable returns are estimated as follows: " STRATEGY STRATEGY 2 STRATEGY 3 A profit of $10,000 with probability 15 and a loss of $1,000 with probs- bility 85 A profit of $1,000 with probability 50, a profit of $500 with
A factory manager is considering whether to replace a temperamental machine. A review of past records indicates the following probability distribution for the number of break- downs of this machine in a week. NUMBER OF BREAKDOWNS PROBABILITY 1 2 3 .10 .26 .42 .16 .06 (a) Find the mean and standard
A store owner stocks an out-of-town newspaper, which is sometimes requested by a small number of customers. Each copy of this newspaper costs him 70 cents, and he sells them for 90 cents each. Any copies left over at the end of the day have no value and are de- stroyed. Any requests for copies that
Students in a large accounting class were asked to rate the course on a scale from 1 (poor) to 5 (excellent). The accompanying table shows proportions of students rating the course in each category. RATING PROPORTION 1 2 3 4 5 .07 19 28 30 .16 Find the mean and standard deviation of the ratings.
A professor teaches a large class, and has scheduled an examination for 7:00 P.M. in a dif- ferent classroom. She estimates the probabilities in the table for the number of students who will call her at home, in the hour before the examination, asking in which classroom it will be held. NUMBER OF
Refer to the information in Exercise 5.(a) Find the mean and variance of the random variable X in part (a). (b) Find the mean and variance of the random variable Y in part (b).
Refer to the information in Exercise 4.(a) Find the mean and standard deviation of the number of riders from this subdivision on this service on a weekday. (b) If the cost of a ride is 50 cents, find the mean and standard deviation of the total pay- ments of riders from this subdivision on this
Refer to the information in Exercise 3.1 (a) Find the mean and standard deviation of the number of paper clips per package. (b) The cost (in cents) of producing a package of clips is 16 +2X, where X is the number of clips in the package. The revenue from selling the package, however many clips it
Refer to the information in Exercise 2.Find the mean and standard deviation of the num- ber of orders for new furnaces in this period of two weeks.
Refer to the information in Exercise 1.Find the mean and variance of the number of re- turns of an automobile for corrections for defects during the warranty period.
Refer to Example 4.2. A player makes a bet in which he or she loses $1 if an odd number does not result and wins $1 if it does. Let the random variable X denote the player's gain (in dollars). Find the mean and standard deviation of X.
A student needs to know details of a class assignment that is due the next day, and decides to call fellow class members for this information. She believes that, for any particular call. the probability of obtaining the necessary information is 40.She decides to continue call-ing class members
(a) A very large shipment of parts contains 10% defectives. Two parts are chosen at ran- dom from the shipment and checked. Let the random variable X denote the number of defectives found. Find the probability function of this random variable. (b) A shipment of twenty parts contains two defectives.
A municipal bus company has started operations in a new subdivision. Records were kept on the numbers of riders from this subdivision on the early-morning service. The accom- panying table shows proportions over all weekdays. NUMBER OF RIDERS PROPORTION 0 1 2 3 4 5 6 7 102 12 .23 31 19 .08 .03 .02
A corporation produces packages of paper clips. The number of clips per package varies, as indicated in the accompanying table. NUMBER OF CLIPS PROPORTION OF PACKAGES 47 48 49 50 51 52 53 .04 .13 21 .29 .20 10 103 (a) Draw the probability function. (b) Calculate and draw the cumulative probability
A company specializes in installing and servicing central heating furnaces. In the pre- winter period, service calls may result in an order for a new furnace. The table shows. estimated probabilities for numbers of new furnace orders generated in this way in the last two weeks of September. NUMBER
An automobile dealer calculates the proportion of new cars sold that have been returned. various numbers of times for the correction of defects during the warranty period. The results are shown in the table. NUMBER OF RETURNS PROPORTION 0 1 2 .28 36 23 09 .04 (a) Draw the probability function. (b)
A record store owner assesses customers entering the store as high school age, college age, or older, and finds 30%, 50%, and 20% respectively of all customers fall into these categories. The owner also found that purchases were made by 20% of high school age customers, by 60% of college age
A restaurant manager classifies customers as well dressed, moderately dressed, or poorly dressed, and finds 50%, 40%, and 10% respectively of all customers fall into these cate- gories. The manager found that wine was ordered by 70% of the well dressed, by 50% of the moderately dressed, and by 30%
A market research group specializes in providing assessments of the prospects of sites for new clothing stores in shopping centers. The group assesses prospects as either good, fair, or poor. The records of requests for assessments made to this group were examined, and it was found that for all
A dean has found that 62% of entering freshmen and 78% of junior college transfers even- tually graduate. Of all entering students, 73% are freshmen, and the remainder are junior college transfers. (a) What is the probability that a randomly chosen entering student is a freshman who will eventually
The accompanying table shows, for 1,000 forecasts of earnings per share made by finan- cial analysts, the numbers of forecasts and outcomes in particular categories (compared with the previous year). OUTCOME FORECAST IMPROVEMENT ABOUT THE SAME WORSE Improvement 210 82 66 About the same 106 153 75
A professor finds that she awards a final grade of A to 20% of the students. Of those who obtain a final grade of A, 70% obtained an A in the midterm examination. Also, 10% of students who failed to obtain a final grade of A earned an A in the midterm exam. What is the probability that a student
Subscriptions to American History Illustrated are classified as gift, previous renewal, di- rect mail, or subscription service. In January 1979, 8% of expiring subscriptions were gift: 41%, previous renewal: 6%, direct mail: and 45% subscription service. The percent- ages of renewals in these four
Of 100 patients with a certain disease, ten were chosen at random to undergo a drug treat- ment that increases the cure rate from 50% for those not given the treatment to 75% for those given the drug treatment. (a) What is the probability that a randomly chosen patient both was cured and was given
A consignment of twelve electronic components contains one component that is faulty. Two components are chosen randomly from this consignment for testing. (a) How many different combinations of two components could be chosen? (b) What is the probability that the faulty component will be one of the
A jury of twelve members is to be selected from a panel consisting of eight men and eight women. (a) How many different jury selections are possible? (b) If the choice is made randomly, what is the probability that a majority of the jury members will be men?
A large corporation organized a ballot for all its workers on a new bonus plan. It was found that 65% of all night-shift workers favored the plan and that 40% of all women workers favored the plan. Also, 50% of all employees are night-shift workers, and 30% of all employees are women. Finally, 20%
In a campus restaurant, it was found that 35% of all customers order hot meals and that 50% of all customers are students. Further, 25% of all customers who are students order hot meals. (a) What is the probability that a randomly chosen customer is both a student and orders a hot meal? (b) If a
Based on a survey of students on a large campus, it was estimated that 35% of the students drink at least once a week in campus bars, and that 40% of all students have grade-point averages of B or better. Further, of those who drink at least once a week in campus bars, 30% have a B average or
Show that the probability of the union of the events A and B can be written P(AUB) = P(A) + P(B)[1 - P(A|B)]
State, giving reasons, whether each of the following claims is true or false: (a) The conditional probability of A given B must be at least as large as the probability of A. (b) An event must be independent of its complement. (c) The probability of A given B must be at least as large as the
State, with reasons, whether each of the following statements is true or false: (a) The probability of the union of two events cannot be less than the probability of their intersection. (b) The probability of the union of two events cannot be more than the sum of their indi- vidual probabilities.
"Bayes' theorem is important, as it provides a rule for moving from a prior probability to a posterior probability." Elaborate on this statement so that it would be well understood by a fellow student who has not yet studied probability.
State, with reasons, whether each of the following statements is true or false: (a) The complement of the union of two events is the intersection of their complements. (b) The sum of the probabilities of collectively exhaustive events must equal 1.(c) The number of combinations of x objects chosen
A student feels that 70% of his college courses have been enjoyable and the remainder have been boring. He has access to student evaluations of professors and finds that 60% of his enjoyable courses and 25% of his boring courses have been taught by professors who had previously received strong
A manufacturer produces boxes of candy, each containing ten pieces. Two machines are used for this purpose. After a large batch has been produced, it is discovered that one of the machines, which produces 40% of the total output, has a fault that has led to the intro- duction of an impurity into
The grades of a freshman college class, obtained at the end of their first year of college. were analyzed. Seventy percent of the students in the top quarter of the college class had graduated in the upper 10% of their high school class, as had 50% of the students in the middle half of the college

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