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The Practice Of Statistics For Business And Economics 4th Edition Layth C. Alwan, Bruce A. Craig - Solutions

Use the Normal approximation. Suppose that we toss a fair coin 200 times. Use the Normal approximation to find the probability that the sample proportion of heads is(a) between 0.4 and 0.6.(b) between 0.45 and 0.55.
Find the mean and the standard deviation. If we toss a fair coin 200 times, the number of heads is a random variable that is binomial.(a) Find the mean and the standard deviation of the sample proportion of heads.(b) Is your answer to part (a) the same as the mean and the standard deviation of the
Do our athletes graduate? Refer to the setting of Exercise 5.9 (page 250).(a) Find the mean number of graduates out of 20 players if 80% of players graduate.(b) Find the standard deviation s of the count X if 80% of players graduate.(c) Suppose now that the 20 players came from a population of
Hispanic representation. Refer to the setting of Exercise 5.10 (page 253).(a) What is the mean number of Hispanics on randomly chosen committees of 10 workers?(b) What is the standard deviation s of the count X of Hispanic members?(c) Suppose now that 10% of the factory workers were Hispanic. Then
Misleading résumés. In Exercise 4.27 (page 190), it was stated that 18.4%of executive job applicants lied on their résumés. Suppose an executive job hunter randomly selects five résumés from an executive job applicant pool. Let X be the number of misleading résumés found in the sample.(a)
Hispanic representation. A factory employs several thousand workers, of whom 30% are Hispanic. If the 10 members of the union executive committee were chosen from the workers at random, the number of Hispanics on the committee X would have the binomial distribution with n 5 10 and p 5 0.3.(a) Use
Do our athletes graduate? A university claims that at least 80% of its basketball players get degrees. To see if there is evidence to the contrary, an investigation examines the fate of 20 players who entered the program over a period of several years that ended six years ago. Of these players, 11
Restaurant survey. You operate a restaurant. You read that a sample survey by the National Restaurant Association shows that 40% of adults are committed to eating nutritious food when eating away from home. To help plan your menu, you decide to conduct a sample survey in your own area. You will use
Find the probabilities.(a) Suppose that X has the B(7, 0.15) distribution. Use software or Table C to find P(X 5 0) and P(X $ 5).(b) Suppose that X has the B(7, 0.85) distribution. Use software or Table C to find P(X 5 7) and P(X # 2).(c) Explain the relationship between your answers to parts (a)
Teaching office software. A company uses a computer-based system to teach clerical employees new office software. After a lesson, the computer presents 10 exercises. The student solves each exercise and enters the answer. The computer gives additional instruction between exercises if the answer is
Customer satisfaction calls. The service department of an automobile dealership follows up each service encounter with a customer satisfaction survey by means of a phone call. On a given day, let X be the number of customers a service representative has to call until a customer is willing to
Card dealing. Define X as the number of red cards observed in the following card dealing scenarios:(a) Deal one card from a standard 52-card deck.(b) Deal one card from a standard 52-card deck, record its color, return it to the deck, shuffle the cards. Repeat this experiment 10 times.
Toss a coin. Toss a fair coin 20 times. Let X be the number of heads that you observe.
Using the Internet to make travel reservations. A recent survey of 1351 randomly selected U.S. residents asked whether or not they had used the Internet for making travel reservations.2 There were 1041 people who answered Yes. The other 310 answered No.(a) What is n?(b) Choose one of the two
Seniors who waived out of the math prerequisite. In a random sample of 250 business students who are in or have taken business statistics, 14% reported that they had waived out of taking the math prerequisite for business statistics due to AP calculus credits from high school. Give n, X, and
Risk for one versus many life insurance policies. It would be quite risky for an insurance company to insure the life of only one 25-year-old man under the terms of Exercise 4.153. There is a high probability that person would live and the company would gain $875 in premiums. But if he were to die,
Life insurance. Assume that a 25-year-old man has these probabilities of dying during the next five years:Age at death 25 26 27 28 29 Probability 0.00039 0.00044 0.00051 0.00057 0.00060(a) What is the probability that the man does not die in the next five years?(b) An online insurance site offers a
Risk pooling in a supply chain. Example 4.39(pages 232–233) compares a decentralized versus a centralized inventory system as it ultimately relates to the amount of safety stock (extra inventory over and above mean demand) held in the system. Suppose that the CEO of ElectroWorks requires a 99%
Blood bag demand. Refer to the distribution of daily demand for blood bags X in Case 4.2 (pages 210–211). Assume that demand is independent from day to day.(a) What is the probability at least one bag will be demanded every day of a given month? Assume 30 days in the month.(b) What is the
Sample surveys for sensitive issues. It is difficult to conduct sample surveys on sensitive issues because many people will not answer questions if the answers might embarrass them. Randomized response is an effective way to guarantee anonymity while collecting information on topics such as student
Find some conditional probabilities. Choose a point at random in the square with sides 0 # x # 1 and 0 # y # 1. This means that the probability that the point falls in any region within the square is the area of that region. Let X be the x coordinate and Y the y coordinate of the point chosen. Find
An interesting case of independence.Independence of events is not always obvious. Toss two balanced coins independently. The four possible combinations of heads and tails in order each have probability 0.25. The events A 5 head on the first toss B 5 both tosses have the same outcome may seem
Wine tasting. In the setting of Exercise 4.143, Taster 1’s rating for a wine is 3. What is the conditional probability that Taster 2’s rating is higher than 3?
Higher education at two-year and four-year institutions. The following table gives the counts of U.S. institutions of higher education classified as public or private and as two-year or four-year:32 Public Private Two-year 1000 721 Four-year 2774 672 Convert the counts to probabilities, and
Bachelor’s degrees by gender. Of the 2,325,000 bachelor’s, master’s, and doctoral degrees given by U.S. colleges and universities in a recent year, 69% were bachelor’s degrees, 28% were master’s degrees, and the rest were doctorates. Moreover, women earned 57% of the bachelor’s degrees,
Slot machines. Slot machines are now video games, with winning determined by electronic random number generators. In the old days, slot machines were like this: you pull the lever to spin three wheels; each wheel has 20 symbols, all equally likely to show when the wheel stops spinning; the three
Wine tasters. Two wine tasters rate each wine they taste on a scale of 1 to 5. From data on their ratings of a large number of wines, we obtain the following probabilities for both tasters’ ratings of a randomly chosen wine:Taster 2 Taster 1 1 2 3 4 5 1 0.03 0.02 0.01 0.00 0.00 2 0.02 0.07 0.06
Some probability distributions. Here is a probability distribution for a random variable X:Value of X 2 3 4 Probability 0.2 0.4 0.4(a) Find the mean and standard deviation for this distribution.(b) Construct a different probability distribution with the same possible values, the same mean, and a
Find the probabilities. Refer to the previous exercise. Find the probabilities for each event.
Roll a pair of dice two times. Consider rolling a pair of fair dice two times. For a given roll, consider the total on the up-faces. For each of the following pairs of events, tell whether they are disjoint, independent, or neither.(a) A 5 2 on the first roll, B 5 8 or more on the first roll.(b) A
A different transformation. Refer to the previous exercise. Now let Y 5 4X2 22.(a) Find the distribution of Y.(b) Find the mean and standard deviation for the distribution of Y.(c) Explain why the rules that you used for part (b) of the previous exercise do not work for this transformation.
Work with a transformation. Here is a probability distribution for a random variable X:Value of X 1 2 Probability 0.4 0.6(a) Find the mean and the standard deviation of this distribution.(b) Let Y 5 4X 22. Use the rules for means and variances to find the mean and the standard deviation of the
Using probability rules. Let P(A) 5 0.7, P(B) 5 0.6, and P(C) 5 0.2.(a) Explain why it is not possible that events A and B can be disjoint.(b) What is the smallest possible value for P(A and B)?What is the largest possible value for P(A and B)? It might be helpful to draw a Venn diagram.(c) If
Larger portfolios. Portfolios often contain more than two investments. The rules for means and variances continue to apply, though the arithmetic gets messier. A portfolio containing proportions a of Biotechnology Fund, b of Information Services Fund, and c of Defense and Aerospace Fund has return
More on diversification. Continuing with the previous exercise, suppose Michael’s primary goal is to seek a portfolio mix of the biotechnology and information services funds that will give him minimal risk as measured by standard deviation of the portfolio. Compute the standard deviations for
Diversification. Currently, Michael is exclusively invested in the Fidelity Biotechnology fund. Even though the mean return for this biotechnology fund is quite high, it comes with greater volatility and risk. So, he decides to diversify his portfolio by constructing a portfolio of 80%
Making glassware. In a process for manufacturing glassware, glass stems are sealed by heating them in a flame. The temperature of the flame varies.Here is the distribution of the temperature X measured in degrees Celsius:Temperature 540° 545° 550° 555° 560°Probability 0.1 0.25 0.3 0.25 0.1(a)
What happens when the correlation is 1? We know that variances add if the random variables involved are uncorrelated (r 5 0), but not otherwise.The opposite extreme is perfect positive correlation(r 5 1). Show by using the general addition rule for variances that in this case the standard
Perfectly negatively correlated investments. Consider the following quote from an online site providing investment guidance: “Perfectly negatively correlated investments would provide 100%diversification, as they would form a portfolio with zero variance, which translates to zero risk.’’
Households and families in government data.In government data, a household consists of all occupants of a dwelling unit, while a family consists of two or more persons who live together and are related by blood or marriage. So all families form households, but some households are not families. Here
Pick 3 and law of large numbers. In Example 4.28 (pages 219–220), the mean payoff for the Tri-State Pick 3 lottery was found to be $0.50. In our discussion of the law of large numbers, we learned that the mean of a probability distribution describes the long-run average outcome. In this exercise,
Difference between heads and tails. In Exercise 4.125, the mean and standard deviation were computed directly from the probability distribution of random variable D. Instead, define X as the number of heads in the three flips, and define Y as the number of tails in the three flips.(a) Find the
Standard deviation of the number of aces. Refer to the previous exercise. Find the standard deviation of the number of aces.
Mean of the distribution for the number of aces. In Exercise 4.98 (page 217), you examined the probability distribution for the number of aces when you are dealt two cards in the game of Texas hold ’em.Let X represent the number of aces in a randomly selected deal of two cards in this game. Here
Difference between heads and tails. Suppose a fair coin is tossed three times.(a) Using the labels of “H’’ and “T,’’ list all the possible outcomes in the sample space.(b) For each outcome in the sample space, define the random variable D as the number of heads minus the number of tails
What’s wrong? In each of the following scenarios, there is something wrong. Describe what is wrong, and give a reason for your answer.(a) If you toss a fair coin three times and get heads all three times, then the probability of getting a tail on the next toss is much greater than one-half.(b) If
What happens if the correlation is not zero? Suppose that X is a random variable with mean 20 and standard deviation 5. Also suppose that Y is a random variable with mean 40 and standard deviation 10. Assume that the correlation between X and Y is 0.5.Find the variance and standard deviation of the
Find some variances and standard deviations. Suppose that X is a random variable with mean 20 and standard deviation 5. Also suppose that Y is a random variable with mean 40 and standard deviation 10. Assume that X and Y are independent.Find the variance and the standard deviation of the random
Find the variance and the standard deviation.A random variable X has the following distribution.X 21 0 1 2 Probability 0.3 0.2 0.2 0.3 Find the variance and the standard deviation for this random variable. Show your work.
Find some means. Suppose that X is a random variable with mean 20 and standard deviation 5. Also suppose that Y is a random variable with mean 40 and standard deviation 10. Find the mean of the random variable Z for each of the following cases. Be sure to show your work.(a) Z 5 2 1 10X.(b) Z 5 10X
Portfolio analysis. Show that if 20%of the portfolio is based on the S&P 500 index, then the mean and standard deviation of the portfolio are indeed the values given in Example 4.41 (page 234).
Managing new-product development process. Exercise 4.113 (page 228)gives the distributions of X, the number of weeks to complete the development of product specifications, and Y, the number of weeks to complete the design of the manufacturing process. You did some useful variance calculations in
Comparing sales. It is unlikely that the daily sales of Tamara and Derek in the previous problem are uncorrelated. They will both sell more during the weekends, for example. Suppose that the correlation between their sales is p 5 0.4. Now what are the mean and standard deviation of the difference X
Comparing sales. Tamara and Derek are sales associates in a large electronics and appliance store. Their store tracks each associate’s daily sales in dollars. Tamara’s sales total X varies from day to day with mean and standard deviation mX 5 $1100 and sX 5 $100 Derek’s sales total Y also
Managing new-product development process. Exercise 4.113(page 228) gives the distribution of time to complete two steps in the new-product development process.(a) Calculate the variance and the standard deviation of the number of weeks to complete the development of product specifications.(b)
Mean return on portfolio. The addition rule for means extends to sums of any number of random variables. Let’s look at a portfolio containing three mutual funds from three different industrial sectors: biotechnology, information services, and defense. The monthly returns on Fidelity Select
Managing a new-product development process. Managers often have to oversee a series of related activities directed to a desired goal or output. As a new-product development manager, you are responsible for two sequential steps of the product development process—namely, the development of product
Find mW. The random variable U has mean mU 5 22, and the random variable V has mean mV 5 22. If W 5 0.5U 1 0.5V, find mW.
Find mY. The random variable X has mean mX 5 8. If Y 5 12 1 7X, what is mY?
Use the Law of Large Numbers applet. The Law of Large Numbers applet animates a graph like Figure 4.17 for rolling dice. Use it to better understand the law of large numbers by making a similar graph.
Find the mean of the probability distribution. You toss a fair coin. If the outcome is heads, you win $5.00; if the outcome is tails, you win nothing.Let X be the amount that you win in a single toss of a coin. Find the probability distribution of this random variable and its mean.
Normal approximation for a sample proportion.A sample survey contacted an SRS of 700 registered voters in Oregon shortly after an election and asked respondents whether they had voted. Voter records show that 56% of registered voters had actually voted. We see in the next chapter that in this
How many close friends? How many close friends do you have? Suppose that the number of close friends adults claim to have varies from person to person with mean m 5 9 and standard deviation s 5 2.4. An opinion poll asks this question of an SRS of 1100 adults. We see in Chapter 6 that, in this
The sum of two uniform random numbers.Generate two random numbers between 0 and 1 and take Y to be their sum. Then Y is a continuous random variable that can take any value between 0 and 2. The density curve of Y is the triangle shown in Figure 4.15.(a) Verify by geometry that the area under this
Uniform numbers between 0 and 2. Many random number generators allow users to specify the range of the random numbers to be produced. Suppose that you specify that the range is to be all numbers between 0 and 2. Call the random number generated Y. Then the density curve of the random variable Y has
Find the probabilities. Let the random variable X be a random number with the uniform density curve in Figure 4.12 (page 214). Find the following probabilities:(a) P(X $ 0.30).(b) P(X 5 0.30).(c) P(0.30 , X , 1.30).(d) P(0.20 # X # 0.25 or 0.7 # X # 0.9).(e) X is not in the interval 0.4 to 0.7.
Spell-checking software. Spell-checking software catches “nonword errors,” which are strings of letters that are not words, as when “the” is typed as“eth.” When undergraduates are asked to write a 250-word essay (without spell-checking), the number X of nonword errors has the following
Use the uniform distribution. Suppose that a random variable X follows the uniform distribution described in Example 4.26 (pages 213–214). For each of the following events, find the probability and illustrate your calculations with a sketch of the density curve similar to the ones in Figure 4.12
Discrete or continuous? In each of the following situations, decide whether the random variable is discrete or continuous, and give a reason for your answer.(a) Your web page has five different links, and a user can click on one of the links or can leave the page. You record the length of time that
How much to order? Faced with the demand for the perishable product in blood, hospital managers need to establish an ordering policy that deals with the trade-off between shortage and wastage.As it turns out, this scenario, referred to as a singleperiod inventory problem, is well known in the area
How large are households? Choose an American household at random, and let X be the number of persons living in the household. If we ignore the few households with more than seven inhabitants, the probability model for X is as follows:Household size X 1 2 3 4 5 6 7 Probability 0.27 0.33 0.16 0.14
Texas hold ’em. The game of Texas hold ’em starts with each player receiving two cards. Here is the probability distribution for the number of aces in twocard hands:Number of aces 0 1 2 Probability 0.8507 0.1448 0.0045(a) Verify that this assignment of probabilities satisfies the requirement
Make a graphical display. Refer to the previous exercise. Use a probability histogram to provide a graphical description of the distribution of X.4.97 Find some probabilities. Refer to Exercise 4.95.(a) Find the probability that a randomly selected student takes three or fewer courses.(b) Find the
How many courses? At a small liberal arts college, students can register for one to six courses.In a typical fall semester, 5% take one course, 5%take two courses, 13% take three courses, 26% take four courses, 36% take five courses, and 15% take six courses. Let X be the number of courses taken in
Two day demand. Refer to the distribution of daily demand for blood bags X in Case 4.2(pages 210–211). Let Y be the total demand over two days.Assume that demand is independent from day to day.(a) List the possible values for Y.(b) From the distribution of daily demand, we find that the
Normal probabilities. Example 4.27 gives the Normal distribution N(50,000, 5500) for the tread life X of a type of tire (in miles). Calculate the following probabilities:(a) The probability that a tire lasts more than 50,000 miles.(b) P(X . 60,000).(c) P(X $ 60,000).
Find the probability. For the uniform distribution described in Example 4.26, find the probability that X is between 0.2 and 0.7.
How many cars? Choose an American household at random and let the random variable X be the number of cars (including SUVs and light trucks) they own. Here is the probability model if we ignore the few households that own more than five cars:Number of cars X 0 1 2 3 4 5 Probability 0.09 0.36 0.35
High demand. Refer to Case 4.2 for the probability distribution on daily demand for blood transfusion bags.(a) What is the probability that the hospital will face a high demand of either 11 or 12 bags? Compute this probability directly using the respective probabilities for 11 and 12 bags.(b) Now
Supplier Quality. A manufacturer of an assembly product uses three different suppliers for a particular component. By means of supplier audits, the manufacturer estimates the following percentages of defective parts by supplier:Supplier 1 2 3 Percent defective 0.4% 0.3% 0.6%Shipments from the
Examined by the IRS. The IRS examines(audits) some tax returns in greater detail to verify that the tax reported is correct. The rates of examination vary depending on the size of the individual’s adjusted gross income. In 2014, the IRS reported the percentages of total returns by adjusted gross
Credit card defaults. The credit manager for a local department store is interested in customers who default (ultimately failed to pay entire balance).Of those customers who default, 88% were late (by a week or more) with two or more monthly payments.This prompts the manager to suggest that future
Successful bids, continued. Draw a Venn diagram that illustrates the relation between events A and B in Exercise 4.84. Write each of the following events in terms of A, B, Ac, and Bc. Indicate the events on your diagram and use the information in Exercise 4.84 to calculate the probability of
Independence? In the setting of the previous exercise, are events A and B independent? Do a calculation that proves your answer.
Successful bids. Consolidated Builders has bid on two large construction projects. The company president believes that the probability of winning the first contract (event A) is 0.6, that the probability of winning the second (event B) is 0.5, and that the probability of winning both jobs (event {A
Success on the GMAT. In the setting of Exercise 4.81, what percent of the customers who score at least 600 on the GMAT are undergraduates? (Write this as a conditional probability.)
Sales to women. In the setting of Exercise 4.80, what percent of sales are made to women? (Write this as a conditional probability.)
Preparing for the GMAT. A company that offers courses to prepare would-be MBA students for the GMAT examination finds that 40% of its customers are currently undergraduate students and 60% are college graduates. After completing the course, 50% of the undergraduates and 70% of the graduates achieve
Telemarketing. A telemarketing company calls telephone numbers chosen at random. It finds that 70%of calls are not completed (the party does not answer or refuses to talk), that 20% result in talking to a woman, and that 10% result in talking to a man. After that point, 30% of the women and 20% of
High school baseball players. It is estimated that 56% of MLB players have careers of three or more years.Using the information in Example 4.22 (pages 201–202), determine the proportion of high school players expected to play three or more years in MLB.
High school football players. Using the information in Example 4.21 (pages 200–201), determine the proportion of high school football players expected to play professionally in the NFL.
Loan officer decision. Considering the information provided in the previous exercise, calculate P(O u D).Show your work. Also, express this probability in words in the context of the loan officer’s decision. If new information about the customer becomes available before the loan officer makes her
Loan officer decision. A loan officer is considering a loan request from a customer of the bank.Based on data collected from the bank’s records over many years, there is an 8% chance that a customer who has overdrawn an account will default on the loan.However, there is only a 0.6% chance that a
Unemployment rates. As noted in Example 4.18 (page 197), in the language of government statistics, you are “in the labor force’’ if you are available for work and either working or actively seeking work.The unemployment rate is the proportion of the labor force (not of the entire population)
Conditional probabilities and independence.Using the information in Exercise 4.73, answer these questions.(a) Given that a vehicle is imported, what is the conditional probability that it is a light truck?(b) Are the events “vehicle is a light truck’’ and “vehicle is imported’’
Sales of cars and light trucks. Motor vehicles sold to individuals are classified as either cars or light trucks (including SUVs) and as either domestic or imported. In a recent year, 69% of vehicles sold were light trucks, 78% were domestic, and 55% were domestic light trucks. Let A be the event
Draw a Venn diagram. Draw a Venn diagram that shows the relation between the events A and B in Exercise 4.70. Indicate each of the following events on your diagram and use the information in Exercise 4.70 to calculate the probability of each event. Finally, describe in words what each event is.(a)
Find a conditional probability. In the setting of the previous exercise, what is the conditional probability that a household is prosperous, given that it is educated? Explain why your result shows that events A and B are not independent.
Education and income. Call a household prosperous if its income exceeds $100,000. Call the household educated if at least one of the householders completed college. Select an American household at random, and let A be the event that the selected household is prosperous and B the event that it is
Draw a different tree diagram for the same setting. Refer to the previous two exercises. Draw a tree diagram to illustrate the probabilities in a situation in which you first identify the gender of the student and then identify the type of institution attended. Explain why the probabilities in this

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