problemsbookfinal

Selected Text Problems with Answersfrom Chapters 1−4 for

THE PRACTICEOF BUSINESSSTATISTICS

DAVID S. MOOREGEORGE P. MCCABE

Purdue University

WILLIAM DUCKWORTHIowa State University

STANLEY SCLOVEUniversity of Illinois at Chicago

W. H. Freeman and CompanyNew York

ISBN: 0-7167-9638-4

© 2002 by W. H. Freeman and Company

No part of this book may be reproduced by any mechanical, photographic, or electronicprocess, or in the form of a phonographic recording, nor may it be stored in a retrievalsystem, transmitted, or otherwise copied for public or private use, without writtenpermission from the publisher.

Printed in the United States of America

First printing 2001

Selected Text Problems with Answersfrom Chapters 1−4 for

THE PRACTICE OF BUSINESS STATISTICSTable of Contents

Chapter 1 Examining Distributions

Exercise 1.4 Occupational deaths. 1Exercise 1.45 Education and income. 2Exercise 1.53 A hot stock? 3Exercise 1.54 Initial public offerings. 5Exercise 1.69 GMAT scores. 6Exercise 1.95 Grading managers. 7

Chapter 2 Examining Relationships

Exercise 2.10 Health and wealth. 8Exercise 2.15 Business starts and failures. 10Exercise 2.18 Where the stores are. 12Exercise 2.25 Mutual fund performance. 14Exercise 2.29 CEO compensation and stock market performance. 15Exercise 2.45 Moving in step? 16Exercise 2.46 Interpreting correlation. 17Exercise 2.48 Cash or credit? 18Exercise 2.55 What's my grade? 20Exercise 2.75 Education and income. 21Exercise 2.100 Are high interest rates bad for stocks? 22Exercise 2.108 Size and selling price of houses. 24

Chapter 3 Producing Data

Exercise 3.11 Sampling by accountants. 26Exercise 3.13 Ring-no-answer. 27Exercise 3.19 Quality control sampling. 28Exercise 3.40 Does charting help investors? 29

Chapter 4 Probability and Sampling Distributions

Exercise 4.24 Stock price movements. 30Exercise 4.26 Car colors. 31Exercise 4.39 How many cars? 32Exercise 4.53 How many rooms. 33Exercise 4.61 Time and motion studies. 34Exercise 4.63 Time and motion studies. 35Exercise 4.70 Diversification. 36Exercise 4.87 Flaws in carpets. 37

Table B Random Digits Table 38

Answers 39

Moore/McCabe/Duckworth/Sclove: The Practice of Business Statistics 1

EXERCISE 1.4 Occupational deaths.

In 1999 there were 6023 job-related deaths in the United States. Among these were 807deaths in agricultural-related jobs (including forestry and fishing), 121 in mining, 1190in construction, 719 in manufacturing, 1006 in transportation and public utilities, 237 inwholesale trade, 507 in retail trade, 105 in finance-related jobs (including insurance andreal estate), 732 in service-related jobs, and 562 in government jobs.

(a) Find the percent of occupational deaths for each of these job categories, rounded to thenearest percent. What percent of job-related deaths were in categories not listed above?

(b) Make a well-labeled bar graph of the distribution of occupational deaths. Be sure toinclude an “other occupations” bar.

(c) Make a well-labeled Pareto chart of these data. What percent of all occupational deathsare accounted for by the first 3 categories in your Pareto chart?

(d) Would it also be correct to use a pie chart to display these data? Explain your answer.

2 Moore/McCabe/Duckworth/Sclove: The Practice of Business Statistics

EXERCISE 1.45 Education and income.

Each March, the Bureau of Labor Statistics (BLS) records the incomes of all adults in asample of 50,000 American households. We are interested in how income varies with thehighest education level a person has reached. Computer software applied to the data fromthe March 2000 survey gives the following results for people aged 25 or over:

It is common to make boxplots of large data sets using the 5% and 95% points in placeof the minimum and maximum. The highest income among the 31,970 people with onlya high school education, for example, is $425,510. It is more informative to see that 95%of this group earned less than $56,294. The 5% and 95% points contain between them themiddle 90% of the observations.

(a) Use this output to make boxplots that compare the income distributions for the fiveeducation groups.

(b) Write a brief summary of the relationship between education and income. For example,do people who start college but don’t get a degree do much better than people with only ahigh school education?


EXERCISE 1.53 A hot stock?

We saw in Example 1.14 that it is usual in the study of investments to use the mean andstandard deviation to summarize and compare investment returns. On the following page,Table 1.10 gives the monthly returns on Philip Morris stock for the period from August1990 to August 2001. (The return on an investment consists of the change in its price plusany cash payments made, given here as a percent of its price at the start of each month.)

(a) Make either a histogram or a stemplot of these data. How did you decide which graphto make?

(b) There are two clear outliers. What are the values of these observations? (The mostextreme observation is explained by news of action against smoking, which depressed thistobacco company stock.) Describe the shape, center, and spread of the data after you omitthe two outliers.

(c) Find the mean monthly return and the standard deviation of the returns. If you invested$100 in this stock at the beginning of a month and got the mean return, how much wouldyou have at the end of the month?

(d) The distribution can be described as “symmetric and single-peaked, with two low out-liers.” If you invested $100 in this stock at the beginning of the worst month in the data(the most extreme outlier), how much would you have at the end of the month? Find themean and standard deviation again, this time leaving out the two low outliers. How muchdid these two observations affect the summary measures? Would leaving out these twoobservations substantially change the median? The quartiles? How do you know, withoutactual calculation? (Returns over longer periods of time, or returns on portfolios containingseveral investments, tend to follow a Normal distribution more closely than these monthlyreturns. So use of the mean and standard deviation is better justified for such data.)


Table 1.10Monthly percent returns on Phillip Morris stock

from August 1990 to August 2001

3.0 -5.7 1.2 4.1 3.2 7.3 7.5 18.7 3.7 -1.82.4 -6.5 6.7 9.4 -2.0 -2.8 -3.4 19.2 -4.8 0.5

-0.6 2.8 -0.5 -4.5 8.7 2.7 4.1 -10.3 4.8 -2.3-3.1 -10.2 -3.7 -26.6 7.2 -2.4 -2.8 3.4 -4.6 17.24.2 0.5 8.3 -7.1 -8.4 7.7 -9.6 6.0 6.8 10.91.6 0.2 -2.4 -2.4 3.9 1.7 9.0 3.6 7.6 3.2

-3.7 4.2 13.2 0.9 4.2 4.0 2.8 6.7 -10.4 2.710.3 5.7 0.6 -14.2 1.3 2.9 11.8 10.6 5.2 13.8

-14.7 3.5 11.7 1.5 2.0 -3.2 -3.9 -4.7 9.8 4.9-8.3 4.8 -3.2 -10.9 0.7 6.4 11.3 -5.1 12.3 10.59.4 -3.6 -12.4 -16.5 -8.9 -0.4 10.0 5.4 -7.3 0.5

-7.4 -22.9 -0.5 -10.6 -9.2 -3.3 5.2 5.4 19.4 3.5-4.9 17.8 0.7 24.4 4.3 16.6 0.0 9.5 -0.4 5.62.6 -2.7 -8.1 4.2


EXERCISE 1.54 Initial public offerings.

During the stockmarket boom of the 1990s, initial public offerings (IPOs) of the stock ofnew companies often produced enormous gains for people who bought the stocks when theyfirst became available. At least that’s what legend says. A study of all 4567 companies thatwent public in the years 1990 to 2000 (excluding very small IPOs) found that on the averagetheir stock prices had either risen 111% or declined 31% by the end of the year 2000.One ofthese numbers is the mean change in price and one is the median change. Which is which,and how can you tell?


EXERCISE 1.69 GMAT scores.

Most graduate schools of business require applicants for admission to take the GraduateManagement Admission Council’s GMAT examination.Total scores on the GMAT for themore than 500,000 people who took the exam between April 1997 and March 2000 areroughly Normally distributed with mean µ = 527 and standard deviation σ = 112.

(a) What percent of test-takers have scores above 500?

(b) What GMAT scores fall in the lowest 25% of the distribution?

(c) How high a GMAT score is needed to be in the highest 5%?


EXERCISE 1.95 Grading managers.

Some companies “grade on a bell curve” to compare the performance of their managersand professional workers. This forces the use of some low performance ratings, so thatnot all workers are graded “above average.” Until the threat of lawsuits forced a change,Ford Motor Company’s “performance management process” assigned 10% A grades, 80%B grades, and 10% C grades to the company’s 18,000 managers. It isn’t clear that the“bell curve” of ratings is really a Normal distribution. Nonetheless, suppose that Ford’sperformance scores are Normally distributed. This year, managers with scores less than 25received C’s and those with scores above 475 received A’s. What are the mean and standarddeviation of the scores?


EXERCISE 2.10 Health and wealth.

On the following page, Figure 2.7 is a scatterplot of data from the World Bank. Theindividuals are all the world’s nations for which data are available. The explanatory variableis a measure of how rich a country is, the gross domestic product (GDP) per person. GDPis the total value of the goods and services produced in a country, converted into dollars.The response variable is life expectancy at birth.We expect people in richer countries tolive longer. Describe the form, direction and shape of the overall pattern. Does the graphconfirm our expectation? The three African nations marked on the graph are outliers;a detailed study would ask what special factors explain the low life expectancy in thesecountries.


Figure 2.7

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EXERCISE 2.15 Business starts and failures.

On the following page, Table 2.4 gives data for number of businesses started and number ofbusinesses failed by state for 1998. We might expect an association to exist between theseeconomic measures.

(a) Make a scatterplot of business starts against business failures. Take business starts asthe explanatory variable.

(b) The plot shows a positive association between the two variables. Why do we say thatthe association is positive?

(c) Find the point for Florida in the scatterplot and circle it.

(d) There is an outlier at the upper right of the plot. Which state is this?

(e) We wonder about clusters and gaps in the data display. There is a relatively clear clusterof states at the lower left of the plot. Four states are outside this cluster. Which states arethese? Are they mainly from one part of the country?


Table 2.4 Business Starts and Failures

State Starts Failures State Starts FailuresAL 2645 546 MT 397 201AK 271 177 NE 565 383AZ 2868 1225 NV 1465 677AR 1091 748 NH 708 322CA 21582 17679 NJ 6412 2024CO 3041 2483 NM 887 585CT 2069 530 NY 13403 4233DE 508 28 NC 4371 846DC 537 75 ND 229 144FL 13029 2047 OH 4829 2524GA 5471 800 OK 1367 990HI 593 781 OR 1823 1109ID 639 441 PA 5525 2641IL 5542 3291 RI 544 150IN 2611 473 SC 2023 410IA 1020 244 SD 281 275KS 967 1140 TN 2835 1369KY 1824 270 TX 10936 6785LA 1849 377 UT 1417 388ME 577 259 VT 261 80MD 3139 1283 VA 3502 860MA 3425 1200 WA 2956 2528MI 4293 1551 WV 623 305MN 2111 1711 WI 2357 1005MS 1347 177 WY 213 166MO 2163 1321


EXERCISE 2.18 Where the stores are.

Target and Wal-Mart are two of the largest retail chains in the United States. Target has977 stores and Wal-Mart has 2624 stores. The file ex02-18.dat has the number of Targetstores and Wal-Mart stores listed by state. (The contents of this file are displayed on thefollowing page.)

(a) Use software to create a scatterplot of the number of Target stores versus the numberof Wal-Mart stores for each of the fifty states.

(b) Identify any unusual observations in your scatterplot by labeling the point with thestate abbreviation. Describe specifically what about the observation makes it stand out inthe scatterplot.

(c) Comment on the form, direction, and strength of the relationship between the numberof Target stores and the number of Wal-Mart stores per state.


Contents of the file ex02-18.dat

State Targets Wal-Marts State Targets Wal-MartsAL 4 82 MT 5 10AK 0 5 NE 9 20AZ 28 39 NV 12 16AR 2 77 NH 3 22CA 151 117 NJ 15 23CO 24 41 NM 8 22CT 3 20 NY 22 65DE 1 5 NC 24 94FL 68 143 ND 4 8GA 30 96 OH 34 88HI 0 5 OK 8 80ID 4 13 OR 11 25IL 52 111 PA 22 78IN 29 83 RI 1 7IA 18 51 SC 8 59KS 10 50 SD 4 8KY 12 74 TN 19 88LA 2 80 TX 90 254ME 0 20 UT 7 15MD 22 30 VT 0 4MA 8 37 VA 25 67MI 48 57 WA 25 26MN 56 39 WV 1 28MS 2 61 WI 26 60MO 18 112 WY 2 9


EXERCISE 2.25 Mutual fund performance.

Many mutual funds compare their performance with that of a benchmark, an index of thereturns on all securities of the kind that the fund buys. The Vanguard International GrowthFund, for example, takes as its benchmark the Morgan Stanley EAFE (Europe, Australasia,Far East) index of overseas stock market performance. Here are the percent returns for thefund and for the EAFE from 1982 (the first full year of the fund’s existence) to 2000.

Year Fund EAFE Year Fund EAFE1982 5.27 −0.86 1992 −5.79 −11.851983 43.08 24.61 1993 44.74 32.941984 −1.02 7.86 1994 0.76 8.061985 56.94 56.72 1995 14.89 11.551986 56.71 69.94 1996 14.65 6.361987 12.48 24.93 1997 4.12 2.061988 11.61 28.59 1998 16.93 20.331989 24.76 10.80 1999 26.34 27.301990 −12.05 −23.20 2000 −8.60 −13.961991 4.74 12.50

Make a scatterplot suitable for predicting fund returns from EAFE returns. Is there a clearstraight-line pattern? How strong is this pattern? (Give a numerical measure.) Are thereany extreme outliers from the straight-line pattern?


EXERCISE 2.29 CEO compensation and stock market performance.

An academic study says that, “The evidence indicates that the correlation between thecompensation of corporate CEOs and the performance of their company’s stock is close tozero.” A business magazine reports this as “A new study shows that companies that paytheir CEOs highly tend to perform poorly in the stock market, and vice versa.” Explainwhy the magazine’s report is wrong. Write a statement in plain language (don’t use theword “correlation”) to explain the study’s conclusion.


EXERCISE 2.45 Moving in step?

One reason to invest abroad is that markets in different countries don’t move in step. WhenAmerican stocks go down, foreign stocks may go up. So an investor who holds both bears lessrisk. That’s the theory. Now we read that “The correlation between changes in Americanand European share prices has risen from 0.4 in the mid-1990s to 0.8 in 2000.”Explain to aninvestor who knows no statistics why this fact reduces the protection provided by buyingEuropean stocks.


EXERCISE 2.46 Interpreting correlation.

The same article mentioned in Exercise 2.45 that claims that the correlation betweenchanges in stock prices in Europe and the United States was 0.8 in 2000 goes on to saythat “Crudely, that means that movements on Wall Street can explain 80% of price move-ments in Europe.” Is this true? What is the correct percent explained if r = 0.8?


EXERCISE 2.48 Cash or credit?

We might expect credit card purchases to differ from cash purchases at the same store.Let’s compare regressions for credit card and cash purchases using the consignment shopdata in Table 2.1 (shown on the following page).

(a) Make a scatterplot of daily gross sales y versus items sold x for credit card purchases.Using a separate plot symbol or color, add daily gross sales and items sold for cash. It issomewhat difficult to compare the two patterns by eye.

(b) Regression can help. Find the equations of the two least-squares regression lines ofgross sales on items sold, for credit card sales and for cash sales. Draw these lines on yourplot. What are the slopes of the two regression lines? Explain carefully what comparingthe slopes says about credit card purchases versus cash purchases.


Table 2.1 Gross Sales and Number of Items Sold


EXERCISE 2.55 What’s my grade?

In Professor Friedman’s economics course, the correlation between the students’ total scoresbefore the final examination and their final examination scores is r = 0.6. The pre-examtotals for all students in the course have mean 280 and standard deviation 30. The finalexam scores have mean 75 and standard deviation 8. Professor Friedman has lost Julie’sfinal exam but knows that her total before the exam was 300. He decides to predict herfinal exam score from her pre-exam total.

(a) What is the slope of the least-squares regression line of final exam scores on pre-examtotal scores in this course? What is the intercept?

(b) Use the regression line to predict Julie’s final exam score.

(c) Julie doesn’t think this method accurately predicts how well she did on the final exam.Calculate r2 and use the value you get to argue that her actual score could have been muchhigher (or much lower) than the predicted value.


EXERCISE 2.75 Education and income.

There is a strong positive correlation between years of schooling completed x and lifetimeearnings y for American men. One possible reason for this association is causation: moreeducation leads to higher-paying jobs. But lurking variables may explain some of the cor-relation. Suggest some lurking variables that would explain why men with more educationearn more.


EXERCISE 2.100 Are high interest rates bad for stocks?

The scatterplot in Figure 2.25 (shown on the following page) suggests that returns oncommon stocks may be somewhat lower in years with high interest rates. Here is part ofthe Excel output for the regression of stock returns on the bill returns for the same years:

(a) What is the equation of the least-squares line? Use this line to predict the percentreturn on stocks in a year when Treasury bills return 5%.

(b) Explain what the slope of the regression line tells us. Does the slope confirm that highinterest rates are in general bad for stocks?

(c) If you knew the return on Treasury bills for next year, do you think you could predictthe return on stocks quite accurately? Use both the scatterplot in Figure 2.25 and theregression output to justify your answer.


Figure 2.25

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EXERCISE 2.108 Size and selling price of houses.

On the following page, Table 2.13 provides information on a random sample of 50 housessold in Ames, Iowa, in the year 2000.

(a) Describe the distribution of selling price with a graph and a numerical summary. Whatare the main features of this distribution?

(b) Make a scatterplot of selling price versus square feet and describe the relationshipbetween these two variables.

(c) Calculate the least-squares regression line for these data. On average, how much doeseach square foot add to the selling price of a house?

(d) What would you expect the selling price of a 1600 square foot house in Ames to be?

(e) What percentage of the variability in these fifty selling prices can be attributed todifferences in square footage?


Table 2.13 Houses sold in Ames, IA

Selling Price Square Footage ($) Age (as of 2000) Selling Price Square Footage ($) Age (as of 2000)268380 1897 1 169900 1686 35131000 1157 15 180000 2054 34112000 1024 35 127000 1386 50112000 935 35 242500 2603 10122000 1236 39 152900 1582 3127900 1248 32 171600 1790 1157600 1620 33 195000 1908 6135000 1124 33 83100 1378 72145900 1248 35 125000 1668 55126000 1139 39 60500 1248 100142000 1329 40 85000 1229 59107500 1040 45 117000 1308 60110000 951 42 57000 892 90187000 1628 1 110000 1981 72

94000 816 43 127250 1098 7099500 1060 24 119000 1858 8078000 800 68 172500 2010 6055790 492 79 123000 1680 8670000 792 80 161715 1670 153600 980 62 179797 1938 1

157000 1629 3 117250 1120 36166730 1889 0 116500 914 4340000 2759 6 117000 1008 23195000 1811 3 177500 1920 32215850 2400 27 132000 1146 37


EXERCISE 3.11 Sampling by accountants.

Accountants use stratified samples during audits to verify a company’s records of suchthings as accounts receivable. The stratification is based on the dollar amount of the itemand often includes 100% sampling of the largest items. One company reports 5000 accountsreceivable. Of these, 100 are in amounts over $50,000; 500 are in amounts between $1000and $50,000; and the remaining 4400 are in amounts under $1000. Using these groups asstrata, you decide to verify all of the largest accounts and to sample 5% of the midsizeaccounts and 1% of the small accounts. How would you label the two strata from whichyou will sample? Use Table B, starting at line 115, to select only the first 5 accounts fromeach of these strata.


EXERCISE 3.13 Ring-no-answer.

A common form of nonresponse in telephone surveys is “ring-no-answer.” That is, a call ismade to an active number, but no one answers. The Italian National Statistical Institutelooked at nonresponse to a government survey of households in Italy during two periods,January 1 to Easter and July 1 to August 31. All calls were made between 7 and 10 p.m.,but 21.4% gave “ring-no-answer” in one period versus 41.5% “ring-no-answer” in the otherperiod.Which period do you think had the higher rate of no answers? Why? Explain whya high rate of nonresponse makes sample results less reliable.


EXERCISE 3.19 Quality control sampling.

A manufacturer of chemicals chooses 3 containers from each lot of 25 containers of a reagentto test for purity and potency. Below are the control numbers stamped on the bottles inthe current lot. Use Table B at line 111 to choose an SRS of 3 of these bottles.

A1096 A1097 A1098 A1101 A1108A1112 A1113 A1117 A2109 A2211A2220 B0986 B1011 B1096 B1101B1102 B1103 B1110 B1119 B1137B1189 B1223 B1277 B1286 B1299


EXERCISE 3.40 Does charting help investors?

Some investment advisors believe that charts of past trends in the prices of securities canhelp predict future prices. Most economists disagree. In an experiment to examine theeffects of using charts, business students trade (hypothetically) a foreign currency at com-puter screens. There are 20 student subjects available, named for convenience A, B, C,. . . , T. Their goal is to make as much money as possible, and the best performances arerewarded with small prizes. The student traders have the price history of the foreign cur-rency in dollars in their computers. They may or may not also have software that highlightstrends. Describe two designs for this experiment, a completely randomized design and amatched pairs design in which each student serves as his or her own control. In both cases,carry out the randomization required by the design.


EXERCISE 4.24 Stock price movements.

You watch the price of Cisco Systems stock for four days. Give a sample space for each ofthese random phenomena:

(a) You record the sequence of up days and down days.

(b) You record the number of up days.


EXERCISE 4.26 Car colors.

Choose a new car or light truck at random and note its color. Here are the probabilities ofthe most popular colors for cars made in North America in 2000:

Color Silver White Black Dark Green Dark Blue Medium RedProbability 0.176 0.172 0.113 0.089 0.088 0.067

What is the probability that the car you choose has any color other than the six listed?What is the probability that a randomly chosen car is either silver or white?


EXERCISE 4.39 How many cars?

Choose an American household at random and let the random variable X be the number ofcars (including SUVs and light trucks) they own. Here is the probability model if we ignorethe few households that own more than 5 cars:

Number of cars X 0 1 2 3 4 5Probability 0.09 0.36 0.35 0.13 0.05 0.02

(a) Verify that this is a legitimate discrete distribution. Display the distribution in aprobability histogram.

(b) Say in words what the event {X ≥ 1} is. Find P (X ≥ 1).

(c) Your company builds houses with two-car garages. What percent of households havemore cars than the garage can hold?


EXERCISE 4.53 How many rooms?

Furniture makers and others are interested in how many rooms housing units have, becausemore rooms can generate more sales. Here are the distributions of the number of rooms forowner-occupied units and renter-occupied units in San Jose, California:

Rooms 1 2 3 4 5 6 7 8 9 10Owned .003 .002 .023 .104 .210 .224 .197 .149 .053 .035Rented .008 .027 .287 .363 .164 .093 .039 .013 .003 .003

(a) Make probability histograms of these two distributions, using the same scales. Whatare the most important differences between the distributions for owner-occupied and rentedhousing units?

(b) Find the mean number of rooms for both types of housing unit. How do the meansreflect the differences you found in (a)?


EXERCISE 4.61 Time and motion studies.

A time and motion study measures the time required for an assembly line worker to performa repetitive task. The data show that the time required to bring a part from a bin to itsposition on an automobile chassis varies from car to car with mean 11 seconds and standarddeviation 2 seconds. The time required to attach the part to the chassis varies with mean20 seconds and standard deviation 4 seconds.

(a) What is the mean time required for the entire operation of positioning and attachingthe part?

(b) If the variation in the worker’s performance is reduced by better training, the standarddeviations will decrease. Will this decrease change the mean you found in (a) if the meantimes for the two steps remain as before?

(c) The study finds that the times required for the two steps are independent. A part thattakes a long time to position, for example, does not take more or less time to attach thanother parts. How would your answer in (a) change if the two variables were dependent withcorrelation 0.8? With correlation 0.3?


EXERCISE 4.63 Time and motion studies.

Find the standard deviation of the time required for the two-step assembly operation studiedin Exercise 4.61, assuming that the study shows the two times to be independent. Redothe calculation assuming that the two times are dependent, with correlation 0.3. Can youexplain in nontechnical language why positive correlation increases the variability of thetotal time?


Portfolio analysis

Here are the means, standard deviations, and correlations for the monthly returns fromthree Fidelity mutual funds for the 36 months ending in December 2000. Because there arenow three random variables, there are three correlations. We use subscripts to show whichpair of random variables a correlation refers to.

W = monthly return on Magellan Fund µW = 1.14% σW = 4.64%X = monthly return on Real Estate Fund µX = 0.16% σX = 3.61%Y = monthly return on Japan Fund µY = 1.59% σY = 6.75%

CorrelationsρWX = 0.19 ρWY = 0.54 ρXY = −0.17

EXERCISE 4.70 Diversification.

Many advisors recommend using roughly 20% foreign stocks to diversify portfolios of U.S.stocks. Michael owns Fidelity Magellan Fund, which concentrates on stocks of large Ameri-can companies. He decides to move to a portfolio of 80% Magellan and 20% Fidelity JapanFund. Show that (based on historical data) this portfolio has both a higher mean returnand less volatility than Magellan alone. This illustrates the beneficial effects of diversifyingamong investments.


EXERCISE 4.87 Flaws in carpets.

The number of flaws per square yard in a type of carpet material varies with mean 1.6 flawsper square yard and standard deviation 1.2 flaws per square yard. The population distri-bution cannot be Normal, because a count takes only whole-number values. An inspectorsamples 200 square yards of the material, records the number of flaws found in each squareyard, and calculates x, the mean number of flaws per square yard inspected. Use the centrallimit theorem to find the approximate probability that the mean number of flaws exceeds2 per square yard.


Table B Random DigitsLine101 19223 95034 05756 28713 96409 12531 42544 82853102 73676 47150 99400 01927 27754 42648 82425 36290103 45467 71709 77558 00095 32863 29485 82226 90056104 52711 38889 93074 60227 40011 85848 48767 52573105 95592 94007 69971 91481 60779 53791 17297 59335106 68417 35013 15529 72765 85089 57067 50211 47487107 82739 57890 20807 47511 81676 55300 94383 14893108 60940 72024 17868 24943 61790 90656 87964 18883109 36009 19365 15412 39638 85453 46816 83485 41979110 38448 48789 18338 24697 39364 42006 76688 08708111 81486 69487 60513 09297 00412 71238 27649 39950112 59636 88804 04634 71197 19352 73089 84898 45785113 62568 70206 40325 03699 71080 22553 11486 11776114 45149 32992 75730 66280 03819 56202 02938 70915115 61041 77684 94322 24709 73698 14526 31893 32592116 14459 26056 31424 80371 65103 62253 50490 61181117 38167 98532 62183 70632 23417 26185 41448 75532118 73190 32533 04470 29669 84407 90785 65956 86382119 95857 07118 87664 92099 58806 66979 98624 84826120 35476 55972 39421 65850 04266 35435 43742 11937121 71487 09984 29077 14863 61683 47052 62224 51025122 13873 81598 95052 90908 73592 75186 87136 95761123 54580 81507 27102 56027 55892 33063 41842 81868124 71035 09001 43367 49497 72719 96758 27611 91596125 96746 12149 37823 71868 18442 35119 62103 39244126 96927 19931 36809 74192 77567 88741 48409 41903127 43909 99477 25330 64359 40085 16925 85117 36071128 15689 14227 06565 14374 13352 49367 81982 87209129 36759 58984 68288 22913 18638 54303 00795 08727130 69051 64817 87174 09517 84534 06489 87201 97245131 05007 16632 81194 14873 04197 85576 45195 96565132 68732 55259 84292 08796 43165 93739 31685 97150133 45740 41807 65561 33302 07051 93623 18132 09547134 27816 78416 18329 21337 35213 37741 04312 68508135 66925 55658 39100 78458 11206 19876 87151 31260136 08421 44753 77377 28744 75592 08563 79140 92454137 53645 66812 61421 47836 12609 15373 98481 14592138 66831 68908 40772 21558 47781 33586 79177 06928139 55588 99404 70708 41098 43563 56934 48394 51719140 12975 13258 13048 45144 72321 81940 00360 02428141 96767 35964 23822 96012 94591 65194 50842 53372142 72829 50232 97892 63408 77919 44575 24870 04178143 88565 42628 17797 49376 61762 16953 88604 12724144 62964 88145 83083 69453 46109 59505 69680 00900145 19687 12633 57857 95806 09931 02150 43163 58636146 37609 59057 66967 83401 60705 02384 90597 93600147 54973 86278 88737 74351 47500 84552 19909 67181148 00694 05977 19664 65441 20903 62371 22725 53340149 71546 05233 53946 68743 72460 27601 45403 88692150 07511 88915 41267 16853 84569 79367 32337 03316


Answers to Problemsselected from

Chapters 1−4 for

THE PRACTICE OF BUSINESS STATISTICS


EXERCISE 1.4 Occupational deaths.(a) 13% agricultural; 2% mining; 20% construction; 12% manufacturing; 17% transportation andutilities; 4% wholesale trade; 8% retail trade; 2% finance, 12% service; 9% government; 1%other.(b) See plot below.

Ag. Mines Const. Manuf. Trans.&

Utils.

Whole-sale

Retail Finance Svc. Govt. Other

05

1015

20

Distribution of occupational deaths

Perc

ent

Moore/McCabe/Duckworth/Sclove: The Practice of Business Statistics42

(c) See plot below. 50%.

Const. Trans.&

Utils.

Ag. Svc. Manuf. Govt. Retail Whole-sale

Mines Finance Other

05

1015

20

Pareto chart of occupational deathsPe

rcen

t

(d) Yes it would because each death can be assigned to exactly one category.


EXERCISE 1.45 Education and income.(a) See plot below.

Inco

me

050

000

1500

00

High school Some college Bachelor's Master's Professional

(b) Median income in March of 2000 appeared to increase with education level, although therewas more income variability (as evidenced by the IQR) in each progressively higher educationgroup. On average those with a higher level of education earn more, but there are plenty ofexceptions to this pattern. For example, the top 25% of earners with only a high school diplomaearn more than the bottom 25% with a Master's degree.


EXERCISE 1.53 A hot stock?(a) The sample size is large enough that a stemplot is probably more difficult to read than ahistogram. Further, plotting these data in a stemplot requires the values to be rounded, whicheliminates one common advantage of this plot. See plot below.

-30 -20 -10 0 10 20

010

2030

40

Monthly percent return on Phillip Morris stock (8/90-8/01)

Percent return

Num

ber o

f mon

ths

(b) The outliers are -26.6 and -22.9. After the outliers have been omitted, the distribution ofmonthly returns is unimodal and approximately symmetric with only some left skew. The centerof the returns appears to be just above zero. The returns range from about -20 to about 25, withwell over half of the returns between -10 and 15.(c) The mean is 1.63, the standard deviation is 8.29. An investment of $100 would be worth$101.63.(d) $73.4. Without the outliers, the mean is 2.04 and the standard deviation is 7.65. The meanincreases by about .41 percentage points, while the standard deviation drops by 0.63 percentagepoints. Since the median and quartiles are robust or resistant to outliers, they would not changevery much if the outliers were omitted.

EXERCISE 1.54 Initial public offeringsThe median is -31% and the mean is +111%. Some very successful companies will inflate themean, but the median is more resistant to extreme values. It was not true that half of allcompanies increased their stock price more than 111% and half by less.

EXERCISE 1.69 GMAT Scores(a) 59.53%(b) 451 and below.(c) Above 711.


EXERCISE 1.95 Grading managersMean is 250; standard deviation is 175.57.

EXERCISE 2.10 Health and wealthThere appears to be a positive relationship between GDP per person and life expectancy at birth.The relationship is not linear but rather curved and, judging by the scatter about the c-shape,moderately strong. The graph supports our expectation of a positive relationship.

EXERCISE 2.15 Business starts and failures(a) See plot below.

Business failures by business starts in 1998 (50 states and DC)

Business starts

Busi

ness

failu

res

0 5000 10000 15000 20000

050

0010

000

1500

0

FL

CA

NY

TX

(b) Because, on average, as the explanatory variable increases, so does the response or dependentvariable. Further, the straight line that might be used to describe these data has a positive slope.(c) See labeled point on plot above.(d) California.(e) The states are California, Florida, New York, and Texas. They are not from one region, butare the four most populous states.


EXERCISE 2.18 Where the stores are.(a) See plot below.

Target stores vs. Wal-Mart stores in 50 states

Wal-Mart stores

Targ

et s

tore

s

0 50 100 150 200 250

050

100

150

TX

CA

(b) Two points stand out: California and Texas. California has a very high number of Targetstores, and a relatively high number of Wal-Marts. Texas has an extremely high number of Wal-Mart stores and more Targets than every state but California.(c) The relationship between Wal-Mart stores and Target stores is positive and approximatelylinear. The strength of the relationship is moderate to weak since the points are not tightlyclustered about a line.


EXERCISE 2.25 Mutual fund performanceSee plot below. A straight line appears to describe the data adequately; the correlation isestimated to be 0.85. There are no extreme outliers, although 1983 and 1986 are relatively farfrom the line.

EAFE return (percent)

Vang

uard

retu

rn (p

erce

nt)

0 20 40 60

010

2030

4050

EXERCISE 2.29 CEO compensation and stock market performance.The magazine is incorrectly interpreting a correlation of zero to mean there is a negativerelationship between compensation and performance. We might explain the study's conclusionsby saying that knowing a company's stock performance provides almost no information aboutwhat the CEO will earn, and knowing a CEO's compensation provides little to no informationabout the company's stock performance.

EXERCISE 2.45 Moving in step?The correlation between American and European stock prices is positive and has become fairlystrong. Therefore stocks in the US and Europe are now to some degree marching in lockstep. Ashift in the prices of shares in the American market is likely to be met with a shift in the samedirection and of a similar magnitude in the European market. Therefore the European marketdoesn't counterbalance moves in the American market.

EXERCISE 2.46 Interpreting correlation.No this is not true. About 64% of the variability in one market is explained by changes in theother market.


EXERCISE 2.48 Cash or credit?(a) See plot below.

Gross sales vs. items sold for credit and cash

Items sold

Gro

ss s

ales

($)

0 10 20 30 40 50

200

400

600

800 = credit

= cash

(b) Gross sales = 240.88 + 10.34(Item count credit card); Gross sales = 64.58 + 12.46(Item countcash); One additional cash purchase is expected to increase the daily gross sales by an estimated$12.46, while an additional credit card purchase is expected to increase the gross sales by anestimated $10.34.

EXERCISE 2.55 What's my grade?(a) The slope is 0.16. The intercept is 30.2. (b) 78.2(c) r2 = 0.36. Only 36% of the variation in final exam scores is explained by the students' pre-exam totals. Other factors (such as time spent preparing for the exam) also influenced final examscores. Students could have done much worse or much better than predicted based on these otherfactors.

EXERCISE 2.75 Education and income.Lurking variables might include health and socioeconomic background. Men from families withmore resources tend to be better educated, and men who are in better health will live longer andearn more.


EXERCISE 2.100 Are high interest rates bad for stocks?(a) Stock return = 16.2352 - 0.5727(Tbill return). 13.37%.(b) A negative slope suggests that as treasury bill returns increase, average stock returns woulddecrease. No because we cannot conclude the true slope is different from zero and therefore wecannot conclude there is a linear relationship between T-bills and stock returns.(c) Neither the scatterplot nor the regression output provide evidence that there is a linearrelationship between these two variables. Knowing the T-bill return for next year would probablynot help us predict stock returns.

EXERCISE 2.108 Size and selling price of houses. (a) The distribution is unimodal with a median of $127,100 but it is heavily skewed right. Theselling prices ranged from $53,600 to $340,000. About one quarter of all homes sold for less than$110,500 and about one quarter for more than $169,100. See plot below.

50000 100000 150000 200000 250000 300000 350000

05

1015

20

Selling price of Ames, IA homes

Selling price ($)


(b) There appears to be a moderate to strong positive linear relationship between the squarefootage of an Ames-area home and its selling price. See plot below.

Selling price vs. square feet of Ames, IA homes

Square feet

Pric

e ($

)

500 1000 1500 2000 2500

5000

015

0000

2500

0035

0000

(c) The estimated intercept is 4786.46 and the estimated slope is 92.82. On average, an additionalsquare foot will add an estimated $92.82 to the selling price of a home.(d) $153,300.10.(e) 69.64%.

EXERCISE 3.11 Sampling by accountants.From the medium-debt accounts strata, sample accounts 417, 494, 322, 247, and 097. From thesmall-debt accounts strata, sample accounts 3698, 1452, 2605, 2480, and 3716.

EXERCISE 3.13 Ring-no-answer.Most likely the higher "ring-no-answer" rate is from the July 1 to August 31 period, sincehouseholds are more likely to be away on vacation during the summer months. A high rate ofnonresponse increases the chances that certain groups are systematically underrepresented in thesample; the sample does not represent the population of interest so much as it represents thosepeople who tend to be at home and are willing to answer the survey questions.

EXERCISE 3.19 Quality control sampling.Bottles 12, 4, and 11. (B0986, A1101, A2220.)


EXERCISE 3.40 Does charting help investors?Ten of the 20 subjects are selected at random and given computers with the software thathighlights trends. The other ten subjects are given computers without the software. After a fixedperiod of trading, compare average profits for the two groups. One possible randomization is toassign students A, B, C, D, F, G, I, M, R, and S to the computers with the trend software andstudents E, H, J, K, L, N, O, P, Q, and T to the computers without the trend software.

For the matched pairs design, ten subject are initially assigned computers with the trendhighlighting software while the other ten are given standard computers. After a fixed tradingperiod, students switch computer types and trade for the same amount of time. One possiblerandomization is to initially assign students C, D, F, H, I, J, L, M, N, and S to the computers withthe trend software and students A, B, E, G, K, O, P, Q, R, and T to the computers without it.

EXERCISE 4.24 Stock price movements.(a) {++++, +++-, ++-+, +-++, -+++, ++--, +--+, --++, -+-+, +-+-, -++-, +---, -+--, --+-, ---+, ----}(b) {0, 1, 2, 3, 4}

EXERCISE 4.26 Car colors.0.295; 0.348.


EXERCISE 4.39 How many cars?(a) The probabilities are all between zero and one and sum to 1. See plot below.

0 1 2 3 4 5

0.0

0.1

0.2

0.3

0.4

Probability histogram for number of cars owned

Cars

Prob

abilit

y

(b) A randomly chosen household has one or more cars. The probability of this event is 0.91.(c) 0.20.


EXERCISE 4.53 How many rooms?(a) The owner-occupied distribution is mounded with a center near 6 while the renter-occupieddistribution is more peaked with a centered near 4. See plots below and on following page.

1 2 3 4 5 6 7 8 9 10

0.0

0.1

0.2

0.3

0.4

Rooms

Prob

abilit

y


1 2 3 4 5 6 7 8 9 10

0.0

0.1

0.2

0.3

0.4

Probability histogram of rooms in rented units

Rooms

Prob

abilit

y

(b) The mean for owner-occupied units is 6.284, the mean for renter-occupied units is 4.187. Thedifference in means is evident in the plots.

EXERCISE 4.61 Time and motion studies.(a) Mean time is 31 seconds.(b) No because a reduction in variation about the mean is not the same as a reduction in averagetime.(c) The overall mean does not depend on either standard deviation or the correlation and wouldnot change.

EXERCISE 4.63 Time and motion studies.4.47; 4.73; Longer times in the first step are associated with longer times in the second step, andshorter times in the first step are associated with shorter times in the second step. Therefore, if thefirst step takes longer than average, the second one is likely to as well, and the two steps couldend up taking much longer than average. The positive correlation causes more spread about themean. If the correlation were negative, the delays or advances would tend to cancel each otherout and there would be less spread about the mean.

EXERCISE 4.70 Diversification.The mean return of the diversified portfolio is 1.23%. The standard deviation is 4.58%.

EXERCISE 4.87 Flaws in carpets<0.0001

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