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Continued part 1, part 2 of ebook Business statistics: For contemporary decision making provide readers with content about: regression analysis and forecasting; simple regression analysis and correlation; multiple regression analysis; building multiple regression models; time-series forecasting and index numbers; nonparametric statistics and quality; analysis of categorical data; nonparametric statistics; statistical quality control;...
UNIT IV REGRESSION ANALYSIS AND FORECASTING In the first three units of the text, you were introduced to basic statistics, distributions, and how to make inferences through confidence interval estimation and hypothesis testing In Unit IV, we explore relationships between variables through regression analysis and learn how to develop models that can be used to predict one variable by another variable or even multiple variables We will examine a cadre of statistical techniques that can be used to forecast values from time-series data and how to measure how well the forecast is CHAPTER 12 Simple Regression Analysis and Correlation LEARNING OBJECTIVES The overall objective of this chapter is to give you an understanding of bivariate linear regression analysis, thereby enabling you to: Calculate the Pearson product-moment correlation coefficient to determine if there is a correlation between two variables Explain what regression analysis is and the concepts of independent and dependent variable Calculate the slope and y-intercept of the least squares equation of a regression line and from those, determine the equation of the regression line Calculate the residuals of a regression line and from those determine the fit of the model, locate outliers, and test the assumptions of the regression model Calculate the standard error of the estimate using the sum of squares of error, and use the standard error of the estimate to determine the fit of the model Calculate the coefficient of determination to measure the fit for regression models, and relate it to the coefficient of correlation Use the t and F tests to test hypotheses for both the slope of the regression model and the overall regression model Calculate confidence intervals to estimate the conditional mean of the dependent variable and prediction intervals to estimate a single value of the dependent variable Determine the equation of the trend line to forecast outcomes for time periods in the future, using alternate coding for time periods if necessary 10 Use a computer to develop a regression analysis, and interpret the output that is associated Najlah Feanny/©Corbis with it Predicting International Hourly Wages by the Price of a Big Mac The McDonald’s Corporation is the leading global foodservice retailer with more than 30,000 local restaurants serving nearly 50 million people in more than 119 countries each day This global presence, in addition to its consistency in food offerings and restaurant operations, makes McDonald’s a unique and attractive setting for economists to make salary and price comparisons around the world Because the Big Mac hamburger is a standardized hamburger produced and sold in virtually every McDonald’s around the world, the Economist, a weekly newspaper focusing on international politics and business news and opinion, as early as 1986 was compiling information about Big Mac prices as an indicator of exchange rates Building on this idea, researchers Ashenfelter and Jurajda proposed comparing wage rates across countries and the price of a Big Mac hamburger Shown below are Big Mac prices and net hourly wage figures (in U.S dollars) for 27 countries Note that net hourly wages are based on a weighted average of 12 professions Country Argentina Australia Brazil Britain Canada Chile China Czech Republic Denmark Euro area Hungary Indonesia Japan Malaysia Big Mac Price (U.S $) Net Hourly Wage (U.S $) 1.42 1.86 1.48 3.14 2.21 1.96 1.20 1.96 4.09 2.98 2.19 1.84 2.18 1.33 1.70 7.80 2.05 12.30 9.35 2.80 2.40 2.40 14.40 9.59 3.00 1.50 13.60 3.10 (continued) Country Mexico New Zealand Philippines Poland Russia Singapore South Africa South Korea Sweden Switzerland Thailand Turkey United States Big Mac Price (U.S $) Net Hourly Wage (U.S $) 2.18 2.22 2.24 1.62 1.32 1.85 1.85 2.70 3.60 4.60 1.38 2.34 2.71 2.00 6.80 1.20 2.20 2.60 5.40 3.90 5.90 10.90 17.80 1.70 3.20 14.30 Managerial and Statistical Questions Is there a relationship between the price of a Big Mac and the net hourly wages of workers around the world? If so, how strong is the relationship? Is it possible to develop a model to predict or determine the net hourly wage of a worker around the world by the price of a Big Mac hamburger in that country? If so, how good is the model? If a model can be constructed to determine the net hourly wage of a worker around the world by the price of a Big Mac hamburger, what would be the predicted net hourly wage of a worker in a country if the price of a Big Mac hamburger was $3.00? Sources: McDonald’s Web site at: http://www.mcdonalds.com/corp/about html; Michael R Pakko and Patricia S Pollard, “Burgernomics: A Big Mac Guide to Purchasing Power Parity,” research publication by the St Louis Federal Reserve Bank at: http://research.stlouisfed.org/publications/review/03/11/ M pakko.pdf; Orley Ashenfelter and Stepán Jurajda, “Cross-Country Comparisons of Wage Rates: The Big Mac Index,” unpublished manuscript, Princeton University and CERGEEI/Charles University, October 2001; The Economist, at: http://www.economist.com/index.html In business, the key to decision making often lies in the understanding of the relationships between two or more variables For example, a company in the distribution business may determine that there is a relationship between the price of crude oil and their own transportation costs Financial experts, in studying the behavior of the bond market, might find it useful to know if the interest rates on bonds are related to the prime 465 466 Chapter 12 Simple Regression Analysis and Correlation interest rate set by the Federal Reserve A marketing executive might want to know how strong the relationship is between advertising dollars and sales dollars for a product or a company In this chapter, we will study the concept of correlation and how it can be used to estimate the relationship between two variables We will also explore simple regression analysis through which mathematical models can be developed to predict one variable by another We will examine tools for testing the strength and predictability of regression models, and we will learn how to use regression analysis to develop a forecasting trend line 12.1 CORRELATION TA B L E Data for the Economics Example Day 10 11 12 Interest Rate Futures Index 7.43 7.48 8.00 7.75 7.60 7.63 7.68 7.67 7.59 8.07 8.03 8.00 221 222 226 225 224 223 223 226 226 235 233 241 PEARSON PRODUCTMOMENT CORRELATION COEFFICIENT (12.1) Correlation is a measure of the degree of relatedness of variables It can help a business researcher determine, for example, whether the stocks of two airlines rise and fall in any related manner For a sample of pairs of data, correlation analysis can yield a numerical value that represents the degree of relatedness of the two stock prices over time In the transportation industry, is a correlation evident between the price of transportation and the weight of the object being shipped? If so, how strong are the correlations? In economics, how strong is the correlation between the producer price index and the unemployment rate? In retail sales, are sales related to population density, number of competitors, size of the store, amount of advertising, or other variables? Several measures of correlation are available, the selection of which depends mostly on the level of data being analyzed Ideally, researchers would like to solve for r, the population coefficient of correlation However, because researchers virtually always deal with sample data, this section introduces a widely used sample coefficient of correlation, r This measure is applicable only if both variables being analyzed have at least an interval level of data Chapter 17 presents a correlation measure that can be used when the data are ordinal The statistic r is the Pearson product-moment correlation coefficient, named after Karl Pearson (1857–1936), an English statistician who developed several coefficients of correlation along with other significant statistical concepts The term r is a measure of the linear correlation of two variables It is a number that ranges from -1 to to +1, representing the strength of the relationship between the variables An r value of +1 denotes a perfect positive relationship between two sets of numbers An r value of -1 denotes a perfect negative correlation, which indicates an inverse relationship between two variables: as one variable gets larger, the other gets smaller An r value of means no linear relationship is present between the two variables r = ©xy - ©(x - x)(y - y) 2©(x - x)2 ©(y - y)2 (©x©y) n = C c ©x - (©x)2 (©y)2 d c ©y d n n Figure 12.1 depicts five different degrees of correlation: (a) represents strong negative correlation, (b) represents moderate negative correlation, (c) represents moderate positive correlation, (d) represents strong positive correlation, and (e) contains no correlation What is the measure of correlation between the interest rate of federal funds and the commodities futures index? With data such as those shown in Table 12.1, which represent the values for interest rates of federal funds and commodities futures indexes for a sample of 12 days, a correlation coefficient, r, can be computed 12.1 Correlation 467 FIGURE 12.1 Five Correlations (a) Strong Negative Correlation (r = –.933) (b) Moderate Negative Correlation (r = –.674) (c) Moderate Positive Correlation (r = 518) (d) Strong Positive Correlation (r = 909) (e) Virtually No Correlation (r = –.004) Examination of the formula for computing a Pearson product-moment correlation coefficient (12.1) reveals that the following values must be obtained to compute r : ©x, ©x 2, ©y, ©y 2, ©xy, and n In correlation analysis, it does not matter which variable is designated x and which is designated y For this example, the correlation coefficient is computed as shown in Table 12.2 The r value obtained (r = 815) represents a relatively strong positive relationship between interest rates and commodities futures index over this 12-day period Figure 12.2 shows both Excel and Minitab output for this problem 468 Chapter 12 Simple Regression Analysis and Correlation TA B L E Computation of r for the Economics Example Futures Index y Interest x Day 10 11 12 7.43 7.48 8.00 7.75 7.60 7.63 7.68 7.67 7.59 8.07 8.03 8.00 © x = 92.93 221 222 226 225 224 223 223 226 226 235 233 241 © y = 2,725 x2 55.205 55.950 64.000 60.063 57.760 58.217 58.982 58.829 57.608 65.125 64.481 64.000 © x = 720.220 (21,115.07) r = y2 48,841 49,284 51,076 50,625 50,176 49,729 49,729 51,076 51,076 55,225 54,289 58,081 © y = 619,207 (92.93)(2725) 12 (92.93)2 (2725)2 c(720.22) d c(619,207) d B 12 12 xy 1,642.03 1,660.56 1,808.00 1,743.75 1,702.40 1,701.49 1,712.64 1,733.42 1,715.34 1,896.45 1,870.99 1,928.00 © xy = 21,115.07 = 815 FIGURE 12.2 Excel Output Excel and Minitab Output for the Economics Example Interest Rate Interest Rate Futures Index Futures Index 0.815 Minitab Output Correlations: Interest Rate, Futures Index Pearson correlation of Interest Rate and Futures Index = 0.815 p-Value = 0.001 12.1 PROBLEMS 12.1 Determine the value of the coefficient of correlation, r, for the following data X Y 18 12 13 11 14 17 21 12.2 Determine the value of r for the following data X Y 158 349 296 510 87 301 110 322 436 550 12.3 In an effort to determine whether any correlation exists between the price of stocks of airlines, an analyst sampled six days of activity of the stock market Using the following prices of Delta stock and Southwest stock, compute the coefficient of correlation Stock prices have been rounded off to the nearest tenth for ease of computation Delta Southwest 47.6 46.3 50.6 52.6 52.4 52.7 15.1 15.4 15.9 15.6 16.4 18.1 12.2 Introduction to Simple Regression Analysis 469 12.4 The following data are the claims (in $ millions) for BlueCross BlueShield benefits for nine states, along with the surplus (in $ millions) that the company had in assets in those states State Claims Surplus Alabama Colorado Florida Illinois Maine Montana North Dakota Oklahoma Texas $1,425 273 915 1,687 234 142 259 258 894 $277 100 120 259 40 25 57 31 141 Use the data to compute a correlation coefficient, r, to determine the correlation between claims and surplus 12.5 The National Safety Council released the following data on the incidence rates for fatal or lost-worktime injuries per 100 employees for several industries in three recent years Industry Textile Chemical Communication Machinery Services Nonferrous metals Food Government Year Year Year 46 52 90 1.50 2.89 1.80 3.29 5.73 48 62 72 1.74 2.03 1.92 3.18 4.43 69 63 81 2.10 2.46 2.00 3.17 4.00 Compute r for each pair of years and determine which years are most highly correlated 12.2 INTRODUCTION TO SIMPLE REGRESSION ANALYSIS TA B L E Airline Cost Data Number of Passengers Cost ($1,000) 61 63 67 69 70 74 76 81 86 91 95 97 4.280 4.080 4.420 4.170 4.480 4.300 4.820 4.700 5.110 5.130 5.640 5.560 Regression analysis is the process of constructing a mathematical model or function that can be used to predict or determine one variable by another variable or other variables The most elementary regression model is called simple regression or bivariate regression involving two variables in which one variable is predicted by another variable In simple regression, the variable to be predicted is called the dependent variable and is designated as y The predictor is called the independent variable, or explanatory variable, and is designated as x In simple regression analysis, only a straight-line relationship between two variables is examined Nonlinear relationships and regression models with more than one independent variable can be explored by using multiple regression models, which are presented in Chapters 13 and 14 Can the cost of flying a commercial airliner be predicted using regression analysis? If so, what variables are related to such cost? A few of the many variables that can potentially contribute are type of plane, distance, number of passengers, amount of luggage/freight, weather conditions, direction of destination, and perhaps even pilot skill Suppose a study is conducted using only Boeing 737s traveling 500 miles on comparable routes during the same season of the year Can the number of passengers predict the cost of flying such routes? It seems logical that more passengers result in more weight and more baggage, which could, in turn, result in increased fuel consumption and other costs Suppose the data displayed in Table 12.3 are the costs and associated number of passengers for twelve 500-mile commercial airline flights using Boeing 737s during the same season of the year We will use these data to develop a regression model to predict cost by number of passengers Usually, the first step in simple regression analysis is to construct a scatter plot (or scatter diagram), discussed in Chapter Graphing the data in this way yields preliminary information about the shape and spread of the data Figure 12.3 is an Excel scatter plot of the data in Table 12.3 Figure 12.4 is a close-up view of the scatter plot produced by 470 Chapter 12 Simple Regression Analysis and Correlation FIGURE 12.3 6.000 Excel Scatter Plot of Airline Cost Data Cost ($1,000) 5.000 4.000 3.000 2.000 1.000 0.000 20 40 60 80 Number of Passengers 100 120 FIGURE 12.4 Close-Up Minitab Scatter Plot of Airline Cost Data 5500 Cost 5000 4500 4000 60 70 80 90 Number of Passengers 100 Minitab Try to imagine a line passing through the points Is a linear fit possible? Would a curve fit the data better? The scatter plot gives some idea of how well a regression line fits the data Later in the chapter, we present statistical techniques that can be used to determine more precisely how well a regression line fits the data 12.3 DETERMINING THE EQUATION OF THE REGRESSION LINE The first step in determining the equation of the regression line that passes through the sample data is to establish the equation’s form Several different types of equations of lines are discussed in algebra, finite math, or analytic geometry courses Recall that among these equations of a line are the two-point form, the pointslope form, and the slope-intercept form In regression analysis, researchers use the slope-intercept equation of a line In math courses, the slope-intercept form of the equation of a line often takes the form y = mx + b where m = slope of the line b = y intercept of the line In statistics, the slope-intercept form of the equation of the regression line through the population points is yN = b + b 1x where yN = the predicted value of y b = the population y intercept b = the population slope 12.3 Determining the Equation of the Regression Line 471 For any specific dependent variable value, yi , yi = b + b 1xi + H i where xi = the value of the independent variable for the ith value yi = the value of the dependent variable for the ith value b = the population y intercept b = the population slope H i = the error of prediction for the ith value Unless the points being fitted by the regression equation are in perfect alignment, the regression line will miss at least some of the points In the preceding equation, H i represents the error of the regression line in fitting these points If a point is on the regression line, H i = These mathematical models can be either deterministic models or probabilistic models Deterministic models are mathematical models that produce an “exact” output for a given input For example, suppose the equation of a regression line is y = 1.68 + 2.40x For a value of x = 5, the exact predicted value of y is y = 1.68 + 2.40(5) = 13.68 We recognize, however, that most of the time the values of y will not equal exactly the values yielded by the equation Random error will occur in the prediction of the y values for values of x because it is likely that the variable x does not explain all the variability of the variable y For example, suppose we are trying to predict the volume of sales (y) for a company through regression analysis by using the annual dollar amount of advertising (x) as the predictor Although sales are often related to advertising, other factors related to sales are not accounted for by amount of advertising Hence, a regression model to predict sales volume by amount of advertising probably involves some error For this reason, in regression, we present the general model as a probabilistic model A probabilistic model is one that includes an error term that allows for the y values to vary for any given value of x A deterministic regression model is y = b + b 1x The probabilistic regression model is y = b + b 1x + H b + b x is the deterministic portion of the probabilistic model, b + b 1x + H In a deterministic model, all points are assumed to be on the line and in all cases H is zero Virtually all regression analyses of business data involve sample data, not population data As a result, b and b are unattainable and must be estimated by using the sample statistics, b0 and b1 Hence the equation of the regression line contains the sample y intercept, b0, and the sample slope, b1 EQUATION OF THE SIMPLE REGRESSION LINE yN = b0 + b1x Where b0 = the sample intercept b1 = the sample slope To determine the equation of the regression line for a sample of data, the researcher must determine the values for b0 and b1 This process is sometimes referred to as least squares analysis Least squares analysis is a process whereby a regression model is developed by producing the minimum sum of the squared error values On the basis of this premise and calculus, a particular set of equations has been developed to produce components of the regression model.* *Derivation of these formulas is beyond the scope of information being discussed here but is presented in WileyPLUS 472 Chapter 12 Simple Regression Analysis and Correlation FIGURE 12.5 Error of the Prediction Minitab Plot of a Regression Line Regression Line Points (X, Y) Examine the regression line fit through the points in Figure 12.5 Observe that the line does not actually pass through any of the points The vertical distance from each point to the line is the error of the prediction In theory, an infinite number of lines could be constructed to pass through these points in some manner The least squares regression line is the regression line that results in the smallest sum of errors squared Formula 12.2 is an equation for computing the value of the sample slope Several versions of the equation are given to afford latitude in doing the computations SLOPE OF THE REGRESSION LINE (12.2) (©x)(©y) n (©x)2 ©x n ©xy - ©(x - x)(y - y) ©xy - nx y b1 = = = ©(x - x) ©x - nx The expression in the numerator of the slope formula 12.2 appears frequently in this chapter and is denoted as SSxy SSxy = ©(x - x)(y - y) = ©xy - (©x)(©y) n The expression in the denominator of the slope formula 12.2 also appears frequently in this chapter and is denoted as SSxx SSxx = ©(x - x)2 = ©x - (©x)2 n With these abbreviations, the equation for the slope can be expressed as in Formula 12.3 ALTERNATIVE FORMULA FOR SLOPE (12.3) b1 = SSxy SSxx Formula 12.4 is used to compute the sample y intercept The slope must be computed before the y intercept y INTERCEPT OF THE REGRESSION LINE (12.4) b0 = y - b1x = ©y (©x) - b1 n n Formulas 12.2, 12.3, and 12.4 show that the following data are needed from sample information to compute the slope and intercept: ©x, ©y, ©x 2, and, ©xy, unless sample means are used Table 12.4 contains the results of solving for the slope and intercept and determining the equation of the regression line for the data in Table 12.3 The least squares equation of the regression line for this problem is yN = 1.57 + 0407x C19-26 Chapter 19 Decision Analysis FIGURE 19.7 th (.65) Decision Tree for the Investment Example—All Options Included $500 No grow $360 Rapid gro wth (.35 Bonds ) $360 th (.65) Don’t buy –$200 No grow Stocks Rapid gro wth (.35 $255 $432.80 $413.84 $100 ) $1100 th (.832) No grow Rapid gro wth (.16 Bonds 8) $432.80 Buy (–$100) Forecast No growth (.625) $100 th (.832) –$200 No grow Stocks $18.40 Rapid gro wth (.16 8) $513.84 $238.80 Forecast Rapid growth (.375) $500 th (.347) No grow Rapid gro wth (.65 Bonds $648.90 3) $1100 $500 $100 th (.347) –$200 No grow Stocks $648.90 Rapid gro wth (.65 3) $1100 Suppose the decision maker had to pay $100 for the forecaster’s prediction The expected monetary value of the decision with information shown in Figure 19.6 is reduced from $513.84 to $413.84, which is still superior to the $360 expected monetary value without sample information Figure 19.7 is the decision tree for the investment information with the options of buying the information or not buying the information included The tree is constructed by combining the decision trees from Figures 19.5 and 19.6 and including the cost of buying information ($100) and the expected monetary value with this purchased information ($413.84) D E M O N S T R AT I O N PROBLEM 19.4 In Demonstration Problem 19.1, the decision makers were faced with the opportunity to increase capacity to meet a possible increase in product demand Here we reduced the decision alternatives and states of nature and altered the payoffs and probabilities Use the following decision table to create a decision tree that displays the decision alternatives, the payoffs, the probabilities, the states of demand, and the expected monetary payoffs The decision makers can buy information about the states of demand for $5 (recall that amounts are in $ millions) Incorporate this fact into your decision Calculate the expected value of sampling information for this problem The decision alternatives are: no expansion or build a new facility The states of demand and prior probabilities are: less demand (.20), no change (.30), or large increase (.50) C19-27 19.4 Revising Probabilities in Light of Sample Information State of Demand Decision Alternative Less (.20) No Change (.30) Large Increase (.50) −$ −$50 $ −$20 $ $65 No Expansion New Facility The state-of-demand forecaster has historically not been accurate 100% of the time For example, when the demand was less, the forecaster correctly predicted it 75 of the time When there was no change in demand, the forecaster correctly predicted it 80 of the time Sixty-five percent of the time the forecaster correctly forecast large increases when large increases occurred Shown next are the probabilities that the forecaster will predict a particular state of demand under the actual states of demand State of Demand Forecast Less No Change Large Increase 75 20 05 10 80 10 05 30 65 Less No Change Large Increase Solution The following figure is the decision tree for this problem when no sample information is purchased 0) Less (.2 $3 No expansion –$3 No change (.30) Large in crease (.50 $2 ) $6 $16.50 0) Less (.2 New facility $16.50 –$50 No change (.30) Large in crease (.50 –$20 ) $65 In light of sample information, the prior probabilities of the three states of demand can be revised Shown here are the revisions for F1 (forecast of less demand), F2 (forecast of no change in demand), and F3 (forecast of large increase in demand) State of Demand Prior Probability For Forecast of Less Demand (F1 ) 20 Less (s1 ) 30 No change (s2 ) Large increase (s3 ) 50 Conditional Probability Joint Probability Revised Probability P(F1 |ss1 ) ϭ 75 P(F1 ∩ s1 ) ϭ 150 150͞ 205 ϭ 732 P(F1 |s2 ) ϭ 10 P(F1 ∩ s2 ) ϭ 030 030͞ 205 ϭ 146 P(F1 |s3 ) ϭ 05 P(F1 ∩ s3 ) ϭ 025 025͞ 205 P(F1 ) ϭ 205 For Forecast of No Change in Demand (F2 ) Less (s1 ) 20 P(F2 |s1 ) ϭ 20 P(F2 ∩ s1 ) ϭ 040 040͞ 430 30 P(F2 |s2 ) ϭ 80 P(F2 ∩ s2 ) ϭ 240 240͞ 430 No change (s2 ) 50 P(F2 |s3 ) ϭ 30 P(F2 ∩ s3 ) ϭ 150 150͞ 430 Large increase (s3 ) P(F2 ) ϭ 430 For Forecast of Large Increase in Demand (F3 ) 20 P(F3 |s1 ) ϭ 05 P(F3 ∩ s1 ) ϭ 010 010͞ 365 Less (s1 ) 30 P(F3 |s2 ) ϭ 10 P(F3 ∩ s2 ) ϭ 030 030͞ 365 No change (s2 ) Large increase (s3 ) 50 P(F3 |s3 ) ϭ 65 P(F3 ∩ s3 ) ϭ 325 325͞ 365 P(F3 ) ϭ 365 ϭ 122 ϭ 093 ϭ 558 ϭ 349 ϭ 027 ϭ 082 ϭ 890 C19-28 Chapter 19 Decision Analysis From these revised probabilities and other information, the decision tree containing alternatives and states using sample information can be constructed The following figure is the decision tree containing the sample information alternative and the portion of the tree for the alternative of no sampling information 0) Less (.2 $3 No expansion –$3 No change (.30) Large increa se (.50 ) $2 $6 $16.50 0) Less (.2 New facility No change (.30) –$20 Large increa se (.50 ) $65 $16.50 Don’t buy –$1.172 No expansion –$1.172 $17.74 New facility Forecast (.205) Less –$31.59 Buy (–$5.00) $2.931 No expansion $22.74 –$50 Forecast (.430) No change $6.875 New facility $6.875 Forecast (.365) Large increase $5.423 No expansion $54.86 New facility $54.86 32) –$3 Less (.7 No change (.146) $2 Large increa se (.12 2) $6 32) –$50 Less (.7 No change (.146) –$20 Large increa se (.12 2) $65 93) –$3 Less (.0 No change (.558) $2 Large increa se (.34 9) $6 93) –$50 Less (.0 No change (.558) –$20 Large increa se (.34 9) $65 27) –$3 Less (.0 No change (.082) $2 Large increa se (.89 0) $6 27) –$50 Less (.0 No change (.082) –$20 Large increa se (.89 0) $65 If the decision makers calculate the expected monetary value after buying the sample information, they will see that the value is $17.74 The final expected monetary value with sample information is calculated as follows EMV at Buy Node: - $1.172(.205) + $6.875(.430) + $54.86(.365) = $22.74 However, the sample information cost $5 Hence, the net expected monetary value at the buy node is $22.74 (EMV) - $5.00 (cost of information) = $17.74 (net expected monetary value) Problems C19-29 The worth of the sample information is Expected Monetary Value of Sample Information = Expected Monetary Value with Sample Information - Expected Monetary Value without Sample Information = $22.74 - $16.50 = $6.24 19.4 PROBLEMS 19.12 Shown here is a decision table from a business situation The decision maker has an opportunity to purchase sample information in the form of a forecast With the sample information, the prior probabilities can be revised Also shown are the probabilities of forecasts from the sample information for each state of nature Use this information to answer parts (a) through (d) State of Nature s (.30) s (.70) Alternative d1 $350 −$100 d2 −$200 $325 State of Nature Forecast s s2 s1 s2 90 10 25 75 a Compute the expected monetary value of this decision without sample information b Compute the expected monetary value of this decision with sample information c Use a tree diagram to show the decision options in parts (a) and (b) d Calculate the value of the sample information 19.13 a A car rental agency faces the decision of buying a fleet of cars, all of which will be the same size It can purchase a fleet of small cars, medium cars, or large cars The smallest cars are the most fuel efficient and the largest cars are the greatest fuel users One of the problems for the decision makers is that they not know whether the price of fuel will increase or decrease in the near future If the price increases, the small cars are likely to be most popular If the price decreases, customers may demand the larger cars Following is a decision table with these decision alternatives, the states of nature, the probabilities, and the payoffs Use this information to determine the expected monetary value for this problem State of Nature Decision Alternative Small Cars Medium Cars Large Cars Fuel Decrease (.60) Fuel Increase (.40) −$225 $125 $350 $425 −$150 −$400 b The decision makers have an opportunity to purchase a forecast of the world oil markets that has some validity in predicting gasoline prices The following matrix gives the probabilities of these forecasts being correct for various states of nature Use this information to revise the prior probabilities and recompute the expected monetary value on the basis of sample information What is the C19-30 Chapter 19 Decision Analysis expected value of sample information for this problem? Should the agency decide to buy the forecast? State of Nature Forecast Fuel Decrease Fuel Increase 75 25 15 85 Fuel Decrease Fuel Increase 19.14 a A small group of investors is considering planting a tree farm Their choices are (1) don’t plant trees, (2) plant a small number of trees, or (3) plant a large number of trees The investors are concerned about the demand for trees If demand for trees declines, planting a large tree farm would probably result in a loss However, if a large increase in the demand for trees occurs, not planting a tree farm could mean a large loss in revenue opportunity They determine that three states of demand are possible: (1) demand declines, (2) demand remains the same as it is, and (3) demand increases Use the following decision table to compute an expected monetary value for this decision opportunity State of Demand Decline (.20) Same (.30) Increase (.50) Don’t Plant Decision Alternative Small Tree Farm Large Tree Farm $20 −$90 −$600 $0 $10 −$150 −$40 $175 $800 b Industry experts who believe they can forecast what will happen in the tree industry contact the investors The following matrix shows the probabilities with which it is believed these “experts” can foretell tree demand Use these probabilities to revise the prior probabilities of the states of nature and recompute the expected value of sample information How much is this sample information worth? State of Demand Forecast Decrease Same Increase Decrease Same Increase 70 25 05 02 95 03 02 08 90 19.15 a Some oil speculators are interested in drilling an oil well The rights to the land have been secured and they must decide whether to drill The states of nature are that oil is present or that no oil is present Their two decision alternatives are drill or don’t drill If they strike oil, the well will pay $1 million If they have a dry hole, they will lose $100,000 If they don’t drill, their payoffs are $0 when oil is present and $0 when it is not The probability that oil is present is 11 Use this information to construct a decision table and compute an expected monetary value for this problem b The speculators have an opportunity to buy a geological survey, which sometimes helps in determining whether oil is present in the ground When the geologists say there is oil in the ground, there actually is oil 20 of the time When there is oil in the ground, 80 of the time the geologists say there is no oil When there is no oil in the ground, 90 of the time the geologists say there is no oil When there is no oil in the ground, 10 of the time the geologists say there is oil Use this information to revise the prior probabilities of oil being present in the ground and compute the expected monetary value based on sample information What is the value of the sample information for this problem? Problems C19-31 Decision Making at the CEO Level The study of CEOs revealed that decision making takes place in many different areas of business No matter what the decision concerns, it is critical for the CEO or manager to identify the decision alternatives Sometimes decision alternatives are not obvious and can be identified only after considerable examination and brainstorming Many different alternatives are available to decision makers in personnel, finance, operations, and so on Alternatives can sometimes be obtained from worker suggestions and input Others are identified through consultants or experts in particular fields Occasionally, a creative and unobvious decision alternative is derived that proves to be the most successful choice Alex Trotman at Ford Motor, in a reorganization decision, chose the alternative of combining two operations into one unit Other alternatives might have been to combine other operations into a unit (rather than the North American and European), create more units, or not reorganize at all At Kodak, CEO George Fisher made the decision that the company would adopt digital and electronic imaging wholeheartedly In addition, he determined that these new technologies would be interwoven with their paper and film products in such a manner as to be “seamless.” Fisher had other alternatives available such as not entering the arena of digital and electronic imaging or entering it but keeping it separated from the paper and film operation Union Pacific was faced with a crisis as it watched Burlington Northern make an offer to buy Santa Fe Pacific The CEO chose to propose a counteroffer He could have chosen to not enter the fray CEOs need to identify as many states of nature that can occur under the decision alternatives as possible What might happen to sales? Will product demand increase or decrease? What is the political climate for environmental or international monetary regulation? What will occur next in the business cycle? Will there be inflation? What will the competitors do? What new inventions or developments will occur? What is the investment climate? Identifying as many of these states as possible helps the decision maker examine decision alternatives in light of those states and calculate payoffs accordingly Many different states of nature may arise that will affect the outcome of CEO decisions made in the 1990s Ford Motor may find that the demand for a “world car” does not material, materializes so slowly that the company wastes their effort for many years, materializes as Trotman foresaw, or materializes even faster The world economy might undergo a depression, a slowdown, a constant growth, or even an accelerated rate of growth Political conditions in countries of the world might make an American “world car” unacceptable The governments of the countries that would be markets for such a car might cause the countries to become more a part of the world economy, stay about the same, slowly withdraw from the world scene, or become isolated States of nature can impact a CEO’s decision in other ways The rate of growth and understanding of technology is uncertain in many ways and can have a great effect on the decision to embrace digital and electronic imaging Will the technology develop in time for the merging of these new technologies and the paper and film operations? Will there be suppliers who can provide materials and parts? What about the raw materials used in digital and electronic imaging? Will there be an abundance, a shortage, or an adequate supply of raw materials? Will the price of raw materials fluctuate widely, increase, decrease, or remain constant? The decision maker should recognize whether he or she is a risk avoider or a risk taker Does propensity toward risk vary by situation? Should it? How the board of directors and stockholders view risk? Will the employees respond to risk taking or avoidance? Successful CEOs may well incorporate risk taking, risk avoidance, and expected value decision making into their decisions Perhaps the successful CEOs know when to take risks and when to pull back In Union Pacific’s decision to make a counteroffer for Santa Fe Pacific, risk taking is evident Of course with the possibility of Burlington Northern growing into a threateningly large competitor, it could be argued that making a counteroffer was actually a risk averse decision Certainly, the decision by a successful company like Ford Motor, which had five of the top 10 vehicles at the time, to reorganize in an effort to make a “world car” is risk taking Kodak’s decision to embrace digital and electronic imaging and merge it with their paper and film operations is a risktaking venture If successful, the payoffs from these CEO decisions could be great The current success of Ford Motor may just scratch the surface if the company successfully sells their “world car” in the twenty-first century On the other hand, the company could experience big losses or receive payoffs somewhere in between Union Pacific’s purchasing of Santa Fe Pacific could greatly increase their market share and result in huge dividends to the company A downturn in transportation, the unforeseen development of some more efficient mode of shipping, inefficient or hopelessly irreversibly poor management in Santa Fe Pacific, or other states of nature could result in big losses to Union Pacific MCI’s decision to wage a war with the Baby Bells did not result in immediate payoffs because of a slowing down of growth due to efforts by AT&T in the long-distance market and MCI’s difficulty in linking to a large cable company CEOs are not always able to visualize all decision alternatives However, creative, inspired thinking along with the brainstorming of others and an extensive investigation of the facts and figures can successfully result in the identification of C19-32 Chapter 19 Decision Analysis most of the possibilities States of nature are unknown and harder to discern However, awareness, insight, understanding, and knowledge of economies, markets, governments, and competitors can greatly aid a decision maker in considering possible states of nature that may impact the payoff of a deci- sion along with the likelihood that such states of nature might occur The payoffs for CEOs range from the loss of thousands of jobs including his/her own, loss of market share, and bankruptcy of the company, to worldwide growth, record stockholder dividends, and fame ETHICAL CONSIDERATIONS Ethical considerations occasionally arise in decision analysis situations The techniques presented in this chapter are for aiding the decision maker in selecting from among decision alternatives in light of payoffs and expected values Payoffs not always reflect all costs, including ethical considerations The decision maker needs to decide whether to consider ethics in examining decision alternatives What are some decision alternatives that raise ethical questions? Some decision alternatives are environmentally damaging in the form of ground, air, or water pollution Other choices endanger the health and safety of workers In the area of human resources, some decision alternatives include eliminating jobs and laying off workers Should the ethical issues involved in these decisions be factored into payoffs? For example, what effects would a layoff have on families and communities? Does a business have any moral obligation toward its workers and its community that should be taken into consideration in payoffs? Does a decision alternative involve producing a product that is detrimental to a customer or a customer’s family? Some marketing decisions might involve the use of false or misleading advertising Are distorted or misleading advertising attacks on competing brands unethical even when the apparent payoff is great? States of nature are usually beyond the control of the decision maker; therefore, it seems unlikely that unethical behavior would be connected with a state of nature However, obtaining sample or perfect information under which to make decisions about states of nature has the usual potential for unethical behavior in sampling In many cases, payoffs other than the dollar values assigned to a decision alternative should be considered In using decision analysis with ethical behavior, the decision maker should attempt to factor into the payoffs the cost of pollution, safety features, human resource loss, and so on Unethical behavior is likely to result in a reduction or loss of payoff SUMMARY Decision analysis is a branch of quantitative management in which mathematical and statistical approaches are used to assist decision makers in reaching judgments about alternative opportunities Three types of decisions are (1) decisions made under certainty, (2) decisions made under uncertainty, and (3) decisions made with risk Several aspects of the decisionmaking situation are decision alternatives, states of nature, and payoffs Decision alternatives are the options open to decision makers from which they can choose States of nature are situations or conditions that arise after the decision has been made over which the decision maker has no control The payoffs are the gains or losses that the decision maker will reap from various decision alternatives These three aspects (decision alternatives, states of nature, and payoffs) can be displayed in a decision table or payoff table Decision making under certainty is the easiest of the three types of decisions to make In this case, the states of nature are known, and the decision maker merely selects the decision alternative that yields the highest payoff Decisions are made under uncertainty when the likelihoods of the states of nature occurring are unknown Four approaches to making decisions under uncertainty are maximax criterion, maximin criterion, Hurwicz criterion, and minimax regret The maximax criterion is an optimistic approach based on the notion that the best possible outcomes will occur In this approach, the decision maker selects the maximum possible payoff under each decision alternative and then selects the maximum of these Thus, the decision maker is selecting the maximum of the maximums The maximin criterion is a pessimistic approach The assumption is that the worst case will happen under each decision alternative The decision maker selects the minimum payoffs under each decision alternative and then picks the maximum of these as the best solution Thus, the decision maker is selecting the best of the worst cases, or the maximum of the minimums The Hurwicz criterion is an attempt to give the decision maker an alternative to maximax and maximin that is somewhere between an optimistic and a pessimistic approach With this approach, decision makers select a value called alpha between and to represent how optimistic they are The maximum and minimum payoffs for each decision alternative are examined The alpha weight is applied to the maximum payoff under each decision alternative and Ϫ a is applied to the minimum payoff These two weighted values are combined Supplementary Problems for each decision alternative, and the maximum of these weighted values is selected Minimax regret is calculated by examining opportunity loss An opportunity loss table is constructed by subtracting each payoff from the maximum payoff under each state of nature This step produces a lost opportunity under each state The maximum lost opportunity from each decision alternative is determined from the opportunity table The minimum of these values is selected, and the corresponding decision alternative is chosen In this way, the decision maker has reduced or minimized the regret, or lost opportunity In decision making with risk, the decision maker has some prior knowledge of the probability of each occurrence of each state of nature With these probabilities, a weighted payoff referred to as expected monetary value (EMV) can be calculated for each decision alternative A person who makes decisions based on these EMVs is called an EMVer The expected monetary value is essentially the average payoff that would occur if the decision process were to be played out over a long period of time with the probabilities holding constant C19-33 The expected value of perfect information can be determined by comparing the expected monetary value if the states of nature are known to the expected monetary value The difference in the two is the expected value of perfect information Utility refers to a decision maker’s propensity to take risks People who avoid risks are called risk avoiders People who are prone to take risks are referred to as risk takers People who use EMV generally fall between these two categories Utility curves can be sketched to ascertain or depict a decision maker’s tendency toward risk By use of Bayes’ theorem, the probabilities associated with the states of nature in decision making under risk can be revised when new information is obtained This information can be helpful to the decision maker However, it usually carries a cost for the decision maker This cost can reduce the payoff of decision making with sample information The expected monetary value with sample information can be compared to the expected monetary value without it to determine the value of sample information KEY TERMS decision alternatives decision analysis decision making under certainty decision making under risk decision making under uncertainty decision table decision trees EMVer expected monetary value (EMV) expected value of perfect information expected value of sample information Hurwicz criterion maximax criterion maximin criterion minimax regret opportunity loss table payoffs payoff table risk avoider risk taker states of nature FORMULA Bayes’ Rule P (Xi ƒY) = P(Xi) # P(Y ƒ Xi) P(X1) # P(Y ƒ X1) + P(X2) # P(Y ƒ X2) + # # # + P(Xn) # P(Y ƒ Xn) SUPPLEMENTARY PROBLEMS CALCULATING THE STATISTICS 19.16 Use the following decision table to complete parts (a) through (d) State of Nature Decision Alternative d1 d2 d3 d4 s1 s2 50 −75 25 75 100 200 40 10 a Use the maximax criterion to determine which decision alternative to select b Use the maximin criterion to determine which decision alternative to select c Use the Hurwicz criterion to determine which decision alternative to select Let a ϭ d Compute an opportunity loss table from these data Use this table and a minimax regret criterion to determine which decision alternative to select 19.17 Use the following decision table to complete parts (a) through (c) State of Nature s (.30) s (.25) s (.20) s (.25) Decision Alternative d1 d2 400 300 250 −100 300 600 100 200 C19-34 Chapter 19 Decision Analysis a Draw a decision tree to represent this decision table b Compute the expected monetary values for each decision and label the decision tree to indicate what the final decision would be c Compute the expected payoff of perfect information Compare this answer to the answer determined in part (b) and compute the value of perfect information 19.18 Shown here is a decision table A forecast can be purchased by the decision maker The forecaster is not correct 100% of the time Also given is a table containing the probabilities of the forecast being correct under different states of nature Use the first table to compute the expected monetary value of this decision without sample information Use the second table to revise the prior probabilities of the various decision alternatives From this and the first table, compute the expected monetary value with sample information Construct a decision tree to represent the options, the payoffs, and the expected monetary values Calculate the value of sample information State of Nature Decision Alternative d1 d2 d3 s (.40) s (.60) $200 −$75 $175 $150 $450 $125 b Construct an opportunity loss table and use minimax regret to select a decision alternative c Compare the results of the maximax, maximin, and minimax regret criteria in selecting decision alternatives 19.20 Some companies use production learning curves to set pricing strategies They price their product lower than the initial cost of making the product; after some period of time, the learning curve takes effect and the product can be produced for less than its selling price In this way, the company can penetrate new markets with aggressive pricing strategies and still make a longterm profit A company is considering using the learning curve to set its price on a new product There is some uncertainty as to how soon, if at all, the production operation will learn to make the product more quickly and efficiently If the learning curve does not drop enough or the initial price is too low, the company will be operating at a loss on this product If the product is priced too high, the sales volume might be too low to justify production Shown here is a decision table that contains as its states of nature several possible learning-curve scenarios The decision alternatives are three different pricing strategies Use this table and the Hurwicz criterion to make a decision about the pricing strategies with each given value of alpha State of Nature State of Nature s1 Forecast s s1 s2 90 10 30 70 No Slow Fast Learning Learning Learning Price Low Decision Alternative Price Medium Price High −$700 −$300 $100 −$400 −$100 $125 $1,200 $550 $150 TESTING YOUR UNDERSTANDING 19.19 Managers of a manufacturing firm decided to add Christmas tree ornaments to their list of production items However, they have not decided how many to produce because they are uncertain about the level of demand Shown here is a decision table that has been constructed to help the managers in their decision situation Use this table to answer parts (a) through (c) State of Demand Decision Alternative (Produce) Small Number Modest Number Large Number Small Moderate Large $200 $100 −$300 $250 $300 $400 $300 $600 $2,000 a Use maximax and maximin criteria to evaluate the decision alternatives a b c d a ϭ 10 a ϭ 50 a ϭ 80 Compare and discuss the decision choices in parts (a) through (c) 19.21 An entertainment company owns two amusement parks in the South They are faced with the decision of whether to open the parks in the winter If they choose to open the parks in the winter, they can leave the parks open during regular hours (as in the summer) or they can open only on the weekends To some extent, the payoffs from opening the park hinge on the type of weather that occurs during the winter season Following are the payoffs for various decision options about opening the park for two different weather scenarios: mild weather and severe weather Use the information to construct a decision tree Determine the expected monetary value and the value of perfect information Supplementary Problems State of the Weather Mild (.75) Open Regular Hours $2,000 Decision Alternative Open Weekends Only $1,200 Not Open at All −$300 Severe (.25) −$2,500 −$200 $100 19.22 A U.S manufacturing company decided to consider producing a particular model of one of its products just for sale in Germany Because of the German requirements, the product must be made specifically for German consumption and cannot be sold in the United States Company officials believe the market for the product is highly price-sensitive Because the product will be manufactured in the United States and exported to Germany, the biggest variable factor in being price competitive is the exchange rate between the two countries If the dollar is strong, German consumers will have to pay more for the product in marks If the dollar becomes weaker against the mark, Germans can buy more U.S products for their money The company officials are faced with decision alternatives of whether to produce the product The states of the exchange rates are: dollar weaker, dollar stays the same, and dollar stronger The probabilities of these states occurring are 35, 25, and 40, respectively Some negative payoffs will result from not producing the product because of sunk development and market research costs and because of lost market opportunity If the product is not produced, the payoffs are Ϫ$700 when the dollar gets weaker, Ϫ$200 when the dollar remains about the same, and $150 when the dollar gets stronger If the product is produced, the payoffs are $1,800 when the dollar gets weaker, $400 when the exchange rates stay about the same, and Ϫ$1,600 when the dollar gets stronger Use this information to construct a decision tree and a decision table for this decision-making situation Use the probabilities to compute the expected monetary values of the decision alternatives On the basis of this information, which decision choice should the company make? Compute the expected monetary value of perfect information and the value of perfect information 19.23 a A small retailer began as a mom-and-pop operation selling crafts and consignment items During the past two years, the store’s volume grew significantly The owners are trying to decide whether to purchase an automated checkout system Their present manual system is slow They are concerned about lost business due to inability to ring up sales quickly The automated system would also offer some accounting and inventory advantages The problem is that the automated system carries a large fixed cost, and the owners feel that sales volume would have to grow to justify the cost C19-35 The following decision table contains the decision alternatives for this situation, the possible states of future sales, prior probabilities of those states occurring, and the payoffs Use this information to compute the expected monetary payoffs for the alternatives State of Sales Reduction (.15) Constant (.35) Increase (.50) Decision Automate −$40,000 −$15,000 $60,000 Alternative Donít Automate $5,000 $10,000 −$30,000 b For a fee, the owners can purchase a sales forecast for the near future The forecast is not always perfect The probabilities of these forecasts being correct for particular states of sales are shown here Use these probabilities to revise the prior state probabilities Compute the expected monetary value on the basis of sample information Determine the value of the sample information State of Sales Forecast Reduction Constant Increase Reduction Constant Increase 60 30 10 10 80 10 05 25 70 19.24 a A city is considering airport expansion In particular, the mayor and city council are trying to decide whether to sell bonds to construct a new terminal The problem is that at present demand for gates is not strong enough to warrant construction of a new terminal However, a major airline is investigating several cities to determine which will become its new hub If this city is selected, the new terminal will easily pay for itself The decision to build the terminal must be made by the city before the airline will say whether the city has been chosen as its hub Shown here is a decision table for this dilemma Use this information to compute expected monetary values for the alternatives and reach a conclusion State of Nature Decision Alternative Build Terminal Don’t Build Terminal City Chosen (.20) City Not Chosen (.80) $12,000 −$1,000 −$8,000 $2,000 b An airline industry expert indicates that she will sell the city decision makers her best “guess” as to whether the city will be chosen as hub for the airline The probabilities of her being right or wrong are given C19-36 Chapter 19 Decision Analysis Use these probabilities to revise the prior probabilities of the city being chosen as the hub and then calculate the expected monetary value by using the sample information Determine the value of the sample information for this problem State of Nature Forecast City Chosen City Not Chosen City Chosen City Not Chosen 45 55 40 60 ANALYZING THE DATABASES A carpet and rug manufacturer in the manufacturing data- Suppose you are the CEO of a hospital in the hospital data- base is faced with the decision of making expenditures in the form of machinery in anticipation of strong growth or not making the expenditures and losing market opportunity Experts claim that there is about a 40% probability of having strong growth in the industry and a 60% probability of not having strong growth in the industry If there is strong growth, the company will realize a payoff of $5,500,000 if the capital expenditure is made Under strong growth, if the company does not make the capital expenditure, it will still realize a $750,000 payoff in new business, which it can handle with existing equipment If there is not strong growth in the industry and the company has made the capital expenditure, the payoff will be Ϫ$2,000,000 If there is not strong growth and the capital expenditure has not been made, the company will receive a payoff of $500,000 Analyze this situation by using the decision analysis techniques presented in this chapter base You are considering expansion of the physical facility What are some decision alternatives to consider? What are some states of nature that can occur in this decision-making environment? How would you go about calculating the payoffs for such a decision? Suppose you are the CEO of a company such as Procter & Gamble in the financial database What are some decisions that you might make in which you would consider decision alternatives? Name three arenas in which you would be making substantial strategic decisions (e.g., marketing, finance, production, and human resources) Delineate at least three decision alternatives in each of these arenas Examine and discuss at least two states of nature that could occur under these decision alternatives in each arena CASE FLETCHER-TERRY: ON THE CUTTING EDGE The Fletcher-Terry Company of Farmington, Connecticut, is a worldwide leader in the development of glass-cutting tools and accessories for professional glaziers, glass manufacturers, glass artisans, and professional framers The company can trace its roots back to 1868 when a young engineer, Samuel Monce, developed and patented a hardened steel tool that could effectively replace expensive diamonds as a cutting device Using this invention as his centerpiece, Monce formed the Monce Company, which went on to become a leader in glass-cutting devices for several decades Meanwhile, in 1894, Monce’s nephew Fred Fletcher got a patent on a replaceablewheel cutter Ten years later he went into business with his father-in-law, Franklin Terry, forming the Fletcher-Terry Company In 1935, the Fletcher-Terry Company bought the Monce Company, thereby combining the assets and knowledge of the two companies For the next four decades, Fletcher-Terry had much success making its traditional product lines of hand-held glass cutters and cutting wheels for the glass, glazing, and hardware markets However, by the 1980s, Fletcher-Terry was facing a crisis Its two largest customers, distributors of cutting devices, decided to introduce their own private-label cutters made overseas By the end of 1982, Fletcher-Terry’s sales of handheld glass cutters were down 45% Fletcher-Terry responded by investing heavily in technology with the hope that automation would cut costs; however, the technology never worked The company then decided to expand its line of offerings by creating private lines through imports, but the dollar weakened and any price advantage was lost Eventually, Fletcher-Terry had to write off this line with a substantial loss Company managers realized that if they did not change the way they did business, the company would not survive They began a significant strategic planning process in which they set objectives and redefined the mission of the company Among the new objectives were to increase market share where the company was already strong, penetrate new markets with new products, provide technological expertise for product development, promote greater employee involvement and growth, and achieve a sales growth rate twice that of the gross domestic product To accomplish these objectives, the company invested in plant and process improvements that reduced costs and improved quality Markets were researched for both old and new products and marketing efforts were launched to reestablish the company’s products as being “the first choice of professionals.” A participatory management system was implemented that encouraged risk taking and creativity among employees Case Following these initiatives, sales growth totaled 82.5% from 1987 and 1993 Fletcher-Terry expanded its offerings with bevel mat cutters, new fastener tools, and a variety of hand tools essential to professional picture framers, and graduated from being a manufacturer of relatively simple hand tools to being a manufacturer of mechanically complex equipment and tools Today, Fletcher-Terry maintains a leadership position in its industry through dedicated employees who are constantly exploring new ideas to help customers become more productive Because of its continuous pursuit of quality, the company earned the Ford Q-101 Quality Supplier Award In August of 2001, Fletcher-Terry introduced its FramerSolutions.com online business-to-business custom mat cutting service especially designed for professional picture framers The mission of Fletcher-Terry is to develop innovative tools and equipment for the markets they serve worldwide and make customer satisfaction their number one priority Discussion Fletcher-Terry managers have been involved in many decisions over the years Of particular importance were the decisions made in the 1980s when the company was struggling to survive Several states of nature took place in the late 1970s and 1980s over which managers had little or no control Suppose the Fletcher-Terry management team wants to reflect on their decisions and the events that surrounded them, and they ask you to make a brief report summarizing the situation Delineate at least five decisions that Fletcher-Terry probably had to make during that troublesome time Using your knowledge of the economic situation both in the United States and in the rest of the world in addition to information given in the case, present at least four states of nature during that time that had significant influence on the outcomes of the managers’ decisions C19-37 At one point, Fletcher-Terry decided to import its own private line of cutters Suppose that before taking such action, the managers had the following information available Construct a decision table and a decision tree by using this information Explain any conclusions reached Suppose the decision for managers was to import or not import If they imported, they had to worry about the purchasing value of the dollar overseas If the value of the dollar went up, the company could profit $350,000 If the dollar maintained its present position, the company would still profit by $275,000 However, if the value of the dollar decreased, the company would be worse off with an additional loss of $555,000 One business economic source reported that there was a 25% chance that the dollar would increase in value overseas, a 35% chance that it would remain constant, and a 40% chance that it would lose value overseas If the company decided not to import its own private label, it would have a $22,700 loss no matter what the value of the dollar was overseas Explain the possible outcomes of this analysis to the management team in terms of EMV, risk aversion, and risk taking Bring common sense into the process and give your recommendations on what the company should given the analysis Keep in mind the company’s situation and the fact that it had not yet tried any solution Explain to company officials the expected value of perfect information for this decision Source: Adapted from “Fletcher-Terry: On the Cutting Edge,” Real-World Lessons for America’s Small Businesses: Insights from the Blue Chip Enterprise Initiative Published by Nation’s Business magazine on behalf of Connecticut Mutual Life Insurance Company and the U.S Chamber of Commerce in association with The Blue Chip Enterprise Initiative, 1994 See also Fletcher, Terry, available at http://www.fletcher-terry.com/ Areas of the Standard Normal Distribution The entries in this table are the probabilities that a standard normal random variable is between and Z (the shaded area) Z TABLE 0.1 Z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.1 0.2 0.3 0.4 0000 0398 0793 1179 1554 0040 0438 0832 1217 1591 0080 0478 0871 1255 1628 0120 0517 0910 1293 1664 0160 0557 0948 1331 1700 0199 0596 0987 1368 1736 0239 0636 1026 1406 1772 0279 0675 1064 1443 1808 0319 0714 1103 1480 1844 0359 0753 1141 1517 1879 0.5 0.6 0.7 0.8 0.9 1915 2257 2580 2881 3159 1950 2291 2611 2910 3186 1985 2324 2642 2939 3212 2019 2357 2673 2967 3238 2054 2389 2704 2995 3264 2088 2422 2734 3023 3289 2123 2454 2764 3051 3315 2157 2486 2794 3078 3340 2190 2517 2823 3106 3365 2224 2549 2852 3133 3389 1.0 1.1 1.2 1.3 1.4 3413 3643 3849 4032 4192 3438 3665 3869 4049 4207 3461 3686 3888 4066 4222 3485 3708 3907 4082 4236 3508 3729 3925 4099 4251 3531 3749 3944 4115 4265 3554 3770 3962 4131 4279 3577 3790 3980 4147 4292 3599 3810 3997 4162 4306 3621 3830 4015 4177 4319 1.5 1.6 1.7 1.8 1.9 4332 4452 4554 4641 4713 4345 4463 4564 4649 4719 4357 4474 4573 4656 4726 4370 4484 4582 4664 4732 4382 4495 4591 4671 4738 4394 4505 4599 4678 4744 4406 4515 4608 4686 4750 4418 4525 4616 4693 4756 4429 4535 4625 4699 4761 4441 4545 4633 4706 4767 2.0 2.1 2.2 2.3 2.4 4772 4821 4861 4893 4918 4778 4826 4864 4896 4920 4783 4830 4868 4898 4922 4788 4834 4871 4901 4925 4793 4838 4875 4904 4927 4798 4842 4878 4906 4929 4803 4846 4881 4909 4931 4808 4850 4884 4911 4932 4812 4854 4887 4913 4934 4817 4857 4890 4916 4936 2.5 2.6 2.7 2.8 2.9 4938 4953 4965 4974 4981 4940 4955 4966 4975 4982 4941 4956 4967 4976 4982 4943 4957 4968 4977 4983 4945 4959 4969 4977 4984 4946 4960 4970 4978 4984 4948 4961 4971 4979 4985 4949 4962 4972 4979 4985 4951 4963 4973 4980 4986 4952 4964 4974 4981 4986 3.0 3.1 3.2 3.3 3.4 4987 4990 4993 4995 4997 4987 4991 4993 4995 4997 4987 4991 4994 4995 4997 4988 4991 4994 4996 4997 4988 4992 4994 4996 4997 4989 4992 4994 4996 4997 4989 4992 4994 4996 4997 4989 4992 4995 4996 4997 4990 4993 4995 4996 4997 4990 4993 4995 4997 4998 3.5 4.0 4.5 5.0 6.0 4998 49997 499997 4999997 499999999 Critical Values from the t Distribution α ta TABLE 0.2 Values of α for One-Tailed Test and α / for Two-Tailed Test df t.100 t.050 t.025 t.010 t.005 t.001 3.078 1.886 1.638 1.533 1.476 6.314 2.920 2.353 2.132 2.015 12.706 4.303 3.182 2.776 2.571 31.821 6.965 4.541 3.747 3.365 63.656 9.925 5.841 4.604 4.032 318.289 22.328 10.214 7.173 5.894 10 1.440 1.415 1.397 1.383 1.372 1.943 1.895 1.860 1.833 1.812 2.447 2.365 2.306 2.262 2.228 3.143 2.998 2.896 2.821 2.764 3.707 3.499 3.355 3.250 3.169 5.208 4.785 4.501 4.297 4.144 11 12 13 14 15 1.363 1.356 1.350 1.345 1.341 1.796 1.782 1.771 1.761 1.753 2.201 2.179 2.160 2.145 2.131 2.718 2.681 2.650 2.624 2.602 3.106 3.055 3.012 2.977 2.947 4.025 3.930 3.852 3.787 3.733 16 17 18 19 20 1.337 1.333 1.330 1.328 1.325 1.746 1.740 1.734 1.729 1.725 2.120 2.110 2.101 2.093 2.086 2.583 2.567 2.552 2.539 2.528 2.921 2.898 2.878 2.861 2.845 3.686 3.646 3.610 3.579 3.552 21 22 23 24 25 1.323 1.321 1.319 1.318 1.316 1.721 1.717 1.714 1.711 1.708 2.080 2.074 2.069 2.064 2.060 2.518 2.508 2.500 2.492 2.485 2.831 2.819 2.807 2.797 2.787 3.527 3.505 3.485 3.467 3.450 26 27 28 29 30 1.315 1.314 1.313 1.311 1.310 1.706 1.703 1.701 1.699 1.697 2.056 2.052 2.048 2.045 2.042 2.479 2.473 2.467 2.462 2.457 2.779 2.771 2.763 2.756 2.750 3.435 3.421 3.408 3.396 3.385 40 50 60 70 80 1.303 1.299 1.296 1.294 1.292 1.684 1.676 1.671 1.667 1.664 2.021 2.009 2.000 1.994 1.990 2.423 2.403 2.390 2.381 2.374 2.704 2.678 2.660 2.648 2.639 3.307 3.261 3.232 3.211 3.195 90 100 150 200 ∞ 1.291 1.290 1.287 1.286 1.282 1.662 1.660 1.655 1.653 1.645 1.987 1.984 1.976 1.972 1.960 2.368 2.364 2.351 2.345 2.326 2.632 2.626 2.609 2.601 2.576 3.183 3.174 3.145 3.131 3.090 wn an Me nt S σ unk nown σ1 , σ kno wn 9.2 z HT for μ 9.3 t HT for μ 9.5 χ2 HT for σ2 σ2 wn no unk mp les ≥3 -sa 10.4 10.5 z HT for p1Ϫ p2 F HT for σ 2, σ 2 10.2 10.1 z HT for μ1−μ2 t HT for μ1−μ2 own 10.1 10.2 z CI for μ Ϫμ t CI for μ Ϫμ 2 10.3 t CI for D Proportion 9.4 z HT for p ple am 1-S s is T est hes pot Hy ples unkn 1 10.4 z CI for p Ϫ p ples 8.2 t CI for μ 8.4 χ CI for σ2 C fid on ce en als n ow ple Sa 1- 8.3 z CI for p Proportion rv te In n) io at m sti (E t Sam Tree Diagram Taxonomy of Inferential Techniques ns t Sam 8.1 z CI for μ σ kno Mea nden know n n an know Me σ un σ1 , σ an s m e ende anc Depe Indep i Var ample s ns 2 σ σ1 , 2 ortio en ,σ Prop 10.3 t HT for D s ce Dep ple an Vari am tS den σ a 2-S Proportions le mp s kn ple Me am s nce nden ia Var depe nden t Var iab In 11.2 One-Way ANOVA In de pe nd en tV ar ia bl es 11.4 Randomized Block Design s Indep e 2-S le e abl ari t V ables n e ari nd epe k V nd Bloc I +1 11.5 Two-Way ANOVA ... 12 7.43 7.48 8.00 7.75 7.60 7.63 7.68 7.67 7.59 8.07 8.03 8.00 © x = 92. 93 22 1 22 2 22 6 22 5 22 4 22 3 22 3 22 6 22 6 23 5 23 3 24 1 © y = 2, 725 x2 55 .20 5 55.950 64.000 60.063 57.760 58 .21 7 58.9 82 58. 829 ... Residuals y - yN (y - yN )2 10 11 12 23 29 29 35 42 46 50 54 64 66 76 78 69 95 1 02 118 126 125 138 178 156 184 176 22 5 -1 3 .22 -. 62 6.38 8.99 1.37 -8 .56 -4 .49 26 .58 -1 7.74 5.80 -2 4. 52 20. 02 174.77 -0 .38... 4. 820 4.700 5.110 5.130 5.640 5.560 22 7 -. 054 123 -. 20 8 061 -. 28 2 157 -. 167 040 -. 144 20 4 0 42 05153 0 029 2 01513 04 326 003 72 079 52 024 65 027 89 00160 020 74 041 62 00176 ©(y - yN) = - 001 ©(y - yN)2