484 Microsoft Excel 2010: Data Analysis and Business Modeling In the Two Way ANOVA with Interaction worksheet, I changed the data from the previous example to the data shown in Figure 57-10. After running the analysis for a two-factor ANOVA with replication, I obtained the results shown in Figure 57-11. FIGURE 57-10 Sa es data w th nteract on between pr ce and advert s ng. FIGURE 57-11 Output for the two factor ANOVA w th nteract on. In this data set, the p-value for interaction is .001. When you see a low p-value (less than .15) for interaction, you do not even check p-values for row and column factors. You simply forecast sales for any price and advertising combination to equal the mean of the three observations involving that price and advertising combination. For example, the best forecast for sales during a month with high advertising and medium price is: 4 34 + 40 + 32 3 106 = 35.555 units= The standard deviation of forecast errors is again the square root of the mean square within ( 17.11 = 4.14) Thus, you can be 95 percent sure that the sales forecast is accurate within 8.26 units. Chapter 57 Randomized Blocks and Two-Way ANOVA 485 Figure 57-12 illustrates why this data exhibits a signicant interaction between price and advertising. For a low and medium price, increased advertising increases sales, but if price is high, increased advertising has no effect on sales. This explains why you cannot use equation 2 to forecast sales when a signicant interaction is present. After all, how can you talk about an advertising effect when the effect of advertising depends on the price? FIGURE 57-12 Pr ce and advert s ng exh b t a s gn cant nteract on n th s set of data. Problems The data for the following problems is in the le Ch57.xlsx. 1. You believe that pressure (high, medium, or low) and temperature (high, medium, or low) inuence the yield of a production process. Given this theory, determine the an- swers to the following problems: ❑ Use the data in the Problem 1 worksheet to determine how temperature and/or pressure inuence the yield of the process. ❑ With high pressure and low temperature, you’re 95 percent sure that process yield will be in what range? 2. You are trying to determine how the particular sales representative and the number of sales calls (one, three, or ve) made to a doctor inuence the amount (in thousands of dollars) that each doctor prescribes of your drug. Use the data in the Problem 2 work- sheet to determine the answers to the following problems: ❑ How do the representative and number of sales calls inuence sales volume? ❑ If Rep 3 makes ve sales calls to a doctor, you’re 95 percent sure she will generate prescriptions within what range of dollars? 3. Answer the questions in Problem 2 by using the data in the Problem 3 worksheet. 4. The le Coupondata.xlsx contains information on sales of peanut butter for weeks when a coupon was given out (or not) and advertising was done (or not) in the Sunday paper. Describe how the coupon and advertising inuence peanut butter sales. 487 Chapter 58 Using Moving Averages to Understand Time Series Question answered in this chapter: ■ I’m trying to analyze the upward trend in quarterly revenues of Amazon.com since 1996. Fourth quarter sales in the U.S. are usually larger (because of Christmas) than sales during the rst quarter of the following year. This pattern obscures the up- ward trend in sales. Is there any way that I can graphically show the upward trend in revenues? Answers to This Chapter’s Question Time series data simply displays the same quantity measured at different points in time. For example, the data in the le Amazon.xlsx, a subset of which is shown in Figure 58-1, displays the time series for quarterly revenues in millions of dollars for Amazon.com. The data covers the time interval from the fourth quarter of 1995 through the third quarter of 2009. To graph this time series, select the range C2:D59, which contains the quarter number (the rst quarter is Quarter 1 and the last is Quarter 57) and Amazon quarterly revenues (in millions of dollars). Then choose Chart on the Insert tab, and choose the second option under the Scatter chart type. (Scatter with Smooth Lines and Markers.) The time series plot is shown in Figure 58-2. FIGURE 58-1 Quarter y revenues for Amazon sa es. 488 Microsoft Excel 2010: Data Analysis and Business Modeling FIGURE 58-2 T me ser es p ot of quarter y toy revenues. There is an upward trend in revenues, but the fact that fourth quarter revenues dwarf revenues during the rst three quarters of each year makes it hard to spot the trend. Because there are four quarters per year, it would be nice to graph average revenues during the last four quarters. This is called a four- period moving average. Using a four-quarter mov- ing average smooths out the seasonal inuence because each average will contain one data point for each quarter. Such a graph is called a moving average graph because the plotted average “moves” over time. Moving average graphs also smooth out random variation, which helps you get a better idea of what is going on with your data. To create a moving average graph of quarterly revenues, you can modify the chart. Select the graph, and then click a data point until all the data points are displayed in blue. Right-click any point, click Add Trendline, and then select the Moving Average option. Set the period equal to 4. Microsoft Excel now creates the four-quarter moving average trend curve that’s shown in Figure 58-3. (See the le Amazonma.xlsx.) For each quarter, Excel plots the average of the current quarter and the last three quarters. Of course, for a four-quarter moving average, the moving average curve starts with the fourth data point. The moving average curve makes it clear that Amazon.com’s revenues had a steady upward trend. In fact the slope of the four-quarter moving average appears to be increasing. In all likelihood , the slope of this moving average graph will eventually level off, resulting in a graph that looks like an S curve. The Excel Trend Curve feature cannot t S curves, but the Excel 2010 Solver can be used to t S curves to data. Chapter 58 Using Moving Averages to Understand Time Series 489 FIGURE 58-3 Four quarter mov ng average trend curve. Problem ■ The le Ch58data.xlsx contains quarterly revenues for GM, Ford, and GE. Construct a four-quarter moving average trend curve for each company’s revenues. Describe what you learn from each trend curve. 491 Chapter 59 Winters’s Method You often need to predict future values of a time series, such as monthly costs or monthly product revenues. This is usually difcult because the characteristics of any time series are constantly changing. Smoothing or adaptive methods are usually best suited for forecasting future values of a time series. In this chapter, I describe the most powerful smoothing method: Winters’s method. To help you understand how Winters’s method works, I’ll use it to forecast monthly housing starts in the United States. Housing starts are simply the number of new homes whose construction begins during a month. I’ll begin by describing the three key characteristics of a time series. Time Series Characteristics The behavior of most time series can be explained by understanding three characteristics: base, trend, and seasonality. ■ The base of a series describes the series’ current level in the absence of any seasonality. For example, suppose the base level for U.S. housing starts is 160,000. In this case, you can believe that if the current month were an average month relative to other months of the year, 160,000 housing starts would occur. ■ The trend of a time series is the percentage increase per period in the base. Thus, a trend of 1.02 means that you estimate that housing starts are increasing by 2 percent each month. ■ The seasonality (seasonal index) for a period tells you how far above or below a typical month you can expect housing starts to be. For example, if the December seasonal index is .8, then December housing starts are 20 percent below a typical month. If the June seasonal index is 1.3, then June housing starts are 30 percent higher than a typical month. Parameter Denitions paAfter observing month t, you will have used all data observed through the end of month t to estimate the following quantities of interest: ■ L t Level of series ■ T t Trend of series ■ S t Seasonal index for current month 492 Microsoft Excel 2010: Data Analysis and Business Modeling The key to Winters’s method is the following three equations, which are used to update L t , T t , and S t . In the following formulas, alp, bet, and gam are called smoothing parameters. You choose the values of these parameters to optimize forecasts. In the following formulas, c equals the number of periods in a seasonal cycle (c 12 months, for example) and x t equals the observed value of the time series at time t. ■ Formula 1: L t alp(x t /s t c )+(1–alp)(L t 1 *T t 1 ) ■ Formula 2: T t bet(L t /L t 1 )+(1–bet)T t 1 ■ Formula 3: S t gam(x t /L t )+(1–gam)s t c Formula 1 indicates that the new base estimate is a weighted average of the current observation (deseasonalized) and the last period’s base updated by the last trend estimate. Formula 2 indicates that the new trend estimate is a weighted average of the ratio of the current base to the last period’s base (this is a current estimate of trend) and the last period’s trend. Formula 3 indicates that you update your seasonal index estimate as a weighted average of the estimate of the seasonal index based on the current period and the previous estimate. Note that larger values of the smoothing parameters correspond to putting more weight on the current observation. You can dene F t,k as your forecast (F) after period t for the period t+k. This results in the formula F t,k L t*( T t) k s t+k c . (I refer to this as formula 4.) This formula rst uses the current trend estimate to update the base k periods forward. Then the resulting base estimate for period t+k is adjusted by the appropriate seasonal index. Initializing Winters’s Method To start Winters’s method, you must have initial estimates for the series base, trend, and seasonal indexes. I used monthly housing starts for the years 1986 and 1987 to initialize Winters’s method. Then I chose smoothing parameters to optimize one-month-ahead forecasts for the years 1988 through 1996. See Figure 59-1 and the le House2.xlsx. Here are the steps I followed. Chapter 59 Winters’s Method 493 FIGURE 59-1 n t a zat on of W nters s method. Step 1 I estimated, for example, the January seasonal index as the average of January housing starts for 1986 and 1987 divided by the average monthly starts for 1986 and 1987. Therefore, copying from G14 to G15:G25 the formula AVERAGE(B2,B14)/AVERAGE($B$2:$B$25) generates the estimates of seasonal indexes. For example, the January estimate is 0.75 and the June estimate is 1.17. Step 2 To estimate the average monthly trend, I took the twelfth root of the 1987 mean starts divided by the 1986 mean starts. I computed this in cell J3 (and copied it to cell D25) with the formula (J1/J2)^(1/12). Step 3 Going into January 1987, I estimated the base of the series as the deseasonalized December 1987 value. This was computed in C25 with the formula (B25/G25). Estimating the Smoothing Constants Now I’m ready to estimate smoothing constants. In column C, I will update the series base; in column D, the series trend; and in column G, the seasonal indexes. In column E, I com- pute the forecast for next month, and in column F, I compute the absolute percentage error for each month. Finally, I use the Solver to choose values for the smoothing constants that minimize the sum of the absolute percentage errors. Here’s the process. Step 1 In G11:I11, I enter trial values (between 0 and 1) for the smoothing constants. Step 2 In C26:C119, I compute the updated series level with formula 1 by copying from C26 to C27:C119 the formula alp*(B26/G14)+(1–alp)*(C25*D25). 494 Microsoft Excel 2010: Data Analysis and Business Modeling Step 3 In D26:D119, I use formula 2 to update the series trend, copying from D26 to D27:D119 the formula bet*(C26/C25)+(1–bet)*D25. Step 4 In G26:G119, I use formula 3 to update the seasonal indexes, copying from G26 to G27:G119 the formula gam*(B26/C26)+(1–gam)*G14. Step 5 In E26:E119, I use formula 4 to compute the forecast for the current month by copying from E26 to E27:E119 the formula (C25*D25)*G14. Step 6 In F26:F119, I compute the absolute percentage error for each month by copying from F26 to F27:F119 the formula ABS(B26-E26)/B26. Step 7 I compute the average absolute percentage error for the years 1988 through 1996 in F21 with the formula AVERAGE(F26:F119). Step 8 Now I use Solver to determine smoothing parameter values that minimize the average absolute percentage error. The Solver Parameters dialog box is shown in Figure 59-2. FIGURE 59-2 So ver Parameters d a og box for W nters s mode . I used smoothing parameters (G11:I11) to minimize the average absolute percentage error (cell F21). The Solver ensures that you nd the best combination of smoothing constants. Smoothing constants must be between 0 and 1. Here, alp .50, bet .01, and gam .27 minimizes the average absolute percentage error. You might nd slightly different values for the smoothing constants, but you should obtain a mean absolute percentage error (MAPE) close to 7.3 percent. In this example, there are many combinations of the smoothing constants that give forecasts having approximately the same MAPE. Our one-month-ahead forecasts are off by an average of 7.3 percent. Chapter 59 Winters’s Method 495 Remarks ■ Instead of choosing smoothing parameters to optimize one-period forecast errors, you could, for example, choose to optimize the average absolute percentage error incurred in forecasting total housing starts for the next six months. ■ If at the end of month t you want to forecast sales for the next four quarters, you would simply add f t,1 +f t,2 +f t,3 +f t,4 . If you want, you could choose smoothing parameters to minimize the absolute percentage error incurred in estimating sales for the next year. Problems All the data for the following problems is in the le Quarterly.xlsx. 1. Use Winters’s method to forecast one-quarter-ahead revenues for Apple. 2. Use Winters’s method to forecast one-quarter-ahead revenues for Amazon.com. 3. Use Winters’s method to forecast one-quarter-ahead revenues for Home Depot. 4. Use Winters’s method to forecast total revenues for the next two quarters for Home Depot. [...]... random var ab e 5 18 Microsoft Excel 2010: Data Analysis and Business Modeling If equal numbers of people prefer Coke to Pepsi and Pepsi to Coke and I ask 100 people whether they prefer Coke to Pepsi, what is the probability that exactly 60 people prefer Coke to Pepsi and the probability that between 40 and 60 people prefer Coke to Pepsi? You have n 100 and p 0.5 You seek the probability that x 60 and. .. government support and Social Security checks), and that spring break reduces customer count Figure 61-5 also shows the improvement in the forecasting a ­ ccuracy The R squared value (RSQ) has improved to 87 percent and the standard error is reduced to 122 customers 506 Microsoft Excel 2010: Data Analysis and Business Modeling FIGURE 61-5  Forecast parameters and forecasts nc ud ng spr ng break and the first... five-card poker hand ■ Number of car accidents you have (hopefully zero!) in a year 509 510 Microsoft Excel 2010: Data Analysis and Business Modeling ■ Number of dots showing on a die ■ Number of free throws out of 12 that Phoenix Sun’s star Steve Nash makes during a basketball game What are the mean, variance, and standard deviation of a random variable? In Chapter 42, “Summarizing Data by Using Descriptive... function for a randomly chosen person’s IQ is shown in Figure 62-2 512 Microsoft Excel 2010: Data Analysis and Business Modeling FIGURE 62-2  Probab ty dens ty funct on for Qs A probability density function (pdf) has the following properties: ■ The value of the pdf is always greater than or equal to 0 ■ The area under the pdf equals 1 ■ The height of the density function for a value x of a random variable... account for the day of the week (5 business days minus 1) ■ Two to account for the types of paydays that occur each month ■ Two to account for whether a particular day follows or precedes a holiday Microsoft Excel 2010 allows only 15 independent variables, so it appears that you’re in trouble 501 502 Microsoft Excel 2010: Data Analysis and Business Modeling FIGURE 61-1  Data used to pred ct cred t un... of size n is drawn from the urn 520 Microsoft Excel 2010: Data Analysis and Business Modeling Let’s look at an example in the file Hypergeom.dist.xlsx, which is shown in Figure 63-3 The Excel 2010 formula HYPERGEOM.DIST(x,n,s,N,0) gives the probability of x successes if n balls are drawn from an urn containing N balls, of which s are marked as “success.” The Excel 2010 formula HYPERGEOM.DIST(x,n,s,N,1)... Copying from cell G25 to G26:G 28 the formula VLOOKUP(D25,season,3)*G25 computes the final forecast for Quarters 21–24 500 Microsoft Excel 2010: Data Analysis and Business Modeling If you think the trend of the series has changed recently, you can estimate the series’ ­ trend based on more recent data For example, you could use the centered ­ oving m averages for Quarters 13– 18 to get a more recent trend... approximately half the time Therefore, a commonly used test to evaluate the randomness of forecast errors is to look at the number of sign changes in the errors If you have n observations, nonrandomness of the errors is indicated if you find either fewer than n–1 – 2 n or more than n–1 + 2 n 5 08 Microsoft Excel 2010: Data Analysis and Business Modeling changes in sign In the Christmas week worksheet, as shown... seasonal indexes helps you better understand a company’s sales pattern The quarterly seasonal indexes for Walmart revenues are as follows: ■ Quarter 1 (January through March): 90 ■ Quarter 2 (April through June): 98 ■ Quarter 3 (July through September): 96 ■ Quarter 4 (October through December): 1.16 497 4 98 Microsoft Excel 2010: Data Analysis and Business Modeling These indexes imply, for example,... 10 defective CD drives? 522 Microsoft Excel 2010: Data Analysis and Business Modeling 2 Using the airline overbooking data: ❑ Determine how the probability of overbooking varies as the number of tickets sold varies from 100 through 115 Hint: Use a one-way data table ❑ Show how the probability of overbooking varies as the number of tickets sold varies from 100 through 115, and the probability that a . Lines and Markers.) The time series plot is shown in Figure 58- 2. FIGURE 58- 1 Quarter y revenues for Amazon sa es. 488 Microsoft Excel 2010: Data Analysis and Business Modeling FIGURE 58- 2 T. a holiday Microsoft Excel 2010 allows only 15 independent variables, so it appears that you’re in trouble. 502 Microsoft Excel 2010: Data Analysis and Business Modeling FIGURE 61-1 Data used. percent and the standard error is reduced to 122 customers. 506 Microsoft Excel 2010: Data Analysis and Business Modeling FIGURE 61-5 Forecast parameters and forecasts nc ud ng spr ng break and