Report

Forecasting Chapter 15 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-1 Chapter Topics ■ Forecasting Components ■ Time Series Methods ■ Forecast Accuracy ■ Time Series Forecasting Using Excel ■ Time Series Forecasting Using QM for Windows ■ Regression Methods Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-2 Forecasting Components ■ A variety of forecasting methods are available for use depending on the time frame of the forecast and the existence of patterns. ■ Time Frames: Short-range (one to two months) Medium-range (two months to one or two years) Long-range (more than one or two years) ■ Patterns: Trend Random variations Cycles Seasonal pattern Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-3 Forecasting Components Patterns (1 of 2) Trend - A long-term movement of the item being forecast. Random variations - movements that are not predictable and follow no pattern. Cycle - A movement, up or down, that repeats itself over a lengthy time span. Seasonal pattern - Oscillating movement in demand that occurs periodically in the short run. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-4 Forecasting Components Patterns (2 of 2) Figure 15.1 (a) Trend; (b) Cycle; (c) Seasonal; (d) Trend with Season Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-5 Forecasting Components Forecasting Methods 1. Times Series - Statistical techniques that use historical data to predict future behavior. 2. Regression Methods - Regression (or causal ) methods that attempt to develop a mathematical relationship between the item being forecast and factors that cause it to behave the way it does. 3. Qualitative Methods - Methods using judgment, expertise and opinion to make forecasts. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-6 Forecasting Components Qualitative Methods “Jury of executive opinion,” a qualitative technique, is the most common type of forecast for long-term strategic planning. Performed by individuals or groups within an organization, sometimes assisted by consultants and other experts, whose judgments and opinions are considered valid for the forecasting issue. Usually includes specialty functions such as marketing, engineering, purchasing, etc., in which individuals have experience and knowledge of the forecasted item. Supporting techniques include the Delphi Method, market research, surveys, and technological forecasting. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-7 Time Series Methods Overview Statistical techniques that make use of historical data collected over a long period of time. Methods assume that what has occurred in the past will continue to occur in the future. Forecasts based on only one factor - time. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-8 Time Series Methods Moving Average (1 of 6) Moving average uses values from the recent past to develop forecasts. This dampens random increases and decreases. Useful for forecasting relatively stable items that do not display any trend or seasonal pattern. Formula for moving average (MA): n Di MAn i1n where: n number of periods in the moving average D data in period i i Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-9 Time Series Methods Moving Average (2 of 6) Example: Instant Paper Clip Supply Company wants to forecast orders for the month of November. Develop three-month and five-month moving averages using the data. Table 15.1 Orders for 10-month period Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-10 Time Series Methods Moving Average (3 of 6) Example: Instant Paper Clip Supply Company wants to forecast orders for the month of November. Three-month moving average: 3 Di MA i1 90 110 130 110 orders 3 3 3 Five-month moving average: 5 Di MA i1 90 110 130 75 50 91 orders 5 5 5 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-11 Time Series Methods Moving Average (4 of 6) Table 15.1 Three- and 5-month moving averages Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-12 Time Series Methods Moving Average (5 of 6) Figure 15.2 Three- and 5-month moving averages Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-13 Time Series Methods Moving Average (6 of 6) Longer-period moving averages react more slowly to changes in demand than do shorter-period moving averages. The appropriate number of periods to use often requires trial-anderror experimentation. A moving average does not react well to changes (trends, seasonal effects, etc.) but is easy to use and inexpensive. Good for short-term forecasting. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-14 Time Series Methods Weighted Moving Average In a weighted moving average, weights are assigned to the most recent data. n WMA W D n i1 i i where W the weight for period i, between 0% and 100% i Wi 1.00 Example: Paper clip company weights 50% for October, 33% for September, 17% for August: 3 WMA W D (.50)(90) (.33)(110) (.17)(130) 103.4 orders i i 3 i1 Determining precise weights and the number of periods requires trial-and-error experimentation. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-15 Time Series Methods Exponential Smoothing (1 of 11) Exponential smoothing weights recent past data more strongly than more distant data. Two forms: simple exponential smoothing and adjusted exponential smoothing. Simple exponential smoothing: Ft + 1 = Dt + (1 - )Ft where: Ft + 1 = the forecast for the next period Dt = actual demand in the present period Ft = the previously determined forecast for the present period = a weighting factor (smoothing constant). Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-16 Time Series Methods Exponential Smoothing (2 of 11) The most commonly used values of are between 0.10 and 0.50. Determination of is usually judgmental and subjective and often based on trial-and -error experimentation. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-17 Time Series Methods Exponential Smoothing (3 of 11) Example: PM Computer Services (see Table 15.4). Exponential smoothing forecasts using smoothing constant of .30. Forecast for period 2 (February): F2 = D1 + (1- )F1 = (.30)(.37) + (1-.30)(.37) = 37 units Forecast for period 3 (March): F3 = D2 + (1- )F2 = (.30)(.40) + (1-.30)(37) = 37.9 units Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-18 Time Series Methods Exponential Smoothing (4 of 11) Table 15.4 Exponential smoothing forecasts, = .30 and = .50 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-19 Time Series Methods Exponential Smoothing (5 of 11) The forecast that uses the higher smoothing constant (.50) reacts more strongly to changes in demand than does the forecast with the lower constant (.30). Both forecasts lag behind actual demand. Both forecasts tend to be consistently lower than actual demand. Low smoothing constants are appropriate for stable data without trend; higher constants appropriate for data with trends. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-20 Time Series Methods Exponential Smoothing (6 of 11) Figure 15.3 Exponential smoothing forecasts Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-21 Time Series Methods Exponential Smoothing (7 of 11) ■ Adjusted exponential smoothing: exponential smoothing with a trend adjustment factor added. Formula where: AFt + 1 = Ft + 1 + Tt+1 T = an exponentially smoothed trend factor Tt + 1 + (Ft + 1 - Ft) + (1 - )Tt Tt = the last period trend factor = smoothing constant for trend ( a value between zero and one). ■ Reflects the weight given to the most recent trend data. ■ Determined subjectively. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-22 Time Series Methods Exponential Smoothing (8 of 11) Example: PM Computer Services exponentially smoothed forecasts with = .50 and = .30 (see Table 15.5 next slide). Adjusted forecast for period 3: T3 = (F3 - F2) + (1 - )T2 = (.30)(38.5 - 37.0) + (.70)(0) = 0.45 AF3 = F3 + T3 = 38.5 + 0.45 = 38.95 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-23 Time Series Methods Exponential Smoothing (9 of 11) Table 15.5 Adjusted exponentially smoothed forecast values Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-24 Time Series Methods Exponential Smoothing (10 of 11) ■ The adjusted forecast is consistently higher than the simple exponentially smoothed forecast. ■ It is more reflective of the generally increasing trend of the data. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-25 Time Series Methods Exponential Smoothing (11 of 11) Figure 15.4 Adjusted exponentially smoothed forecast Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-26 Time Series Methods Linear Trend Line (1 of 5) ■ When demand displays an obvious trend over time, a least squares regression line , or linear trend line, can be used to forecast. ■ Formula: y a bx where: a intercept (at period 0) b slope of the line x the time period y forecast for demand for period x Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall b xy nxy x2 nx a y bx where: n number of periods x nx y ny 15-27 Time Series Methods Linear Trend Line (2 of 5) Example: PM Computer Services (see Table 15.6) x 78 6.5 12 y 557 46.42 12 1.72 b xy2 nxy2 3,867 (12)(6.5)(46.42) 2 650 126.5 x nx a y bx 46.42 (1.72)(6.5) 35.2 y 35.2 1.72x linear trend line for period 13, x 13, y 35.2 1.72(13) 57.56 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-28 Time Series Methods Linear Trend Line (3 of 5) Table 15.6 Least squares calculations Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-29 Time Series Methods Linear Trend Line (4 of 5) ■ A trend line does not adjust to a change in the trend as does the exponential smoothing method. ■ This limits its use to shorter time frames in which the trend will not change. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-30 Time Series Methods Linear Trend (5 of 5) Figure 15.5 Linear trend line Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-31 Time Series Methods Seasonal Adjustments (1 of 4) ■ A seasonal pattern is a repetitive up-and-down movement in demand. ■ Seasonal patterns can occur on a quarterly, monthly, weekly, or daily basis. ■ A seasonally adjusted forecast can be developed by multiplying the normal forecast by a seasonal factor. ■ A seasonal factor can be determined by dividing the actual demand for each seasonal period by total annual demand: Si =Di/D Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-32 Time Series Methods Seasonal Adjustments (2 of 4) ■ Seasonal factors lie between zero and one and represent the portion of total annual demand assigned to each season. ■ Seasonal factors are multiplied by annual demand to provide adjusted forecasts for each period. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-33 Time Series Methods Seasonal Adjustments (3 of 4) Example: Wishbone Farms Table 15.7 Demand for turkeys at Wishbone Farms S1 = D1/ D = 42.0/148.7 = 0.28 S2 = D2/ D = 29.5/148.7 = 0.20 S3 = D3/ D = 21.9/148.7 = 0.15 S4 = D4/ D = 55.3/148.7 = 0.37 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-34 Time Series Methods Seasonal Adjustments (4 of 4) Multiply forecasted demand for an entire year by seasonal factors to determine the quarterly demand. Forecast for entire year (trend line for data in Table 15.7): y = 40.97 + 4.30x = 40.97 + 4.30(4) = 58.17 Seasonally adjusted forecasts: SF1 = (S1)(F5) = (.28)(58.17) = 16.28 SF2 = (S2)(F5) = (.20)(58.17) = 11.63 SF3 = (S3)(F5) = (.15)(58.17) = 8.73 SF4 = (S4)(F5) = (.37)(58.17) = 21.53 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-35 Forecast Accuracy Overview Forecasts will always deviate from actual values. Difference between forecasts and actual values are referred to as forecast error. We would like forecast error to be as small as possible. If forecast error is large, either the technique being used is the wrong one, or the parameters need adjusting. Measures of forecast errors: Mean Absolute deviation (MAD) Mean absolute percentage deviation (MAPD) Cumulative error (E bar) Average error, or bias (E) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-36 Forecast Accuracy Mean Absolute Deviation (1 of 7) MAD is the average absolute difference between the forecast and actual demand. The most popular and simplest-to-use measures of forecast error. Formula: Dt Ft MAD n where: t the period number D demand in period t t F the forecast for period t t n the total number of periods Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-37 Forecast Accuracy Mean Absolute Deviation (2 of 7) Example: PM Computer Services (see Table 15.8). Compare accuracies of different forecasts using MAD: Dt Ft 53.41 MAD 4.85 n 11 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-38 Forecast Accuracy Mean Absolute Deviation (3 of 7) Table 15.8 Computational values for MAD and error Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-39 Forecast Accuracy Mean Absolute Deviation (4 of 7) The lower the value of MAD relative to the magnitude of the data, the more accurate the forecast. When viewed alone, MAD is difficult to assess. MAD must be considered in light of magnitude of the data. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-40 Forecast Accuracy Mean Absolute Deviation (5 of 7) Can be used to compare the accuracy of different forecasting techniques working on the same set of demand data (PM Computer Services): Exponential smoothing ( = .50): MAD = 4.04 Adjusted exponential smoothing ( = .50, = .30): MAD = 3.81 Linear trend line: MAD = 2.29 The linear trend line has the lowest MAD; increasing from .30 to .50 improved the smoothed forecast. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-41 Forecast Accuracy Mean Absolute Deviation (6 of 7) A variation on MAD is the mean absolute percent deviation (MAPD). Measures the absolute error as a percentage of demand rather than per period. Eliminates the problem of interpreting the measure of accuracy relative to the magnitude of the demand and forecast values. Formula: Dt Ft 53.41 MAPD .103 or 10.3% 520 Dt Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-42 Forecast Accuracy Mean Absolute Deviation (7 of 7) MAPD for other three forecasts: Exponential smoothing ( = .50): MAPD = 8.5% Adjusted exponential smoothing ( = .50, = .30): MAPD = 8.1% Linear trend: MAPD = 4.9% Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-43 Forecast Accuracy Cumulative Error (1 of 2) Cumulative error is the sum of the forecast errors (E =et). A relatively large positive value indicates the forecast is biased low, a large negative value indicates the forecast is biased high. If the preponderance of errors are positive, the forecast is consistently low; and vice versa. The cumulative error for a trend line is always almost zero, and is therefore not a good measure for this method. The cumulative error for PM Computer Services can be read directly from Table 15.8. E = et = 49.31, indicating the forecasts are frequently below actual demand. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-44 Forecast Accuracy Cumulative Error (2 of 2) Cumulative error for other forecasts: Exponential smoothing ( = .50): E = 33.21 Adjusted exponential smoothing ( = .50, =.30): E = 21.14 Average error (bias) is the per-period average of cumulative error. Average error for the exponential smoothing forecast: et 49.31 E n 4.48 11 A large positive value of average error indicates a forecast is biased low; a large negative error indicates it is biased high. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-45 Forecast Accuracy Example Forecasts by Different Measures Table 15.9 Comparison of forecasts for PM Computer Services Results consistent for all forecasts: Larger value of alpha is preferable. Adjusted forecast is more accurate than exponential smoothing. Linear trend is more accurate than all the others. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-46 Time Series Forecasting Using Excel (1 of 4) =B3*B8+(1-B3)*C8 =C9+D9 =B9-E9 =ABS(B9-E9) =G21/11 =SUM(F9:F20) Exhibit 15.1 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-47 Time Series Forecasting Using Excel (2 of 4) To access this window, click on “Data” on the toolbar ribbon and then the “Data Analysis” add-in Exhibit 15.2 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-48 Time Series Forecasting Using Excel (3 of 4) Demand values = 0.5 Cells in which the forecasted values will be placed Exhibit 15.3 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-49 Time Series Forecasting Using Excel (4 of 4) Exhibit 15.4 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-50 Exponential Smoothing Forecast with Excel QM Click “Add-Ins” to access forecasting macros Input problem data in cells B7 and B10:B21 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 15.5 15-51 Time Series Forecasting Solution with QM for Windows (1 of 2) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 15.6 15-52 Time Series Forecasting Solution with QM for Windows (2 of 2) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 15.7 15-53 Regression Methods Overview Time series techniques relate a single variable being forecast to time. Regression is a forecasting technique that measures the relationship of one variable to one or more other variables. The simplest form of regression is linear regression. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-54 Regression Methods Linear Regression Linear regression relates demand (dependent variable ) to an independent variable. y a bx a y bx b xy nxy 2 2 x nx where: x nx mean of the x data y ny mean of the y data Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-55 Regression Methods Linear Regression Example (1 of 3) State University Athletic Department. Wins 4 6 6 8 6 7 5 7 Attendance 36,300 40,100 41,200 53,000 44,000 45,600 39,000 47,500 x (wins) 4 6 6 8 6 7 5 7 49 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall y (attendance, 1,000s) 36.3 40.1 41.2 53.0 44.0 45.6 39.0 47.5 346.7 xy 145.2 240.6 247.2 424.0 264.0 319.2 195.0 332.5 2,167.7 x2 16 36 36 64 36 49 25 49 311 15-56 Regression Methods Linear Regression Example (2 of 3) x 49 6.125 8 y 346.9 43.34 8 b xy nxy (2,167.70 (8)(6.125)(43.34) 4.06 2 (311) (8)(6.125)2 x2 nx a y bx 43.34 (.406)(6.125) 18.46 Therefore, y 18.46 4.06x Attendance forecast for x 7 wins is y 18.46 4.06(7) 46.88 or 46,880 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-57 Regression Methods Linear Regression Example (3 of 3) Figure 15.6 Linear regression line Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-58 Regression Methods Correlation (1 of 2) Correlation is a measure of the strength of the relationship between independent and dependent variables. Formula: r n xy x y 2 2 2 2 n x x n y y Value lies between +1 and -1. Value of zero indicates little or no relationship between variables. Values near 1.00 and -1.00 indicate a strong linear relationship. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-59 Regression Methods Correlation (2 of 2) Value for State University example: r (8)(2,167.7) (49)(346.7) .948 2 2 (8)(311) (49) (8)(15,224.7) (346.7) Since the value is close to one, we have evidence of a strong linear relationship. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-60 Regression Methods Coefficient of Determination The coefficient of determination is the percentage of the variation in the dependent variable that results from the independent variable. Computed by squaring the correlation coefficient, r. For the State University example: r = .948, r2 = .899 This value indicates that 89.9% of the amount of variation in attendance can be attributed to the number of wins by the team, with the remaining 10.1% due to other, unexplained, factors. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-61 Regression Analysis with Excel (1 of 6) =INTERCEPT(B5:B12,A5:A12) =CORREL(B5:B12,A5:A12) Exhibit 15.8 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-62 Regression Analysis with Excel (2 of 6) Click on “Insert” to access “Charts” Click on “Scatter” Exhibit 15.9 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-63 Regression Analysis with Excel (3 of 6) Exhibit 15.10 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-64 Regression Analysis with Excel (4 of 6) Exhibit 15.11 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-65 Regression Analysis with Excel (5 of 6) Exhibit 15.12 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-66 Regression Analysis with QM for Windows (6 of 6) Exhibit 15.13 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-67 Multiple Regression with Excel (1 of 4) Multiple regression relates demand to two or more independent variables. General form: y = 0 + 1x1 + 2x2 + . . . + kxk where 0 = the intercept 1 . . . k = parameters representing contributions of the independent variables x1 . . . xk = independent variables Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-68 Multiple Regression with Excel (2 of 4) State University example revisited; does the addition of promotional and advertising expenditures to wins improve the prediction of attendance? Wins Promotion ($) Attendance 4 29,500 36,300 6 55,700 40,100 6 71,300 41,200 8 87,000 53,000 6 75,000 44,000 7 72,000 45.600 5 55,300 39,000 7 81,600 47,500 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-69 Multiple Regression with Excel (3 of 4) r2, the coefficient of determination Regression equation coefficients for x1 and x2 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 15.14 15-70 Multiple Regression with Excel (4 of 4) Includes x1 and x2 columns Exhibit 15.15 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-71 Example Problem Solution Computer Software Firm (1 of 4) Problem Statement: For the data below, develop an exponential smoothing forecast using = .40, and an adjusted exponential smoothing forecast using = .40 and = .20. Compare the accuracy of the forecasts using MAD and cumulative error. Period 1 2 3 4 5 6 7 8 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Units 56 61 55 70 66 65 72 75 15-72 Example Problem Solution Computer Software Firm (2 of 4) Step 1: Compute the Exponential Smoothing Forecast. Ft+1 = Dt + (1 - )Ft Step 2: Compute the Adjusted Exponential Smoothing Forecast AFt+1 = Ft +1 + Tt+1 Tt+1 = (Ft +1 - Ft) + (1 - )Tt Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-73 Example Problem Solution Computer Software Firm (3 of 4) Period 1 2 3 4 5 6 7 8 9 Dt 56 61 55 70 66 65 72 75 Ft 56.00 58.00 56.80 62.08 63.65 64.18 67.31 70.39 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall AFt 56.00 58.40 56.88 63.20 64.86 65.26 68.80 72.19 Dt - Ft Dt - AFt 5.00 5.00 -3.00 -3.40 13.20 13.12 3.92 2.80 1.35 0.14 7.81 6.73 7.68 6.20 35.97 30.60 15-74 Example Problem Solution Computer Software Firm (4 of 4) Step 3: Compute the MAD Values Dt Ft 41.97 MAD(Ft ) 5.99 n 7 Dt AFt 37.39 MAD( AFt ) 5.34 n 7 Step 4: Compute the Cumulative Error. E(Ft) = 35.97 E(AFt) = 30.60 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-75 Example Problem Solution Building Products Store (1 of 5) For the following data: Develop a linear regression model Determine the strength of the linear relationship using correlation. Determine a forecast for lumber given 10 building permits in the next quarter. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-76 Example Problem Solution Building Products Store (2 of 5) Quarter Building Permits, x Lumber Sales (1,000s of board ft), y 1 2 3 4 5 6 7 8 9 10 8 12 7 9 15 6 5 8 10 12 12.6 16.3 9.3 11.5 18.1 7.6 6.2 14.2 15.0 17.8 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-77 Example Problem Solution Building Products Store (3 of 5) Step 1: Compute the Components of the Linear Regression Equation. x 92 9.2 10 y 128.6 12.86 10 b xy nxy (1,290.3) (10)(9.2)(12.86) 2 2 2 (932) (10)(9.2) x nx 1.25 a y bx 12.86 (1.25)(9.2) 1.36 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-78 Example Problem Solution Building Products Store (4 of 5) Step 2: Develop the Linear regression equation. y = a + bx, y = 1.36 + 1.25x Step 3: Compute the Correlation Coefficient. r n xy x y 2 2 2 2 n x x n y y r (10)(1,290.3) (92)(128.6) 2 2 (10)(932) (92) (10)(1,810.48)(128.6) .925 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-79 Example Problem Solution Building Products Store (5 of 5) Step 4: Calculate the forecast for x = 10 permits. Y = a + bx = 1.36 + 1.25(10) = 13.86 or 1,386 board ft Copyright © 2012 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-80 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall 15-81