商务与经济统计习题答案第8版中文版SBE8SM18

1、forecastingchapter 18forecastinglearning objectives1.understand that the long-run success of an organization is often closely related to how well management is able to predict future aspects of the operation. 2.know the various components of a time series.3.be able to use smoothing techniques such a

2、s moving averages and exponential smoothing.4.be able to use the least squares method to identify the trend component of a time series.5.understand how the classical time series model can be used to explain the pattern or behavior of the data in a time series and to develop a forecast for the time s

3、eries.6.be able to determine and use seasonal indexes for a time series.7.know how regression models can be used in forecasting.8.know the definition of the following terms:time seriesmean squared errorforecastmoving averagestrend componentweighted moving averagescyclical componentsmoothing constant

4、seasonal componentseasonal constantirregular componentsolutions:1.a.weektime-seriesvalue forecast forecast error (error)21 82133154171252551615116 916-749 75forecast for week 7 is (17 + 16 + 9 ) / 3 = 14b.mse = 75 / 3 = 25c.smoothing constant = .3.week ttime-series valueytforecast ftforecast error y

5、t - ft squared error (yt - ft)218213 8.005.0025.00315 9.006.0036.0041710.206.8046.2451611.564.4419.716912.45-3.4511.90 138.85138.85forecast for week 7 is .2(9) + .8(12.45) = 11.76d.for the a = .2 exponential smoothing forecast mse = 138.85 / 5 = 27.77. since the three-week moving average has a small

6、er mse, it appears to provide the better forecasts.e.smoothing constant = .4.week ttime-series valueytforecast ftforecast error yt - ft squared error (yt - ft)218213 8.0 5.0 25.0031510.0 5.0 25.0041712.0 5.0 25.0051614.0 2.0 4.006914.8 -5.833.64112.64mse = 112.64 / 5 = 22.53. a smoothing constant of

7、 .4 appears to provide better forecasts.forecast for week 7 is .4(9) + .6(14.8) = 12.482.a.weektime-series value4-week moving average forecast(error)25-week moving average forecast(error)211722131942351820.004.0061620.2518.0619.6012.9672019.001.0019.400.3681819.251.5619.201.4492218.0016.0019.009.001

8、02019.001.0018.801.44111520.0025.0019.2017.64122218.7510.5619.00 9.0077.1851.84b.mse(4-week) = 77.18 / 8 = 9.65mse(5-week) = 51.84 / 7 = 7.41c.for the limited data provided, the 5-week moving average provides the smallest mse.3.a.weektime-seriesvalueweighted moving average forecastforecasterror(erro

9、r)211722131942319.333.6713.4751821.33-3.3311.0961619.83-3.8314.6772017.832.174.7181818.33-0.330.1192218.333.6713.47102020.33-0.330.11111520.33-5.3328.41122217.834.17 17.39103.43b.mse = 103.43 / 9 = 11.49prefer the unweighted moving average here.c.you could always find a weighted moving average at le

10、ast as good as the unweighted one. actually the unweighted moving average is a special case of the weighted ones where the weights are equal. 4.weektime-series valueforecasterror(error)211722117.004.0016.0031917.401.602.5642317.565.4429.5951818.10-0.100.0161618.09-2.094.3772017.882.124.4981818.10-0.

11、100.0192218.093.9115.29102018.481.522.31111518.63-3.6313.18122218.273.73 13.91 101.72101.72mse = 101.72 / 11 = 9.25a = .2 provided a lower mse; therefore a = .2 is better than a = .15.a.f13 = .2y12 + .16y11 + .64(.2y10 + .8f10) = .2y12 + .16y11 + .128y10 + .512f10f13 = .2y12 + .16y11 + .128y10 + .51

12、2(.2y9 + .8f9) = .2y12 + .16y11 + .128y10 + .1024y9 + .4096f9f13 = .2y12 + .16y11 + .128y10 + .1024y9 + .4096(.2y8 + .8f8) = .2y12 + .16y11 + .128y10 + .1024y9 + .08192y8 + .32768f8b.the more recent data receives the greater weight or importance in determining the forecast. the moving averages metho

13、d weights the last n data values equally in determining the forecast.6.a.monthyt3-month moving averages forecast(error)2a = 2forecast(error)218028280.004.0038480.4012.9648382.00 1.0081.123.5358383.00 0.0081.502.2568483.33 0.4581.804.8478583.33 2.7982.247.6288484.00 0.0082.791.4698284.33 5.4383.031.0

14、6108383.67 0.4582.830.03118483.00 1.0082.861.30128383.00 0.0083.09 0.0111.1239.06mse(3-month) = 11.12 / 9 = 1.24mse(a = .2) = 39.06 / 11 = 3.55use 3-month moving averages.b.(83 + 84 + 83) / 3 = 83.37.a.monthtime-series value3-month moving average forecast(error)24-month moving average forecast(error

15、)219.529.339.449.69.400.0459.89.430.149.450.1269.79.600.019.530.0379.89.700.019.630.03810.59.770.539.730.5999.910.000.019.950.00109.710.070.149.980.08119.610.030.189.970.14129.69.730.029.920.101.081.09mse(3-month) = 1.08 / 9 = .12mse(4-month) = 1.09 / 8 = .14use 3-month moving averages.b. forecast =

16、 (9.7 + 9.6 + 9.6) / 3 = 9.63c. for the limited data provided, the 5-week moving average provides the smallest mse.8.a.monthtime-series value3-month moving average forecast(error)2a = .2forecast(error)212402350240.0012100.003230262.001024.004260273.33177.69255.6019.365280280.000.00256.48553.19632025

17、6.674010.69261.183459.797220286.674444.89272.952803.708310273.331344.69262.362269.579240283.331877.49271.891016.9710310256.672844.09265.511979.3611240286.672178.09274.411184.0512230263.331110.89267.53 1408.5017,988.5227,818.49mse(3-month) = 17,988.52 / 9 = 1998.72mse(a = .2) = 27,818.49 / 11 = 2528.

18、95based on the above mse values, the 3-month moving averages appears better. however, exponential smoothing was penalized by including month 2 which was difficult for any method to forecast. using only the errors for months 4 to 12, the mse for exponential smoothing is revised tomse(a = .2) = 14,694

19、.49 / 9 = 1632.72thus, exponential smoothing was better considering months 4 to 12.b. using exponential smoothing,f13 = ay12 + (1 - a)f12 = .20(230) + .80(267.53) = 260 9.a.smoothing constant = .3.month ttime-series valueytforecast ftforecast error yt - ft squared error (yt - ft)21105.02135.0105.003

20、0.00900.003120.0114.006.0036.004105.0115.80-10.80116.64590.0112.56-22.56508.956120.0105.7914.21201.927145.0110.0534.951221.508140.0120.5419.46378.699100.0126.38-26.38695.901080.0118.46-38.461479.1711100.0106.92-6.9247.8912110.0104.855.15 26.52 total5613.18mse = 5613.18 / 11 = 510.29forecast for mont

21、h 13: f13 = .3(110) + .7(104.85) = 106.4b.smoothing constant = .5month ttime-series valueytforecast ftforecast error yt - ftsquared error (yt - ft)211052135105 30.00 900.003120.5(135) + .5(105) = 120 0.00 0.004105.5(120) + .5(120) = 120-15.00 225.005 90.5(105) + .5(120) = 112.50-22.50 506.256120.5(9

22、0) + .5(112.5) = 101.25 18.75 351.567145.5(120) + .5(101.25) =110.63 34.371181.308140.5(145) + .5(110.63) = 127.81 12.19 148.609100.5(140) + .5(127.81) = 133.91-33.911149.8910 80.5(100) + .5(133.91) = 116.95-36.951365.3011100.5(80) + .5(116.95) = 98.48 1.52 2.3112110.5(100) + .5(98.48) = 99.24 10.76

23、 115.78 5945.99mse = 5945.99 / 11 = 540.55forecast for month 13: f13 = .5(110) + .5(99.24) = 104.62conclusion: a smoothing constant of .3 is better than a smoothing constant of .5 since the mse is less for 0.3.10.a/b.weektime-series valuea = .2forecast(error)2a = .3forecast(error)217.3527.407.35.002

24、57.35.002537.557.36.03617.36.036147.567.40.02567.42.019657.607.43.02897.46.019667.527.46.00367.50.000477.527.48.00167.51.000187.707.48.04847.51.036197.627.53.00817.57.0025107.557.55.00007.58.0009.1548.1178c. mse(a = .2) = .1548 / 9 = .0172mse(a = .3) = .1178 / 9 = .0131use a = .3.f11 = .3y10 + .7f10

25、 = .3(7.55) + .7(7.58) = 7.5711.a.methodforecastmse3-quarter80.732.534-quarter80.552.81the 3-quarter moving average forecast is better because it has the smallest mse.b.methodforecastmsea = .480.402.40a = .580.572.01the a = .5 smoothing constant is better because it has the smallest mse.c.the a = .5

26、 is better because it has the smallest mse.12.the following values are needed to compute the slope and intercept:tt = 4.7 + 2.1tforecast: t6 = 4.7 + 2.1(6) = 17.313.the following values are needed to compute the slope and intercept:computation of slope:computation of intercept:equation for linear tr

27、end: tt = 207.467 - 3.514tforecast: t6 = 207.467 - 3.514(7) = 182.8714.the following values are needed to compute the slope and intercept:computation of slope:computation of intercept:equation for linear trend: tt = 20.7466 - 0.3514tconclusion: enrollment appears to be decreasing by an average of ap

28、proximately 351 students per year.15.the following values are needed to compute the slope and intercept:computation of slope:computation of intercept:equation for linear trend: tt = 28,800 + 421.429 t16.a linear trend model is not appropriate. a nonlinear model would provide a better approximation.1

29、7.a. a linear trend appears to be reasonable.b. the following values are needed to compute the slope and intercept:computation of slope:computation of intercept:equation for linear trend: tt = 19.993 + 1.774 tconclusion: the firm has been realizing an average cost increase of $1.77 per unit per year

30、.18.a. the following values are needed to compute the slope and intercept:computation of slope:computation of intercept:equation for linear trend: tt = .365 + .193 tforecast: tt = .365 + .193(11) = $2.49b. over the past ten years the earnings per share have been increasing at the average rate of $.1

31、93 per year. although this is a positive indicator of walgreens performance. more information would be necessary to conclude “good investment.”19.a. the following values are needed to compute the slope and intercept:computation of slope:computation of intercept:7.5833 - 0.0514(3.5) = 7.4033equation

32、for linear trend: tt = 7.4033 + 0.0514 tthe number of applications is increasing by approximately 1630 per year.b. 1996: tt = 7.4033 + 0.0514(7) = 7.7633 or about 7.76%1997: tt = 7.4033 + 0.0514(8) = 7.8148 or about 7.81%20.a. the following values are needed to compute the slope and intercept:comput

33、ation of slope:computation of intercept:4184.1 - 397.545(5.5) = 1997.6equation for linear trend: tt = 1997.6 + 397.545 tb. t11 = 1997.6 + 397.545(11) = 6371 t12 = 1997.6 + 397.545(12) = 676821.a.the following values are needed to compute the slope and intercept:computation of slope:computation of in

34、tercept:(118.2/6) - 7.7714(21/6) = -7.5equation for linear trend: tt = -7.5 + 7.7714tb.7.7714 ($m) per yearc.1998 forecast: t8 = -7.5 + 7.7714 (7) = 46.922. a.yearquarterytfour-quartermoving averagecenteredmoving average114223.50333.7504.00454.1254.252164.5004.75235.0005.25355.3755.50475.8756.253176

35、.3756.50266.6256.753648 b.yearquarterytcenteredmoving averageseasonal-irregularcomponent11422333.7500.8000 454.1251.2121 2164.5001.3333 235.0000.6000 355.3750.9302 475.8751.1915 3176.3751.0980 266.6250.9057 3648quarterseasonal-irregularcomponent valuesseasonal indexadjusted seasonal index11.3333,1.0

36、9801.21571.20502.60000,.90570.75290.74633.80000,.90320.86510.867541.2121,1.19151.20181.19124.0355note: adjustment for seasonal index = 4.000 / 4.0355 = 0.991223.a.four quarter moving averages beginning with(1690 + 940 + 2625 + 2500) / 4 = 1938.75other moving averages are 1966.252002.501956.252052.50

37、2025.002060.001990.002123.75b.quarterseasonal-irregularcomponent valuesseasonal indexadjusted seasonal index10.9040.9000.90200.90020.4480.5260.49700.48631.3441.4531.39851.39641.2751.1641.21951.217 4.0070note: adjustment for seasonal index = 4.000 / 4.007 = 0.9983 c. the largest seasonal effect is in

38、 the third quarter which corresponds to the back-to-school demand during july, august, and september of each year. 24.monthseasonal-irregularcomponent valuesseasonal indexadjusted seasonal index10.720.700.710.70720.800.750.780.77730.830.820.830.82740.940.990.970.96651.011.021.021.01661.251.361.311.3

39、0571.491.511.501.49481.191.261.231.22590.980.970.980.976100.981.000.990.986110.930.940.940.936120.780.80 0.790.787 12.05notes:1. adjustment for seasonal index = 12 / 12.05 = 0.9962. the adjustment is really not necessary in this problem since it implies more accuracy than is warranted. that is, the

40、seasonal component values and the seasonal index were rounded to two decimal places.25.a.use a twelve period moving averages. after centering the moving averages, you should obtain the following seasonal indexes:hour seasonal indexhourseasonal index10.77171.20720.86480.99430.95490.85041.392100.64751

41、.571110.57961.667120.504 b.the hours of july 18 are number 37 to 48 in the time series. thus the trend component for 7:00 a.m. on july 18 (period 37) would bet37 = 32.983 + .3922(37) = 47.49a summary of the trend components for the twelve hours on july 18 is as follows:hourtrend componenthourtrend c

42、omponent147.49749.85247.89850.24348.28950.63448.671051.02549.061151.42649.461251.81c.multiply the trend component in part b by the seasonal indexes in part a to obtain the twelve hourly forecasts for july 18. for example, 47.49 x (.771) = 36.6 or rounded to 37, would be the forecast for 7:00 a.m. on

43、 july 18th.the seasonally adjusted hourly forecasts for july 18 are as follows:hourforecasthourforecast13776024185034694346810335771130682122626.a.yes, there is a seasonal effect over the 24 hour period.time periodseasonal index12 - 4 a.m.1.6964 - 8 a.m.1.4588 - 120.71112 - 4 p.m.0.3264 - 8 p.m.0.44

44、88 - 121.362b.time periodforecast12 - 4 p.m.166,761.134 - 8 p.m.146,052.9927.a.monthtime-series value3-month moving average forecastforecast error(error)2134.8750235.6250334.6875433.562535.0625-1.5002.2500532.625034.6250-2.0004.0000634.000033.62500.37500.1406733.625033.39580.22920.0525835.062533.41671.64582.7088934.062534.2292-0.16670.02781034.125034.2500-0.12500.01561133.250034.4167-1.16671.36111232.062533.8125-1.75003.0625note: mse = 13.6189/9 = 1.51forecast for december is (34.1250 + 33.2500 + 32.0625) / 3 = 33.1458b.the weighted moving average forecasts