Chapter 21
21.1 Time series Moving average
48
41 (48+41+37)/3 = 42.00
37 (41+37+32)/3 = 36.67
32 (37+32+36)/3 = 35.00
36 (32+36+31)/3 = 33.00
31 (36+31+43)/3 = 36.67
43 (31+43+52)/3 = 42.00
52 (43+52+60)/3 = 51.67
60 (52+60+48)/3 = 53.33
48 (60+48+41)/3 = 49.67
41 (48+41+30)/3 = 39.67
30
21.3
21.5 Time series Moving average
16
22
19 (16+22+19+24+30)/5 = 22.2
24 (22+19+24+30+26)/5 = 24.2
30 (19+24+30+26+24)/5 = 24.6
26 (24+30+26+24+29)/5 = 26.6
24 (30+26+24+29+21)/5 = 26.0
29 (26+24+29+21+23)/5 = 24.6
21 (24+29+21+23+19)/5 = 23.2
23 (29+21+23+19+15)/5 = 21.4
19
15
21.7 Time series Exponentially smoothed time series
12 12
18 .1(18) +. 9(12) = 12.60
16 .1(16) +. 9(12.60) = 12.94
24 .1(24) +. 9(12.94) = 14.05
17 .1(17) +. 9(14.05) = 14.34
16 .1(16) +. 9(14.34) = 14.51
25 .1(25) +. 9(14.51) = 15.56
21 .1(21) +. 9(15.56) = 16.10
23 .1(23) + .9(16.10) = 16.79
14 .1(14) + .9(16.79) = 16.51
21.9
There appears to be a gradual upward trend.
20.11 Time series Exponentially smoothed time series
38 38
43 .8(43) +. 2(38) = 42.00
42 .8(42) +. 2(42.00) = 42.00
45 .8(45) +. 2(42.00) = 44.40
46 .8(46) +. 2(44.40) = 45.68
48 .8(48) +. 2(45.68) = 47.54
50 .8(50) +. 2(47.54) = 49.51
49 .8(49) +. 2(49.51) = 49.10
46 .8(46) + .2(49.10) = 46.62
45 .8(45) + .2(46.62) = 45.32
21.13 & 21.14
Sales 3-Day moving average 5-Day moving average
43
45 (43+45+22)/3 = 36.67
22 (45+22+25)/3 = 30.67 (43+45+22+25+31)/5 = 33.2
25 (22+25+31)/3 = 26.00 (45+22+25+31+51)/5 = 34.8
31 (25+31+51)/3 = 35.67 (22+25+31+51+41)/5 = 34.0
51 (31+51+41)/3 = 41.00 (25+31+51+41+37)/5 = 37.0
41 (51+41+37)/3 = 43.00 (31+51+41+37+22)/5 = 36.4
37 (41+37+22)/3 = 33.33 (51+41+37+22+25)/5 = 35.2
22 (37+22+25)/3 = 28.00 (41+37+22+25+40)/5 = 33.0
25 (22+25+40)/3 = 29.00 (37+22+25+40+57)/5 = 36.2
40 (25+40+57)/3 = 40.67 (22+25+40+57+30)/5 = 34.8
57 (40+57+30)/3 = 42.33 (25+40+57+30+33)/5 = 37.0
30 (57+30+33)/3 = 40.00 (40+57+30+33+37)/5 = 39.4
33 (30+33+37)/3 = 33.33 (57+30+33+37+64)/5 = 44.2
37 (33+37+64)/3 = 44.67 (30+33+37+64+58)/5 = 44.4
64 (37+64+58)/3 = 53.00 (33+37+64+58+33)/5 = 45.0
58 (64+58+33)/3 = 51.67 (37+64+58+33+38)/5 = 46.0
33 (58+33+38)/3 = 43.00 (64+58+33+38+25)/5 = 43.6
38 (33+38+25)/3 = 32.00
25
c There appears to be a seasonal (weekly) pattern.
21.15a Sales 4-quarter moving average Centered moving average
18
22
(18+22+27+31)/4 = 24.50
27 (24.50+28.25)/2 = 26.375
(22+27+31+33)/4 = 28.25
31 (28.25+27.75)/2 = 28.000
(27+31+33+20)/4 = 27.75
33 (27.75+30.50)/2 = 29.125
(31+33+20+38)/4 = 30.50
20 (30.50+29.25)/2 = 29.875
(33+20+38+26)/4 = 29.25
38 (29.25+27.25)/2 = 28.250
(20+38+26+25)/4 = 27.25
26 (27.25+31.25)/2 = 29.250
(38+26+25+36)/4 = 31.25
25 (31.25+32.75)/2 = 32.000
(26+25+36+44)/4 = 32.75
36 (32.75+33.50)/2 = 33.125
(25+36+44+29)/4 = 33.50
44 (33.50+37.50)/2 = 35.500
(36+44+29+41)/4 = 37.50
29 (37.50+36.75)/2 = 37.125
(44+29+41+33)/4 = 36.75
41 (36.75+38.75)/2 = 37.750
(29+41+33+52)/4 = 38.75
33 (38.75+42.75)/2 = 40.750
(41+33+52+25)/4 = 42.75
52
45
b
c There appears to be a gradual trend of increasing sales.
21.16 & 21.17
Sales Exponentially smoothed w = .4 Exponentially smoothed w = .8
18 18 18
22 .4(22)+.6(18) = 19.60 .8(22)+.2(18) = 21.20
27 .4(27)+.6(19.6) = 22.56 .8(27)+.2(21.2) = 25.84
31 .4(31)+.6(22.56) = 25.94 .8(31)+.2(25.84) = 29.97
33 .4(33)+.6(25.94) = 28.76 .8(33)+.2(29.97) = 32.39
20 .4(20)+.6(28.76) = 25.26 .8(20)+.2(32.39) = 22.48
38 .4(38)+.6(25.26) = 30.35 .8(38)+.2(22.48) = 34.90
26 .4(26)+.6(30.35) = 28.61 .8(26)+.2(34.90) = 27.78
25 .4(25)+.6(28.61) = 27.17 .8(25)+.2(27.78) = 25.56
36 .4(36)+.6(27.17) = 30.70 .8(36)+.2(25.56) = 33.91
44 .4(44)+.6(30.70) = 36.02 .8(44)+.2(33.91) = 41.98
29 .4(29)+.6(36.02) = 33.21 .8(29)+.2(41.98) = 31.60
41 .4(41)+.6(33.21) = 36.33 .8(41)+.2(31.60) = 39.12
33 .4(33)+.6(36.33) = 35.00 .8(33)+.2(39.12) = 34.22
52 .4(52)+.6(35.00) = 41.80 .8(52)+.2(34.22) = 48.44
45 .4(45)+.6(41.80) = 43.08 .8(45)+.2(48.44) = 45.69
21.17 b
21.19
The linear trend model appears to be best.
21.21
The quadratic trend line fits slightly better.
21.23
Year / Quarter / y / /1 / 1 / 55 / 46.6 / 1.179
2 / 44 / 45.6 / 0.965
3 / 46 / 44.5 / 1.033
4 / 39 / 43.5 / 0.897
2 / 1 / 41 / 42.4 / 0.967
2 / 38 / 41.3 / 0.919
3 / 37 / 40.3 / 0.919
4 / 30 / 39.2 / 0.765
3 / 1 / 43 / 38.2 / 1.127
2 / 39 / 37.1 / 1.051
3 / 39 / 36.0 / 1.082
4 / 35 / 35.0 / 1.001
4 / 1 / 36 / 33.9 / 1.061
2 / 32 / 32.9 / 0.974
3 / 30 / 31.8 / 0.943
4 / 25 / 30.7 / 0.813
5 / 1 / 50 / 29.7 / 1.685
2 / 25 / 28.6 / 0.874
3 / 24 / 27.6 / 0.871
4 / 22 / 26.5 / 0.830
Quarter
Year / 1 / 2 / 3 / 4 / Total
1 / 1.179 / 0.965 / 1.033 / 0.897
2 / 0.967 / 0.919 / 0.919 / 0.765
3 / 1.127 / 1.051 / 1.082 / 1.001
4 / 1.061 / 0.974 / 0.943 / 0.813
5 / 1.685 / 0.874 / 0.871 / 0.830
Average / 1.204 / 0.957 / 0.970 / 0.861 / 3.991
Seasonal
Index / 1.207 / 0.959 / 0.972 / 0.863 / 4.000
21.25a
b = -485,079 + 250Year
21.27
= 143 + 7.42 t
Year / Quarter / Period t / y / =143+7.42t /1996 / 1 / 1 / 184 / 150.42 / 1.223
2 / 2 / 173 / 157.84 / 1.096
3 / 3 / 160 / 165.26 / 0.968
4 / 4 / 189 / 172.68 / 1.095
1997 / 1 / 5 / 191 / 180.1 / 1.061
2 / 6 / 185 / 187.52 / 0.987
3 / 7 / 184 / 194.94 / 0.944
4 / 8 / 200 / 202.36 / 0.988
1998 / 1 / 9 / 205 / 209.78 / 0.977
2 / 10 / 192 / 217.2 / 0.884
3 / 11 / 200 / 224.62 / 0.890
4 / 12 / 229 / 232.04 / 0.987
1999 / 1 / 13 / 236 / 239.46 / 0.986
2 / 14 / 219 / 246.88 / 0.887
3 / 15 / 211 / 254.3 / 0.830
4 / 16 / 272 / 261.72 / 1.039
2000 / 1 / 17 / 280 / 269.14 / 1.040
2 / 18 / 261 / 276.56 / 0.944
3 / 19 / 275 / 283.98 / 0.968
4 / 20 / 322 / 291.4 / 1.105
2001 / 1 / 21 / 331 / 298.82 / 1.108
2 / 22 / 301 / 306.24 / 0.983
3 / 23 / 306 / 313.66 / 0.976
4 / 24 / 351 / 321.08 / 1.093
Quarter
Year / 1 / 2 / 3 / 4 / Total
1996 / 1.223 / 1.096 / .968 / 1.095
1997 / 1.060 / .987 / .944 / .988
1998 / .977 / .884 / .890 / .987
1999 / .986 / .887 / .830 / 1.039
2000 / 1.040 / .944 / .968 / 1.105
2001 / 1.108 / .983 / .976 / 1.093
Average / 1.066 / .964 / .929 / 1.051 / 4.010
Seasonal
Index / 1.063 / .962 / .927 / 1.048 / 4.000
21.29 Regression line: = 90.4 + 2.02 t
Year / Quarter / t / y / =90.4 + 2.02t /1996 / 1 / 1 / 106 / 92.42 / 1.147
2 / 2 / 92 / 94.44 / 0.974
3 / 3 / 65 / 96.46 / 0.674
4 / 4 / 121 / 98.48 / 1.229
1997 / 1 / 5 / 115 / 100.5 / 1.144
2 / 6 / 100 / 102.52 / 0.975
3 / 7 / 73 / 104.54 / 0.698
4 / 8 / 135 / 106.56 / 1.267
1998 / 1 / 9 / 114 / 108.58 / 1.050
2 / 10 / 105 / 110.6 / 0.949
3 / 11 / 79 / 112.62 / 0.701
4 / 12 / 140 / 114.64 / 1.221
1999 / 1 / 13 / 121 / 116.66 / 1.037
2 / 14 / 111 / 118.68 / 0.935
3 / 15 / 82 / 120.7 / 0.679
4 / 16 / 163 / 122.72 / 1.328
Quarter
Year / 1 / 2 / 3 / 4 / Total
1996 / 1.147 / .974 / .674 / 1.229
1997 / 1.144 / .975 / .698 / 1.267
1998 / 1.050 / .949 / .701 / 1.221
1999 / 1.037 / .935 / .679 / 1.328
Average / 1.095 / .959 / .688 / 1.261 / 4.003
Seasonal
Index / 1.094 / .958 / .688 / 1.260 / 4.000
21.31 Model 1:
Model 2:
Model 2 is more accurate.
Model 1: SSE = (6.0 - 7.5)2 + (66.6 - 6.3)2 + (7.3 - 5.4)2 + (9.4 - 8.2)2
= 2.25 + .09 + 3.61 + 1.44 = 7.39
Model 2: SSE = (6.0 - 6.3)2 + (6.6 - 6.7)2 + 7.3 - 7.1)2 + (9.4 - 7.5)2
= .09 + .01 + .04 + 3.61 = 3.75
Model 2 is more accurate.
21.33 Technique 1:
SSE = (19 - 22)2 + (24 - 27)2 + (32 - 31)2 + (38 - 35)2
= 4 + 9 + 1 + 1 + 9 = 24
Technique 2:
SSE = (19 - 22)2 + (24 - 24)2 + (28 - 26)2 + (32 - 28)2 + (28 - 30)2
= 9 + 0 + 4 + 16 + 64 = 93
Technique 3:
SSE = (19 - 17)2 + (24 - 20)2 + (28 - 25)2 + (32 - 31)2 + (38 - 39)2
= 4 + 16 + 9 + 1+ 1 = 31
By both measures, technique 1 is the most accurate.
21.35
Day / t / / SI / ForecastSunday / 29 / 186.7 / 1.5 / 280.1
Monday / 30 / 189.0 / .4 / 75.6
Tuesday / 31 / 191.3 / .5 / 95.7
Wednesday / 32 / 193.6 / .6 / 116.2
Thursday / 33 / 195.9 / .7 / 137.1
Friday / 34 / 198.2 / 1.4 / 277.5
Saturday / 35 / 200.5 / 1.9 / 381.0
21.37 `
21.39
Day / t / / SI / Forecast1 / 21 / 24.49 / .675 / 16.53
2 / 22 / 24.85 / .892 / 22.17
3 / 23 / 25.22 / .864 / 21.79
4 / 24 / 25.58 / 1.212 / 31.01
5 / 25 / 25.95 / 1.357 / 35.21
21.41
Quarter / t / / SI / Forecast1 / 17 / 81.81 / 0.839 / 68.64
2 / 18 / 82.99 / 1.058 / 87.80
3 / 19 / 84.17 / 1.270 / 106.90
4 / 20 / 85.35 / 0.833 / 71.10
1 / 21 / 86.53 / 0.839 / 72.60
2 / 22 / 87.71 / 1.058 / 92.80
3 / 23 / 88.89 / 1.270 / 112.89
4 / 24 / 90.07 / 0.833 / 75.03
21.43 a = -16.08 + .959(-231.1) = -237.7
-214.7
21.45
Day / t / / SI / Forecast1 / 29 / 193.14 / 1.404 / 271.17
2 / 30 / 194.80 / 0.517 / 100.71
3 / 31 / 196.46 / 0.515 / 101.18
4 / 32 / 198.12 / 0.621 / 123.03
5 / 33 / 199.78 / 0.675 / 134.85
6 / 34 / 201.44 / 1.145 / 230.65
7 / 35 / 203.10 / 2.123 / 431.18
21.47
21.49 = 20.21 + .732t
21.51
Quarter / t / / SI / Forecast1 / 21 / 35.58 / 0.646 / 23.00
2 / 22 / 36.31 / 1.045 / 37.95
3 / 23 / 37.05 / 1.405 / 52.05
4 / 24 / 37.78 / 0.904 / 34.14
505