
Concept explainers
(a)
Interpretation: The forecast for next quarter, using a four period moving averages needs to be calculated.
Concept Introduction: The moving average method takes the average of the recent number of observations in any time series. The average is taken based on the k number of previous observations.
(a)

Explanation of Solution
B.
Following formula can be used to calculate the forecast values as per moving average method
The forecast for next quarter using a four-period moving average is as follows:
Year | Quarter | Sales | Forecast |
1 | 1 | 4 | |
1 | 2 | 2 | |
1 | 3 | 1 | |
1 | 4 | 5 | |
2 | 1 | 6 | 3 |
2 | 2 | 4 | 3.5 |
2 | 3 | 4 | 4 |
2 | 4 | 14 | 4.75 |
3 | 1 | 10 | 7 |
3 | 2 | 3 | 8 |
3 | 3 | 5 | 7.75 |
3 | 4 | 16 | 8 |
4 | 1 | 12 | 8.5 |
4 | 2 | 9 | 9 |
4 | 3 | 7 | 10.5 |
4 | 4 | 22 | 11 |
5 | 1 | 18 | 12.5 |
5 | 2 | 10 | 14 |
5 | 3 | 13 | 14.25 |
5 | 4 | 35 | 15.75 |
6 | 1 | Forecast | 19 |
Therefore, the forecast for next quarter, using a four-period moving average is 19.
(b)
Interpretation: The good value of a for a single exponential smoothing model needs to be determined .
Concept Introduction: Single Exponential Smoothing is a method which computes the weighted average of previous sales data to forecast the future sales value.
(b)

Explanation of Solution
Following formula can be used to calculate the forecasted values:
The Mean Square Error for various values of a is as follows:
At a = 0.1
Forecast | |||||
Year | Quarter | Sales | Alpha = 0.1 | Error | Squared Error |
1 | 1 | 4 | |||
1 | 2 | 2 | 4 | 2 | 4 |
1 | 3 | 1 | 3.8 | 2.8 | 7.84 |
1 | 4 | 5 | 3.52 | 1.48 | 2.1904 |
2 | 1 | 6 | 3.668 | 2.33 | 5.438224 |
2 | 2 | 4 | 3.9012 | 0.1 | 0.00976144 |
2 | 3 | 4 | 3.91108 | 0.09 | 0.007906766 |
2 | 4 | 14 | 3.919972 | 10.1 | 101.6069645 |
3 | 1 | 10 | 4.9279748 | 5.07 | 25.72543963 |
3 | 2 | 3 | 5.4351773 | 2.44 | 5.93008858 |
3 | 3 | 5 | 5.1916596 | 0.19 | 0.036733398 |
3 | 4 | 16 | 5.1724936 | 10.8 | 117.2348942 |
4 | 1 | 12 | 6.2552443 | 5.74 | 33.00221844 |
4 | 2 | 9 | 6.8297198 | 2.17 | 4.710115974 |
4 | 3 | 7 | 7.0467479 | 0.05 | 0.002185362 |
4 | 4 | 22 | 7.0420731 | 15 | 223.739578 |
5 | 1 | 18 | 8.5378658 | 9.46 | 89.53198432 |
5 | 2 | 10 | 9.4840792 | 0.52 | 0.266174285 |
5 | 3 | 13 | 9.5356713 | 3.46 | 12.00157356 |
5 | 4 | 35 | 9.8821041 | 25.1 | 630.9086924 |
6 | 1 | 12.393894 | SUM | 1264.182935 | |
MSE | 66.53594394 |
At a = 0.2
Forecast | |||||
Year | Quarter | Sales | Alpha = 0.2 | Error | Squared Error |
1 | 1 | 4 | |||
1 | 2 | 2 | 4 | 2 | 4 |
1 | 3 | 1 | 3.6 | 2.6 | 6.76 |
1 | 4 | 5 | 3.08 | 1.92 | 3.6864 |
2 | 1 | 6 | 3.464 | 2.54 | 6.431296 |
2 | 2 | 4 | 3.9712 | 0.03 | 0.00082944 |
2 | 3 | 4 | 3.97696 | 0.02 | 0.000530842 |
2 | 4 | 14 | 3.981568 | 10 | 100.3689797 |
3 | 1 | 10 | 5.9852544 | 4.01 | 16.11818223 |
3 | 2 | 3 | 6.7882035 | 3.79 | 14.35048591 |
3 | 3 | 5 | 6.0305628 | 1.03 | 1.062059718 |
3 | 4 | 16 | 5.8244503 | 10.2 | 103.5418127 |
4 | 1 | 12 | 7.8595602 | 4.14 | 17.14324172 |
4 | 2 | 9 | 8.6876482 | 0.31 | 0.097563671 |
4 | 3 | 7 | 8.7501185 | 1.75 | 3.062914867 |
4 | 4 | 22 | 8.4000948 | 13.6 | 184.9574208 |
5 | 1 | 18 | 11.120076 | 6.88 | 47.33335619 |
5 | 2 | 10 | 12.496061 | 2.5 | 6.230318954 |
5 | 3 | 13 | 11.996849 | 1 | 1.006312832 |
5 | 4 | 35 | 12.197479 | 22.8 | 519.9549713 |
6 | 1 | 16.757983 | SUM | 1036.106677 | |
MSE | 54.53193036 |
At a = 0.3
Forecast | |||||
Year | Quarter | Sales | Alpha = 0.3 | Error | Squared Error |
1 | 1 | 4 | |||
1 | 2 | 2 | 4 | 2 | 4 |
1 | 3 | 1 | 3.4 | 2.4 | 5.76 |
1 | 4 | 5 | 2.68 | 2.32 | 5.3824 |
2 | 1 | 6 | 3.376 | 2.62 | 6.885376 |
2 | 2 | 4 | 4.1632 | 0.16 | 0.02663424 |
2 | 3 | 4 | 4.11424 | 0.11 | 0.013050778 |
2 | 4 | 14 | 4.079968 | 9.92 | 98.40703488 |
3 | 1 | 10 | 7.0559776 | 2.94 | 8.667267892 |
3 | 2 | 3 | 7.9391843 | 4.94 | 24.39554175 |
3 | 3 | 5 | 6.457429 | 1.46 | 2.12409936 |
3 | 4 | 16 | 6.0202003 | 9.98 | 99.59640172 |
4 | 1 | 12 | 9.0141402 | 2.99 | 8.915358615 |
4 | 2 | 9 | 9.9098982 | 0.91 | 0.827914653 |
4 | 3 | 7 | 9.6369287 | 2.64 | 6.953393015 |
4 | 4 | 22 | 8.8458501 | 13.2 | 173.0316597 |
5 | 1 | 18 | 12.792095 | 5.21 | 27.12227379 |
5 | 2 | 10 | 14.354467 | 4.35 | 18.96137891 |
5 | 3 | 13 | 13.048127 | 0.05 | 0.002316168 |
5 | 4 | 35 | 13.033689 | 22 | 482.5188362 |
6 | 1 | 19.623582 | SUM | 973.5909376 | |
MSE | 51.2416283 |
At a = 0.4
Forecast | |||||
Year | Quarter | Sales | Alpha = 0.4 | Error | Squared Error |
1 | 1 | 4 | |||
1 | 2 | 2 | 4 | 2 | 4 |
1 | 3 | 1 | 3.2 | 2.2 | 4.84 |
1 | 4 | 5 | 2.32 | 2.68 | 7.1824 |
2 | 1 | 6 | 3.392 | 2.61 | 6.801664 |
2 | 2 | 4 | 4.4352 | 0.44 | 0.18939904 |
2 | 3 | 4 | 4.26112 | 0.26 | 0.068183654 |
2 | 4 | 14 | 4.156672 | 9.84 | 96.89110612 |
3 | 1 | 10 | 8.0940032 | 1.91 | 3.632823802 |
3 | 2 | 3 | 8.8564019 | 5.86 | 34.29744345 |
3 | 3 | 5 | 6.5138412 | 1.51 | 2.291715033 |
3 | 4 | 16 | 5.9083047 | 10.1 | 101.8423142 |
4 | 1 | 12 | 9.9449828 | 2.06 | 4.223095632 |
4 | 2 | 9 | 10.76699 | 1.77 | 3.12225256 |
4 | 3 | 7 | 10.060194 | 3.06 | 9.364786175 |
4 | 4 | 22 | 8.8361163 | 13.2 | 173.2878344 |
5 | 1 | 18 | 14.10167 | 3.9 | 15.19697856 |
5 | 2 | 10 | 15.661002 | 5.66 | 32.0469421 |
5 | 3 | 13 | 13.396601 | 0.4 | 0.157292447 |
5 | 4 | 35 | 13.237961 | 21.8 | 473.5863558 |
6 | 1 | 21.942776 | SUM | 973.0225869 | |
MSE | 51.2117151 |
At a = 0.5
Forecast | |||||
Year | Quarter | Sales | Alpha = 0.5 | Error | Squared Error |
1 | 1 | 4 | |||
1 | 2 | 2 | 4 | 2 | 4 |
1 | 3 | 1 | 3 | 2 | 4 |
1 | 4 | 5 | 2 | 3 | 9 |
2 | 1 | 6 | 3.5 | 2.5 | 6.25 |
2 | 2 | 4 | 4.75 | 0.75 | 0.5625 |
2 | 3 | 4 | 4.375 | 0.38 | 0.140625 |
2 | 4 | 14 | 4.1875 | 9.81 | 96.28515625 |
3 | 1 | 10 | 9.09375 | 0.91 | 0.821289063 |
3 | 2 | 3 | 9.546875 | 6.55 | 42.86157227 |
3 | 3 | 5 | 6.2734375 | 1.27 | 1.621643066 |
3 | 4 | 16 | 5.6367188 | 10.4 | 107.3975983 |
4 | 1 | 12 | 10.818359 | 1.18 | 1.396274567 |
4 | 2 | 9 | 11.40918 | 2.41 | 5.804146767 |
4 | 3 | 7 | 10.20459 | 3.2 | 10.26939607 |
4 | 4 | 22 | 8.6022949 | 13.4 | 179.4985014 |
5 | 1 | 18 | 15.301147 | 2.7 | 7.283805028 |
5 | 2 | 10 | 16.650574 | 6.65 | 44.23013094 |
5 | 3 | 13 | 13.325287 | 0.33 | 0.105811545 |
5 | 4 | 35 | 13.162643 | 21.8 | 476.8701419 |
6 | 1 | 24.081322 | SUM | 998.398592 | |
MSE | 52.54729432 |
At a = 0.6
Forecast | |||||
Year | Quarter | Sales | Alpha = 0.6 | Error | Squared Error |
1 | 1 | 4 | |||
1 | 2 | 2 | 4 | 2 | 4 |
1 | 3 | 1 | 2.8 | 1.8 | 3.24 |
1 | 4 | 5 | 1.72 | 3.28 | 10.7584 |
2 | 1 | 6 | 3.688 | 2.31 | 5.345344 |
2 | 2 | 4 | 5.0752 | 1.08 | 1.15605504 |
2 | 3 | 4 | 4.43008 | 0.43 | 0.184968806 |
2 | 4 | 14 | 4.172032 | 9.83 | 96.58895501 |
3 | 1 | 10 | 10.068813 | 0.07 | 0.004735201 |
3 | 2 | 3 | 10.027525 | 7.03 | 49.38610931 |
3 | 3 | 5 | 5.81101 | 0.81 | 0.657737298 |
3 | 4 | 16 | 5.324404 | 10.7 | 113.9683495 |
4 | 1 | 12 | 11.729762 | 0.27 | 0.073028789 |
4 | 2 | 9 | 11.891905 | 2.89 | 8.363112465 |
4 | 3 | 7 | 10.156762 | 3.16 | 9.965145423 |
4 | 4 | 22 | 8.2627047 | 13.7 | 188.713281 |
5 | 1 | 18 | 16.505082 | 1.49 | 2.234780134 |
5 | 2 | 10 | 17.402033 | 7.4 | 54.79008896 |
5 | 3 | 13 | 12.960813 | 0.04 | 0.001535613 |
5 | 4 | 35 | 12.984325 | 22 | 484.6899351 |
6 | 1 | 26.19373 | SUM | 1034.121562 | |
MSE | 54.42745061 |
At a = 0.7
Forecast | |||||
Year | Quarter | Sales | Alpha = 0.7 | Error | Squared Error |
1 | 1 | 4 | |||
1 | 2 | 2 | 4 | 2 | 4 |
1 | 3 | 1 | 2.6 | 1.6 | 2.56 |
1 | 4 | 5 | 1.48 | 3.52 | 12.3904 |
2 | 1 | 6 | 3.944 | 2.06 | 4.227136 |
2 | 2 | 4 | 5.3832 | 1.38 | 1.91324224 |
2 | 3 | 4 | 4.41496 | 0.41 | 0.172191802 |
2 | 4 | 14 | 4.124488 | 9.88 | 97.52573726 |
3 | 1 | 10 | 11.037346 | 1.04 | 1.076087554 |
3 | 2 | 3 | 10.311204 | 7.31 | 53.45370276 |
3 | 3 | 5 | 5.1933612 | 0.19 | 0.037388544 |
3 | 4 | 16 | 5.0580084 | 10.9 | 119.7271812 |
4 | 1 | 12 | 12.717403 | 0.72 | 0.514666355 |
4 | 2 | 9 | 12.215221 | 3.22 | 10.33764448 |
4 | 3 | 7 | 9.9645662 | 2.96 | 8.788652906 |
4 | 4 | 22 | 7.8893699 | 14.1 | 199.1098827 |
5 | 1 | 18 | 17.766811 | 0.23 | 0.054377128 |
5 | 2 | 10 | 17.930043 | 7.93 | 62.88558655 |
5 | 3 | 13 | 12.379013 | 0.62 | 0.385624871 |
5 | 4 | 35 | 12.813704 | 22.2 | 492.2317348 |
6 | 1 | 28.344111 | SUM | 1071.391237 | |
MSE | 56.38901248 |
At a = 0.8
Forecast | |||||
Year | Quarter | Sales | Alpha = 0.8 | Error | Squared Error |
1 | 1 | 4 | |||
1 | 2 | 2 | 4 | 2 | 4 |
1 | 3 | 1 | 2.4 | 1.4 | 1.96 |
1 | 4 | 5 | 1.28 | 3.72 | 13.8384 |
2 | 1 | 6 | 4.256 | 1.74 | 3.041536 |
2 | 2 | 4 | 5.6512 | 1.65 | 2.72646144 |
2 | 3 | 4 | 4.33024 | 0.33 | 0.109058458 |
2 | 4 | 14 | 4.066048 | 9.93 | 98.68340234 |
3 | 1 | 10 | 12.01321 | 2.01 | 4.053012894 |
3 | 2 | 3 | 10.402642 | 7.4 | 54.7991074 |
3 | 3 | 5 | 4.4805284 | 0.52 | 0.26985076 |
3 | 4 | 16 | 4.8961057 | 11.1 | 123.2964691 |
4 | 1 | 12 | 13.779221 | 1.78 | 3.165627849 |
4 | 2 | 9 | 12.355844 | 3.36 | 11.26169048 |
4 | 3 | 7 | 9.6711688 | 2.67 | 7.135143001 |
4 | 4 | 22 | 7.5342338 | 14.5 | 209.2583926 |
5 | 1 | 18 | 19.106847 | 1.11 | 1.225109736 |
5 | 2 | 10 | 18.221369 | 8.22 | 67.590914 |
5 | 3 | 13 | 11.644274 | 1.36 | 1.837993339 |
5 | 4 | 35 | 12.728855 | 22.3 | 496.0039097 |
6 | 1 | 30.545771 | SUM | 1104.256079 | |
MSE | 58.11874101 |
At a = 0.9
Forecast | |||||
Year | Quarter | Sales | Alpha = 0.9 | Error | Squared Error |
1 | 1 | 4 | |||
1 | 2 | 2 | 4 | 2 | 4 |
1 | 3 | 1 | 2.2 | 1.2 | 1.44 |
1 | 4 | 5 | 1.12 | 3.88 | 15.0544 |
2 | 1 | 6 | 4.612 | 1.39 | 1.926544 |
2 | 2 | 4 | 5.8612 | 1.86 | 3.46406544 |
2 | 3 | 4 | 4.18612 | 0.19 | 0.034640654 |
2 | 4 | 14 | 4.018612 | 9.98 | 99.62810641 |
3 | 1 | 10 | 13.001861 | 3 | 9.011170664 |
3 | 2 | 3 | 10.300186 | 7.3 | 53.29271739 |
3 | 3 | 5 | 3.7300186 | 1.27 | 1.612852726 |
3 | 4 | 16 | 4.8730019 | 11.1 | 123.8100876 |
4 | 1 | 12 | 14.8873 | 2.89 | 8.336502365 |
4 | 2 | 9 | 12.28873 | 3.29 | 10.81574514 |
4 | 3 | 7 | 9.328873 | 2.33 | 5.423649459 |
4 | 4 | 22 | 7.2328873 | 14.8 | 218.0676175 |
5 | 1 | 18 | 20.523289 | 2.52 | 6.366986015 |
5 | 2 | 10 | 18.252329 | 8.25 | 68.10093183 |
5 | 3 | 13 | 10.825233 | 2.17 | 4.729611994 |
5 | 4 | 35 | 12.782523 | 22.2 | 493.6162714 |
6 | 1 | 32.778252 | SUM | 1128.731901 | |
MSE | 59.40694213 |
After evaluating the MSE of all
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