If we use the multiplicative decomposition method to analyze the quarterly bicyc
ID: 3295461 • Letter: I
Question
If we use the multiplicative decomposition method to analyze the quarterly bicycle sales data given in Table 15.3 below, we find that the quarterly seasonal factors are .42, 1.19, 1.66, and .60. Furthermore, if we use a statistical software package to fit a straight line to the deseasonalized sales values, we find that the estimate of the trend is
trt = 22.61 + .59t
In addition, we find that the half-lengths of 95 percent prediction intervals for the deseasonalized sales values in the four quarters of the next year are, respectively, 2.88, 2.89, 2.96, and 3.06.
(a) Calculate point predictions of bicycle sales in the four quarters of the next year.
(b) Calculate approximate 95 percent prediction intervals for bicycle sales in the four quarters of the next year. (Round your answers to 2 decimal places.)
Point Predictions = sn × tr t sn tr = 22.61 + .59t sn × tr 17 .42 18 1.19 19 1.66 20 .60 TA BLE 15.3 Quarterly Sales of the TRK-50 Mountain Bike Os BikeSales Quarter 1 (Winter) 2 (Spring) 3 (Summer) 4 (Fall) Sales, yt 10 31 43 16 Year 6 45 17 13 34 48 19 10 12 13 14 15 16 15 37 51 21
Explanation / Answer
a. The point prediction for bicycle sales in the four quarters of the next year are as follows:
tr=22.61+0.59t
Substitute t with 17, 18, 19 and 20 respectively to obtain the values of tr.
32.64, 33.23, 33.82, 34.41
Multiply the values of tr with sn that are 0.42, 1.19, 1.66, and 0.60 to obtain the point estimate.
13.7088, 39.5437, 56.1412, 20.646
b. The 95% prediction interval is ythat+-1.96sigmahat
Substitute ythat with sn*tr for t=17, 18, 19 and 20, sigmahat are 2.88, 2.89, 2.96, 3.06 respectively.
yhat17=13.7088+-1.96(2.88)=(8.06, 19.35)
yhat18=39.5437+-1.96(2.89)=(33.88,45.21)
yhat19=56.1412+-1.96(2.96)=(50.34,61.94)
yhat20=20.646+-1.96(3.06)=(14.65,26.64)
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