Choosing an appropriate value of the smoothing constant α is a key decision when applying the exponential smoothing method. When relevant historical data exist, one approach to making this decision is to apply the method retrospectively to these data with different values of α and then choose the value of α that gives the smallest MAD. Use this approach for choosing α with each of the following time series representing monthly sales. In each case, use an initial estimate of 50 and compare α = 0.1, 0.2, 0.3, 0.4, and 0.5.

(a) 51 48 52 49 53 49 48 51 50 49

(b) 52 50 53 51 52 48 52 53 49 52

(c) 50 52 51 55 53 56 52 55 54 53