The following problem provides an example of the computation of exponentially smoothed and adjusted exponentially smoothed forecasts.
A computer software firm has experienced the following demand for its Personal Finance software package:
Period  Units 

1  56 
2  61 
3  55 
4  70 
5  66 
6  65 
7  72 
8  75 
Develop an exponential smoothing forecast, using a = .40, and an adjusted exponential smoothing forecast, using a = .40 and b = .20. Compare the accuracy of the two forecasts, using MAD and cumulative error.
Step 1.  Compute the Exponential Smoothing Forecast F _{ t } _{ +1 } = a D _{ t } + (1 a ) F _{ t } For period 2 the forecast ( assuming that F _{ 1 } = 56) is F _{ 2 } = a D _{ 1 } + (1 a ) F _{ 1 } = (.40)(56) + (.60)(56) = 56 For period 3 the forecast is F _{ 3 } = (.40)(61) + (.60)(56) = 58 The remaining forecasts are computed similarly and are shown in the table on page 704.  
Step 2.  Compute the Adjusted Exponential Smoothing Forecast Starting with the forecast for period 3 (because F _{ 1 } = F _{ 2 } and we will assume that T _{ 2 } = 0),
 
Step 3.  Compute the MAD Values
 
Step 4.  Compute the Cumulative Error Because both MAD and the cumulative error are less for the adjusted forecast, it would appear to be the most accurate. The following problem provides an example of the computation of a linear regression forecast. 
A local building products store has accumulated sales data for twobyfour lumber (in board feet) and the number of building permits in its area for the past 10 quarters :
Quarter  Building Permits, x  Lumber Sales (1,000s of bd. ft.), y 

1  8  12.6 
2  12  16.3 
3  7  9.3 
4  9  11.5 
5  15  18.1 
6  6  7.6 
7  5  6.2 
8  8  14.2 
9  10  15.0 
10  12  17.8 
Develop a linear regression model for these data and determine the strength of the linear relationship by using correlation. If the model appears to be relatively strong, determine the forecast for lumber, given 10 building permits in the next quarter.
Step 1.  Compute the Components of the Linear Regression Equation

Step 2.  Develop the Linear Regression Equation y = a + bx y = 1.36 + 1.25 x 
Step 3.  Compute the Correlation Coefficient Thus, there appears to be a strong linear relationship. 
Step 4.  Calculate the Forecast for x = 10 Permits
