Question

Favors Distribution Company purchases small imported trinkets in bulk, packages

them, and sells them to retail stores. They are conducting an inventory control study of all their

items. The following data for the year 2006 are for one such item, which is not seasonal.




Month Jan Feb Mar Apr May Jun Jul Aug
Sales 15 18 14 12 19 10 11 15


Use trend projection to estimate the relationship between time and sales (state the
trend line equation in this problem). Show the calculations of the intercept and the slope.

Interpret the meaning of the intercept and the slope in this problem.
Calculate forecasts for March 2007, May 2007, and June 2008

Answer 1

Step-by-step explanation:

To estimate the relationship between time and sales using trend projection, we can use the least squares method to find the equation of the trend line. The equation of a straight line is y = mx + b, where y represents sales and x represents time (in this case, the month).

First, we need to assign numbers to the months: Jan = 1, Feb = 2, Mar = 3, and so on.

Next, we calculate the sum of the months (Σx), the sum of the sales (Σy), the sum of the squared months (Σx^2), and the sum of the products of months and sales (Σxy):

Σx = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36

Σy = 15 + 18 + 14 + 12 + 19 + 10 + 11 + 15 = 114

Σx^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 = 204

Σxy = (1 × 15) + (2 × 18) + (3 × 14) + (4 × 12) + (5 × 19) + (6 × 10) + (7 × 11) + (8 × 15) = 475

Next, we calculate the slope (m) and intercept (b) using the following formulas:

m = (nΣxy - ΣxΣy) / (nΣx^2 - (Σx)^2)

b = (Σy - mΣx) / n

where n is the number of data points (in this case, 8).

Using the above formulas:

m = (8 × 475 - 36 × 114) / (8 × 204 - (36)^2) = 0.074

b = (114 - 0.074 × 36) / 8 = 13.381

The trend line equation for this problem is:

Sales = 0.074 × Month + 13.381

Interpretation:

The intercept (13.381) represents the estimated sales at the beginning of the time period (January 2006). It suggests that even at the start of the study, the item had an estimated sales value of 13.381 units.

The slope (0.074) represents the average increase in sales per month. It indicates that, on average, the item's sales increased by approximately 0.074 units per month during the study period.

Calculating forecasts:

To calculate forecasts for specific months, substitute the corresponding month values into the trend line equation.

For March 2007 (Month = 15):

Sales = 0.074 × 15 + 13.381 = 14.751

For May 2007 (Month = 17):

Sales = 0.074 × 17 + 13.381 = 14.997

For June 2008 (Month = 30):

Sales = 0.074 × 30 + 13.381 = 15.491

Therefore, the forecasted sales for March 2007 is approximately 14.751 units, for May 2007 is approximately 14.997 units, and for June 2008 is approximately 15.491 units.