Question

You have joined a northern mail order company selling winter coats. You have the coat sales by quarter for the last three years.

Year 1 Qtr 1, 24 Winter Coats Qtr 2, 12 Qtr 3, 20 Qtr 4, 36 Year 2 Qtr 1, 28 Winter Coats Qtr 2, 10 Qtr 3, 22 Qtr 4, 40 Year 3 Qtr 1, 32 Winter coats Qtr 2, 14 Qtr 3, 27 Qtr 4, 44

Use linear regression to forecast the total coats to be sold in year 4 in thousands. For the equation Y = aX + b give "a".   ____ (two decimals) Give "b" ____ (two decimals)

Give the forecast for the fourth year? ____ (two decimals)Next use the quarters to generate seasonal factors. Give the season factor for quarter one? ____ (two decimals)

Give the season factor for quarter two? _____ (two decimals)Give the season factor for quarter three? _____ (two decimals) Give the season factor for quarter four? ______ (two decimals) Give the forecasted sales for quarter one? ______ (All answers remaining to two decimals) Quarter two? ______Quarter three? ______Quarter four? _____

Answer 1

Data:

Year	Quarter	Winter Coats	Yearly Sales
Year 1	Qtr 1	24	92
Year 1	Qtr 2	12
Year 1	Qtr 3	20
Year 1	Qtr 4	36
Year 2	Qtr 1	28	100
Year 2	Qtr 2	10
Year 2	Qtr 3	22
Year 2	Qtr 4	40
Year 3	Qtr 1	32	117
Year 3	Qtr 2	14
Year 3	Qtr 3	27
Year 3	Qtr 4	44
Total			309

For better understanding let us rearrange the data:

Year	Sales
1	92
2	100
3	117
Total = 6	Total = 309

The linear Regression equation is Y = aX+b

Y = Sales

X = Year

a = slope of the line

b = intercept

The values of a and b can be found by using the least square method.

Year (X)	Sales (Y)	XY	X^2
1	92	92	1
2	100	200	4
3	117	351	9
6	309	643	14

The last row is the total of the values.

X̅ = ∑X /n = 6/3 = 2

Y̅ = ∑Y/n = 309/3 = 103

a = (∑XY – nX̅Y̅) / (∑(X^2)-n (X̅^2)) = (643-3*2*103)/(14-3*2^2) = 12.5

b = Y̅ - a X̅ = 103-12.5*2 = 78

Linear Regression Equation is

Y = 12.5X+78

Forecast for 4th year is (X = 4)

Y = 12.5*4+78 = 128

Before calculating the seasonal factors, let us rearrange the data:

Year	1	2	3	Sum
Quarter
1	24	28	32	84
2	12	10	14	36
3	20	22	27	69
4	36	40	44	120
Sum	92	100	117	309

Seasonal Factor = Sum of Sales of a quarter across the years/grand total of sales

Seasonal Factor for Quarter 1 = (24+28+32+84)/309 = 0.271845

Similarly, the seasonal factor for rest quarters are calculated:

Year	1	2	3	Sum	Seasonal Factors
Quarter
1	24	28	32	84	0.271845
2	12	10	14	36	0.116505
3	20	22	27	69	0.223301
4	36	40	44	120	0.38835
Sum	92	100	117	309

Forecast for each quarter in year 4 is achieved by multiplying the seasonal factors with year 4 sales forecast achieved from linear regression

Year 4 Quarter 1 sales forecast = 0.271845*128 = 34.79

Year 4 Quarter 2 sales forecast = 0.116505*128 = 14.91

Year 4 Quarter 3 sales forecast = 0.223301*128 = 28.58

Year 4 Quarter 4 sales forecast = 0.38835*128 = 49.71

a = 12.5

b = 78

Forecast for 4th year = 128

Seasonal Factor of Qtr 1 = 0.27

Seasonal Factor of Qtr 2 = 0.12

Seasonal Factor of Qtr 3 = 0.22

Seasonal Factor of Qtr 4 = 0.39

Forecasted Sales for:

Quarter 1 = 34.79

Quarter 2 = 14.91

Quarter 3 = 28.58

Quarter 4 = 49.71