Question

Fit a straight line trend by the method of least squares to the following data.

Year	1980	1981	1982	1983	1984	1985	1986	1987
Sales	50.3	52.7	49.3	57.3	56.8	60.7	62.1	58.7

Answer 1

Let x denote the years and y denote the sales. Since number of years is even, 

we take X = (x - 1983.5)/0.5

The trend values are obtained by substituting the years for x, and given in the table.

When x= 1980, Y_t = 55.9875 + 0.83(-7) = 50.1775 

When x = 1981, Y_t = 55.9875 – 5(0.83) = 51.8375 

When x = 1982, Y_{​​​​​​​t} = 55.9875 – 3(0.83) = 53.4975 

When x= 1983, Y_{​​​​​​​t} = 55.9875 – (0.83) = 55.1575 

When x = 1984, Y_{​​​​​​​t} = 55.9875 + (0.83) = 56.8175 

When x = 1985, Y_{​​​​​​​t} = 55.9875 + 3(0.83) = 58.4775 

When x = 1986, Y_{​​​​​​​t} = 55.9875 + 5(0.83) = 60.1375 

When x = 1987, Y_{​​​​​​​t} = 55.9875 + 7(0.83) = 61.7975

We find that ΣYt = ΣY = 447.9