Example of Regression Equation
An investment company advertised the sale of pieces of land at different prices. The given table shows the pieces of land their costs and acreage
Piece of land
|
(x)Acreage Hectares
|
(y) Cost £ 000
|
xy
|
x2
|
A
|
2.3
|
230
|
529
|
5.29
|
B
|
1.7
|
150
|
255
|
2.89
|
C
|
4.2
|
450
|
1890
|
17.64
|
D
|
3.3
|
310
|
1023
|
10.89
|
E
|
5.2
|
550
|
2860
|
27.04
|
F
|
6.0
|
590
|
3540
|
36
|
G
|
7.3
|
740
|
5402
|
53.29
|
H
|
8.4
|
850
|
7140
|
70.56
|
J
|
5.6
|
530
|
2969
|
31.36
|
|
Σx =44.0
|
Σy = 4400
|
Σxy= 25607
|
Σx2 = 254.96
|
Required
Find out the regression equations of
i. y on x and thus estimate the cost of a piece of land along with 4.5 hectares
ii. Estimate the expected average if the piece of land costs of £ 900,000
Σy = an + bΣxy
Σxy = a∑x + bΣx2
By substituting of the suitable values in the above equations we have
4400 = 9a + 44b ........ (i)
25607 = 44a + 254.96b ........(ii)
By using multiplying equation .... (i) by 44 and equation ...... (ii) by 9 we have
193600 = 396a + 1936b ........ (iii)
230463 = 396a + 2294.64b ........(iv)
By using subtraction of equation .... (iii) from equation ...... (iv) We have
36863 = 358.64b
102.78 = b
by substituting for b in equation........ (i)
4400 = 9a + 44( 102.78)
4400 - 4522.32 = 9a
-122.32 = 9a
-13.59 = a
Hence the equation of the regression line of y on x is
Y = 13.59 + 102.78x
When the acreage or hectares is 4.5 then the cost
(y) = -13.59 + (102.78 x 4.5)
= 448.92
= £ 448, 920
Note that
Where the regression equation is described by
y= a + bx
Whereas a is the intercept on the y axis and
b is the slope of the line or else regression coefficient
n is the sample size
After that,
intercept a = (Σy - bΣx)/n
Slope b =