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Compute the appropriate regression equation for the following data:
x (Dependent Variable) | 10 | 12 | 13 | 17 | 18 |
y (Independent Variable) | 5 | 6 | 7 | 9 | 13 |
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Solution
X = x_{i} | Y = y_{i} | x_{i}^{2} | y_{i}^{2} | x_{i}y_{i} | |
10 | 5 | 100 | 25 | 50 | |
12 | 6 | 144 | 36 | 72 | |
13 | 7 | 169 | 49 | 91 | |
17 | 9 | 289 | 81 | 153 | |
18 | 13 | 324 | 169 | 234 | |
Total | 70 | 40 | 1026 | 360 | 600 |
From the table, we have,
n = 5, Σx_{i} = 70, Σy_{i} = 40, Σx_{i}y_{i} = 600, Σx_{i}^{2} = 1026, Σy_{i}^{2} = 360
`barx = (sumx_"i")/"n" = 70/5` = 14, `bary = (sumy_"i")/"n" = 40/5` = 8
Now, for regression equation of X on Y
b_{xy} = `(sumx_"i"y_"i" - "n"bar(x) bar(y))/(sumy_"i"^2 - "n" bar(y)^(-2))`
= `(600 - 5 xx 14 xx 8)/(360 - 5(8)^2)`
= `(600 - 560)/(360 - 320)`
= `40/40`
= 1
Also, a' = `barx - "b"_(xy) bary` = 14 – 1 (8)
= 14 – 8
= 6
∴ The regression equation of X on Y is
X = a' + b_{xy} Y
∴ X = 6 + Y
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