ΣX_{1} = 272, ΣX_{2}= 441, ΣY= 147, ΣX_{1}^{2}= 7428, ΣX_{2}^{2}=19461, ΣY^{2} = 2173, ΣX1Y = 4013, ΣX_{1}X_{2 }= 12005, ΣX_{2}Y = 6485, n=10.

Fit a regression equation Y on X1 and X2. Interpret the regression coefficients.

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Login NowIt is a linear function of one dependent variable with two or more independent variables. With the help of two or more independent variables the value of dependent variable is predicted. For example, if we wish to test the hypothesis the whether that whether or not ‘pass grade’ of students depends on many causes such as previous task mark, study hours, IQ, …. then we can test a regression of cause (pass grade) with effect variables. This test will give us which causes are really significant is generating effect variable and among the significant cause variables their relative value responsible to generate the effect variables. If we assume more than one cause (called Y or dependent variable), it is known as multiple regression. If we assume that the relation between Y and X’s is linear it is called multiple linear regression.

Let us consider three variable Y, X_{1} and Y_{2} in which Y is dependent variable, X_{1} and Y_{2} are independent variables, then the mathematical form of the linear relationship of Y with X1 and X2 is expressed as

Y = b_{0} + b_{1}X_{1} + b_{2}X_{2} + ε

Where Y = Dependent variable

X_{1} and X_{2} = Independent variable or explanatory variable or regressors

b_{0} = Intercept and is called average value of Y and X_{1} and X_{2} are zero

b_{1} = Regression coefficient of Y on X_{1} keeping X_{2} constant. It measures the amount of change in Y per unit change in X_{1} holding X_{2} constant.

b_{2} = Regression coefficient of Y on X_{2} keeping X_{1} constant. It measures the amount of change in Y per unit change in X_{2} holding X_{1} constant.

ε = Random error

Random error(ε) is not created from mistake. It is a technical term that denotes the excess of value from real by model estimation. error is also called Residual.

To fit regression equation y = b_{0} + b_{1}x_{1} + b_{2}x_{2}

Σy = nb_{0} + b_{1}Σ x_{1} + b_{2}Σ x_{2}

or, 147 = 10b_{0} + 272b_{1} + 441b_{2} …………(i)

Σyx_{1} = b_{0}Σx_{1} + b_{1}Σx_{1}^{2} + b_{2}Σ x_{1}x_{2}

or, 4013 = 272b_{0} + 7428b_{1} + 12005b_{2 …………(ii)}

Σyx_{2} = b_{0}Σx_{2} + b_{1}Σ x_{1}x_{2 }+ b_{2}Σx_{2}^{2}

or, 6485= 441b_{0} + 12005b_{1} + 19461b_{2 …………(iii)}

To find b_{0 }, b_{1} and b_{2} using Cramer’s rule

Coefficient of b_{0} |
Coefficient of b_{1} |
Coefficient of b_{2} |
Constant |

10 | 272 | 441 | 147 |

272 | 7428 | 12005 | 4013 |

441 | 12005 | 19461 | 6485 |

Now, \(D = \begin{vmatrix}10 & 272 & 441\\ 272 & 7428 & 12005\\ 441 & 12005 & 19461\end{vmatrix}\)

= 10(144556308 – 144120025) – 272(5293392 – 5294205) + 441(3265360 – 3275748)

= 2858

\(D_1 = \begin{vmatrix}147 & 272 & 441\\ 4013 & 7428 & 12005\\ 6485 & 12005 & 19461\end{vmatrix}\)

= 147(144556308 – 144120025) – 272(78096993 – 77852425) + 441(48176065 – 48170580)

= 29990

\(D_2 = \begin{vmatrix}10 & 147 & 441\\ 272 & 4013 & 12005\\ 441 & 6485 & 19461\end{vmatrix}\)

= 10(78096993 – 77852425) – 147(5293392 – 5294205) – 441(1763920 – 1769733)

= 1658

\(D_1 = \begin{vmatrix}10 & 272 & 147\\ 272 & 7428 & 4013\\ 441 & 12005 & 6485\end{vmatrix}\)

= 10(48170580 – 48176065) – 272(1763920 – 1769733) + 147(3265360 – 3275748)

= -750

Now,

b_{0} = \(\frac{D_1}{D} = \frac{29990}{2825}\) = 10.493

b_{1} = \(\frac{D_2}{D} = \frac{1685}{2825}\) = 0.58

b_{2} = \(\frac{D_3}{D} = \frac{-750}{2825}\) = -0.262

Hence regression equation is y = b_{0} + b_{1}x_{1} + b_{2}x_{2} = 10.493 + 0.58x_{1} – 0.262x_{2}

Here, b_{1} = 0.58 it means y changes by 0.58 per unit change x_{1} keeping x_{2} constant.

Here, b_{2} = -0.262 it means y changes(decreases) by 0.262 per unit change x_{2} keeping x_{1} constant.

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