Suppose we are given following information with n=7, multiple regression model is

\(\hat{A} = 8.15 + 0.6X_{1} + 0.54X_{2}\)

Here , Total sum of square = 1493, and Sum of square due to error = 91
Find i) R2 and interpret it. ii) Test the overall significance of model

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TSS = 1493

SSE = 91

SSR = TSS – SSE = 1493 – 91 = 1402

MSE = SSE / k = 1402 / 2 = 701

MSE = SSE / n-k-1 = 91/7-2-1 = 22.75

R2 = SSR / TSS = 1402 / 1493 = 0.939 = 93.9%

It means 93.9% variation in y is explained by x1 and x2

To test overall significance of regression model

Let β1 and β2 be population regression coefficients of Y on X1 keeping X2 constant and population regression coefficient of Y on X2 keeping X1 constant.

Problem to test

H0 : β1 = β2 = 0

H1 : At least one β1 is different from zero, i = 1,2

Test Statistic

F = \(\frac{MSR}{MSE}\) = \(\frac{701}{22.75}\) = 30.81

Critical Value

At α = 0.05 level of significance, critical value is Fα(k, n-k-1) =6.944

Decision

F = 30.81 > Ftabulated = 6.944, reject H0 at 5% level of significance

Conclusion

There is linear relationship of dependent variable y with at least one of the independent variable xs

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