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Hello students.
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Today we will discuss about this question.
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In this question we need to show that.
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So here we need to show that or we need to prove that in a multiple linear regression model to test the overall significance of model, alternative formula of the test statistic, that is, f is equals to r squared by k, divided by 1 minus r squared by n minus k, 1.
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So here we need to prove this.
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So first of all, consider a multiple linear regression model that is y -i's that is equals to beta 0 plus beta 1 xi plus beta 2xi x2i plus up to plus bk x -k i plus ui.
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Now, beta i, that is regression coefficient, yi that is response variable, xi, that is observation.
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So here, ui, that are random observation.
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Now consider the set of linear hypothesis about the element of beta that is formulated as at zero such that r beta that is equals to smaller.
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So here, r must have full row rank that is there is no linear dependencies between the hypothesis.
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So firstly we have to replace the unknown vector beta by ols.
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So vector b and the arbitrary vector rb.
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So therefore here e of capital rb that is equals to r beta.
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So here, variants of rb, that is equals to e of r b minus beta, b minus beta, dash, r -dash.
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So that is equals to we can write a square x -diss -x minus 1r -dash.
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So here we have to determine the sampling distribution of r -b.
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So here, sine b is multivariate norms as rb that followed the normally distribution, that is r beta, a square x -dus, x, raise to minus 1, r -dash.
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So therefore, here we under add 0 such that r beta that is equal to r, r bta, that is equals to r, r b, r b, b, b, r b, b, b, r b, that is normally distributed over 0, a square r x -dose -x -x -raise -to -minus -1 -r -r -dus.
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Thus, we can write r -b -minus -r, a -square -r -x -dus -x -2 -1 -r -dus -r -b -1 -r -b minus -r is normally distributed as x -q -square...