We have a population of items with multiple measurable attributes.

Every two attribute have a correlation we call rho and a linear relation Y=beta0 and beta1*x

We may test the Hypothesis: rho=0

based on sample r        $t^*=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$

t^*=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}

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We may test the Hypothesis: beta1=0  for one Iv

t=\frac{beta1}{S_{beta1}}=\frac{beta1}{\frac{\sqrt{\frac{SSE}{n-k-1}}}{{S_{x}\sqrt{n-1}}}}\sim T_{n-K-1}\sim \forall
\textup{ k number of vars}

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We may test the Hypothesis: beta1=0  for two Iv

\sqrt{\frac{SSE}{n-k-1}}

\frac{\sqrt{\frac{SSE}{n-k-1}}}{{S_{x}\sqrt{(1-{r_{x_{1}x_{2}}^{2})n-1}}}}

t=\frac{beta1}{S_{beta1}}=\frac{beta1}{\frac{\sqrt{\frac{SSE}{n-k-1}}}{{S_{x}\sqrt{(1-{r_{x_{1}x_{2}}^{2})n-1}}}}}\sim T_{n-K-1}\sim \forall
\textup{ k number of vars}

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We may test the Hypothesis: beta1=beta2=beta3=0

which means that the whole model is useless.

F=\frac{\frac{SSR}{number of Iv}}{\frac{S_{SSE}}{n-number of Iv-1}}=\frac{MSR}{MSE}\sim F_{k,n-k-1}