simple linear regression problem / estimating T value

Question

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which formula is used for this problem?

Answer 1

In this diagram, you have for $j=0,1$

$\square$ The estimated coefficients $\widehat{\beta}_j$

$\square$ The estimated standard error of the coefficients $\widehat{\sigma}_j$

$\square$ The $t-$value $t_j$

$\square$ The $p-$value $p_j=\mathbb{P}[T > |t|]=2\mathbb{P}[T > t]$ (where I used that the $t-$Student distribution is symmetric. The degrees of freedom of this distribution is $\text{df}=n-k-1$, where $n$ is the number of observations and $k$ the number of covariates, so in this case $\text{df}=11$.

You can easily obtain the $t-$values as $$t_j=\frac{\widehat{\beta}_j}{\widehat{\sigma}_j}$$ and calculate the $p-$values.