I have 1 dependent variable and 3 independent variables.
I run multiple regression, and find that the p value for one of the independent variables is higher than 0.05 (95% is my confidence level).
I take that variable out and run it again. Both remaining independent variables have $p$-value less than 0.05 so I conclude I have my model.
Am I correct in thinking that initially, my null hypothesis is
$$H_0= β_1=β_2 = \dots =β_{k-1} = 0$$
and that the alternative hypothesis is
$$H_1=\textrm{At least one } β \neq 0 \textrm{ whilst } p<0.05$$
And that after the first regression, I do not reject, as one variable does not meet my confidence level needs...
So I run it again, and then reject the null as all $p$-values are significant?
Is what I have written accurate?
Edit: Thanks to Bob Jansen for improving this aesthetics of this post.