I have one question. In the formula for calculating the confidence interval, I don't see the degrees of freedom in the formula.
In my book, the formula is : $$\text{Confidence interval:} \qquad \mu = \bar{x} \pm t_{\alpha/2} \frac{s}{\sqrt{n}}$$
Why it's not this instead ? $$\text{Confidence interval:} \qquad \mu = \bar{x} \pm t_{\alpha/2,n-1} \frac{s}{\sqrt{n}}$$
Interesting question. I would say that this is a matter of style or convenience (or both!).
I think the author(s) of you book felt that the degrees of freedom should be "clear from context" and didn't feel like including it in the formula, possibly for typographical convenience.
I pulled up some notes I have and it looks like the author chose to include what you pointed out, except in a different style,
$$\hat \beta\pm t_{n-2}(1-\alpha/2)s_{\hat\beta_i}.$$
The formula is not calculating the same thing as yours, but as you can see, it is possible to include the degrees of freedom in the notation.