I am reading a book about linear regression in which there is a plot showing the regression line and two curves they call a confidence band (also can be seen here). As I learn something last week that a set of samples may result in a statistical quantity called the confidence level which used to describe the "precision" of estimation of the population. However, here shows two curves of confidence level at each variable X of the fitting line. It is quite confusing why there will be a curve instead of a single number. My best guess is at each X, there could be tons of measurement so many $\hat{Y}$'s returned, and there will be confidence level for those $\hat{Y}$'s. If this is correct, why there are local minima of confidence curves. The local minima indicate there is some point at which you do the measurement, results will be more "precise"? I don't see how this happen.
My second question is how to compute the confidence curves. Though many software like R and Mplus work out the problem easily, I want to know how to compute that by hand calculation. I know the thread states the algebra and I borrow the math here
$$ s_{\hat{Y}_{X}} = s_{Y|X}\sqrt{\frac{1}{n}+\frac{\left(X-\bar{X}\right)^{2}}{\sum_{i=1}^{n}{\left(X_{i}-\bar{X}\right)^{2}}}} $$
$$ s_{Y|X} = \sqrt{\frac{\sum_{i=1}^{n}{\left(Y_{i}-\hat{Y}\right)^{2}}}{n-2}} $$
$$ \hat{Y} \pm t_{\nu=n-2, \alpha/2}s_{\hat{Y}} $$
I am trying to understand those expressions and notation. My understanding is $X$ is the independent variable we choose for the fitting line and $X_i$ is the independent variable from data. $\hat{Y}$ is the measurement on the fitted line when $X$ is given while $Y_i$ is the measurement from data. I don't understand what $Y$ stands for. From the math, it seems that we compute something called $S_{\hat{Y}_X}$ from data and fitted values then use the very last formula to compute the curve. In the last expression, the $t$ is the student distribution and $\alpha$ related to the confidence level we choose and I assume there is a typo that $S_\hat{Y}$ should be $S_{\hat{Y}_X}$?
Very last question. I don't remember which book I read before but there is an example to introduce a cross term in the regression model something like
$$ Y = a_0 + a_1X + a_2TX $$
Where $Y$, $X$ and $T$ are given as data and by regression, $a_0$, $a_1$ and $a_2$ are computed. But instead of plotting Y again X, the author concerns the plot $Z=a_2T$ against $T$ only. In this case, if we want to plot the confidence band as well, should I just repeat the same calculation but replace $\hat{Y}$ with $\hat{Z}$, $X$ with $T$ and $\overline{X}$ with $\overline{T}$?
