#econ #stats 
how do I do a hypothesis test if my hypothesis is that the difference is 0? Is it two hypothesis tests for each direction?  or a hypothesis test for the absolute value? How do I pick the arbitrary point? Or do I just run the hypothesis that it's not zero and let my null be that it's zero? I just feel like that's disingenuous.

In scikit learn, there's a function that allows for automated linear regression. Does it do a linear regression on the full population, or does it assume that the population is just a sample, and do the estimated regression. They're both super similar, but one of them is an n-1. Does it even matter for large samples like n=1000?

$\bar X$ vs $\mu_x$ vs $\mu_{\hat x}$ vs $\hat X$
X bar is the mean of the random variable X, which is the same as the mean of the population and the mean of the sample means, right? Which symbol do I use for the mean of the sample means, for a single sample mean, or the population mean? Is there a difference between the hat and the bar? I'm just conflating it all.
Ok it looks like the hat specifically means that it's an estimator???

**Yes**, the $\hat{hat}$ literally means it's an estimator. The $\bar{bar}$ in $\bar x$ means that it's simply the most common estimator for the mean of that variable. In other words,  $\hat \mu = \bar X$. Often, the lowercase version of a greek letter signifies an alternate estimator. $\hat\sigma^2$ is a good estimator of the standard deviation($\sigma^2$), but $s^2$ is better, since it's unbiased.
Capital X, Y, etc is a Random Variable. lowercase x, y, etc means the actual numbers. For example $\bar x = 17$ and $s^2=41^2$.
btw, $s^2$ is better than $\sigma^2$ because it's the *unbiased* approximation.
And, $\hat \mu = \bar X$