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#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