I've spent some time trying to understand what it is that statisticians are doing when they get a point estimate, and relate it back to some population parameter. I can do the calculations and interpret them, but I still don't really understand what underlies it.
Here is what I know, and where I'm stuck.
You are looking for some mean of something. In the true population, there will be a mean (µ) and a standard deviation from that mean (σ).
We obviously can't look at the entire population, but we take a sample of an amount of people (n) and for them, we find a mean (xbar) and standard deviation for them too (s).
Now, we want to try and estimate µ from our sample. xbar is our best estimate (the point estimate), but this will have some error. This is where I'm confused, this is why
- Our best estimate of the error is the standard error (the standard deviation of the sampling distributions), but I'm not sure why. Why do we use a sampling distribution at all when, in practice, we never take more than one sample?