How do you infer about the population from a sample? This how-to is the bulk of my question.
Now, do you need just one sample to infer about the population? Or do you need a multitude of samples? Do you need the sample mean and standard deviation? Or do you need the mean of sample means and the standard deviation of the standard deviations amongst these multitude of samples? Every reference I find leaves this point open to assumption.
Do you then need a large sample size? Is that relevant? Or do you need a large sample set of samples? I mean, is having a large multitude of small samples sufficient? Or is that the same as fewer samples of a larger size?
Suppose then you had the population standard deviation. According to my reference, you can use the standard normal distribution and a z-score test to infer about the population. But it doesnt say how.
Likewise, if your sample is large enough (or if the quantity of samples is large enough, Im not sure), you can also infer about a population from a sample. This time you dont need pop standard deviation, but you do need the sample standard deviation (or the standard deviations of the standard deviations of the samples, Im not sure which).
But if you dont know your population standard deviation and your sample size is small, then use the t-test. But my reference also doesnt tell me how to do that either... up until now there was no mention of the student t-distribution.
What exactly are you infering about the population? Whether you use the z-score or the t-score test, either way, you still need to know something about the population... be it the standard deviation or the mean. So it seems to me that you already need to have tallied up the population in order to make any sense of the samples deviation. What exactly are you inferring about the population from the sample?.. When you already know these particular variables about the population anyway, what is the point?
What I mean to say is... if you know neither the population standard deviation NOR the population mean... but you only have one (or more) samples... THEN how do you infer about the population?