One-tailed and two-tailed hypothesis tests

OK, so you're on the right lines with your thinking, but there's an important misapprehension lurking in here, so let's correct that!

You're concerned with a statistical model where you're assuming that the observations you're collecting are drawn from a probability distribution with unknown, but fixed, parameters, and you're interested in drawing some conclusion about those population parameters from those observations. So, important point: the underlying population parameters are fixed: they are not random variables, so you can't make probability statements about them. What is random are the observations that you have drawn, and any statistics that you calculate from them.

Let's think about a single population parameter, and let's call it , and let's assume it represents the population mean. From your sample of observations, if you want to make inferences about the population mean, a good guess is that we should look at the sample mean as a statistic, and see how it compares with our idea of what the underlying population mean might be. For the range of probability distributions you're used to, it's possible to work out theoretically what the sampling distribution of the sample mean is, given a particular value of the population mean. That is, the distribution of sample mean values you'd get if you repeatedly draw samples from the underlying population, and calculated their means.

So suppose we hypothesize that the population parameter , and we want to use the sample mean to decide whether this is sensible. We know the sampling distribution of the sample mean, given that , so we can work out probability statements about the sample mean, on the assumption that the population mean is .

The basic strategy is that you work out the probability that the value of the sample mean is "as extreme, or more extreme" as its observed value relative to . In other words, if the observed value of the sample mean is "far away" from , you infer that this is unlikely, and therefore reject the hypothesis that .

Now that's the "simple" case, where you assume that the underlying population parameter takes a particular value. More complicated is where you assume that the underlying population parameter takes a range of possible values: , for example. But the idea is the same: if you assume this, what's the probability of seeing the observed value of the sample mean?

Nearly: you are finding the probability that the sample mean takes the value it does (or more extreme one way), under the assumption that the population mean is . You reject that assumption if the probability is too small. In this case, "too small" is often taken to be 5%.

Again, you are finding the probability that the sample mean takes the value it does (or more extreme two ways), under the assumption that the population mean is . In this case, you're interested in whether the sample mean is much more than , or much less, so you need to take account of two tails of the distribution.

Yes, this is what you'd do (only you never "accept H0", you "do not reject H0" - remember that it's quite possible that you don't have enough evidence to reject H0 even when it is wrong - absence of evidence is not evidence of absence).

There are a couple of points here:

In a two-tailed test, you are implicitly doing two statistical tests; you are testing whether the test statistic is "too large", and you are testing if it is "too small". If you remember what the level of a test is, it's the probability of rejecting H0 when it is in fact true. If you do two tests, then you've got two chances to reject H0, so you have to add up the probabilities of rejecting a true H0 for both tests. For a one-sided test, you only have one chance, so you can "spend" all of your in one go.

The other point is that the assumptions underlying a two-sided test are different from those underlying a one-sided test. For a one-sided test you are allowing your underlying population parameter a much wider range of possibilities, compared to a two sided test. This affects the whole probability structure of your problem. So when you say that with that value of 3.2% you'd reject H0 in one case, and you would not reject it in the other, the point is that they are different H0's, with different assumptions. So you're not accepting a particular H0 in one case and rejecting it in the other, you're accepting or rejecting different statements