Second, the type of sampling that we use is crucial.(Yes, we can think about our test’s “robustness” to these assumptions, but that’s a different issue.) For instance, we might assume that the population values follow a Normal distribution, with a known variance, but a mean that is unknown and whose value we’re testing. First, notice that there’s this idea that we know (or can reasonably assume) quite a lot about the underlying population.However, there are some key aspects to this hypothesis testing story that I’ve highlighted above, and which are worth remembering when we consider (below) a different possible way of proceeding. Now, you probably didn’t need to be reminded of all of this. The answer to this question is, of course, the familiar p-value. Or – by asking, “if the null hypothesis really were true, how likely is it that I’d see a value for my test statistic that’s as extreme (or more extreme) than what I’ve actually observed here (again, taking into account the assumptions about the underlying population and the sampling method that was used). If not, we would not reject the null hypothesis. If it is surprising, we would reject the null hypothesis.Then, we ask – “if in fact the null hypothesis were actually true, is the value of our test statistic “surprising”, or not?.Typically, this estimator would have to be transformed ( e.g., “standardized”) to make it “pivotal” – that is, having a sampling distribution that does not depend on any other unknown parameters. Usually, this would be a statistic that had already been found to be a “good” estimator of the parameter under test. We combine the sample values into a single statistic.For example, we might use simple random sampling, so that all sample values are mutually independent of each other. Then we take a carefully constructed sample from the population of interest.We call this the “alternative hypothesis”. We also state clearly what situation will prevail if the hypothesis to be tested is not true.
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