![]() ![]() ![]() (That’s what matters – not that we have a standardized distribution!) However, if we transform x* into z = (x* – μ) / (σ / √(n)), this new statistic has a sampling distribution that is N. Using x* as our test statistic sounds like a good idea, but its sampling distribution is N, so it depends on the unknown value of μ. Then, you’ll recall that the sample average, x* is the most efficient unbiased estimator of μ. ![]() Suppose that the underlying population is N, where σ 2 is known, and we obtain a sample of n observations using simple random sampling. Let’s consider a simple, specific, example. Finally, we have to be able to precisely specify the null and alternative hypotheses ( in advance of seeing the sample data), and these hypotheses have to relate to some parameter(s) in the population. ![]() If we mess up with either of these two things, then the sampling distribution of our test statistic will differ from what we think it is, and the probabilities that are associated with the implementation of Step 6 will be wrong. In reality, much (most) of the data that we use have been obtained using “complex surveys”, involving multi-level cluster sampling and stratified sampling. Usually, simple random sampling is assumed.
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