Question

In a study we expect that the sample mean will vary from the population mean . This can either occur because ??of 1) sampling variation or because 2) the population mean was different than previously expected. How can we decide between these two options?

Answer 1

In a study, we expect that the sampling mean will vary from the population mean.

Now, this can occur either because of the sampling variation, or also because the population mean could have been different from what was previously expected.

Now, to decide between these two options, we have to use the knowledge that the sampling distribution of the sample mean will have mean same as the population mean.

So, we would take all samples that are possible to take from the population, and create the sampling distribution of the sample means. Even if taking all possible samples is impractical, we have to take as many as possible, so that at least the Central Limit Theorem can be implemented.

We note the mean, ie. the expected value of this sampling distribution. This value is nothing but the population mean, as the sample mean is an unbiased estimator of the population mean, ie. expected value of sampling mean is the population mean.

What comes as the expected value is the population mean.

If it is same as the population mean decided previously, then the variation in sampling mean, which in fact is a sample statistic, is due to the sampling variation. If the expected value is different from the population mean expected, we have to check our decision regarding the population mean.