We have just learned that we can estimate the likelihood of drawing a random sample with a particular mean from a population, when we know the population parameters. This is the z-test. With this, we take the first step in inferential statistics, decision-making about hypotheses based on probability.

The problem with the z-test is that we don’t usually know the population parameters. After all, this is usually why we are undertaking the research in the first place! What we need, then, is a way to estimate the parameters.

We usually don’t know the population standard deviation, sigma. As a result, we can’t calculate the standard error using the formula with which we are familiar.

When we collect data, we have information about the sample. Not knowing anything more, this is the best information we have about the population. We can use the sample standard deviation to estimate the population standard deviation, and therefore, the standard error.

We will use the following formula to estimate the standard error:

We can rewrite the z-test, using the estimated standard error:

We will call this the t-ratio, instead of z, because the sampling distribution upon which it is based is ever-so-slightly different than the normal distribution.

With the t-ratio, we can generate an estimate of the population mean — remember, we usually don’t know this when we conduct a study. We refer to this as the confidence interval, and define it as:

This is our best estimate of the interval into which the population mean, mu, falls.

At last! Our first example of how useful this inferential stuff (probability, sampling distributions, standard scores) is.