In this sub-competency we learn how to carry out a hypothesis test for a parameter from just one population. We will learn two techniques, one for investigating a population proportion (*p*) and one for investigating a population mean (*μ*). Recall from sub-competency 9 that there are four main steps in the process:

- State the null and alternative hypotheses.
- Calculate the
*test statistic*, which tells us how “different” our sample results are from what is being claimed in the null hypotheses. - Determine the probability and significance level of our sample producing the results that it did. In other words, determine the probability, or likelihood, that a population with a parameter equal to the value stated in the null hypothesis would include a sample with the characteristic(s) of our sample.
- Make and write the appropriate conclusions based upon the level of significance α for the hypothesis test.

The main computations will occur in Steps 2 and 3 above. If we are testing a population proportion, we’ll use the basic *z*-score as our test statistic, as it compares a sample result with the (claimed) population value *relative to the standard deviation of the sample*. To spoil all the fun, our test statistic is:

A major difference occurs when we study a population mean. Most of you will need to return to sub-competency 8 (confidence intervals) where the *t*-distribution was introduced. Although population proportions follow standard normal distributions, population means, on the other hand, follow a somewhat wider distribution called the *t*-distribution. Therefore, our test statistic will be:

Both of these values tell us how different, in terms of the standard deviation, our sample statistic is from the claimed value of the population parameter.

In sub-competency 8 we first created confidence intervals about a population proportion and then we modified the formulas slightly to create a confidence interval about a population mean. We follow the same path with hypothesis tests.