1. Intro

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:

  1. State the null and alternative hypotheses.
  2. Calculate the test statistic, which tells us how “different” our sample results are from what is being claimed in the null hypotheses.
  3. 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.
  4. 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.