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.