When you conduct a hypothesis test, the exact things that you do in the test will depend on some particular aspects of the test. For example, as we just saw, the critical values and region of rejection are affected by whether a test is one-sided or two-sided. And as we will see later in this chapter, when testing the population mean μ, our method will depend upon whether or not the population standard deviation σ is known.
However, across all kinds of hypothesis tests the same general principles apply and the same general steps are followed. So we will now provide an outline of the main components in the methodology of hypothesis tests. Later in this chapter, we will see this methodology in action as we test particular population parameters.
You begin a hypothesis test by deciding what null and alternative hypotheses are included in the test. It is important to write down the null and alternative hypotheses because it will make it clear what claim is being tested. That is, it will make clear what population parameter is being tested and the value that is being tested for it. It will also make it obvious whether the test is one-sided or two-sided.
The null hypothesis H0 will always state that the population parameter assumes some specific value. The alternative hypothesis HA will always state that the population parameter assumes some value other than this specific value. Here are two examples of null and alternative hypotheses that we have seen in this chapter.
An example of hypotheses for a two-sided test:
H0: π = 0.5
HA: π ≠ 0.5
An example of hypotheses for a one-sided test:
H0: μ = 60
HA: μ < 60
As we talked about in the previous section, an important part of a hypothesis test is to assume that the null hypothesis is true. In fact it is this assumption that allows you to describe the sampling distribution of the population parameter that is being tested, which in turn allows you to relate the sampling distribution to the standard normal distribution.
While this step doesn't require a lot of work, it is very important and you should get in the habit of writing the assumption down. So underneath the null and alternative hypotheses you might write down something like:
Assume π = 0.5.
As discussed in this section, α will be a number that reflects how much evidence you need to reject the null hypothesis. Being a probability, α will be between 0 and 1. The lower α is (that is, the closer it is to 0) the more evidence is required before you agree to reject the null hypothesis. The higher α is (that is, the closer it is to 1) the less evidence is required before you reject the null hypothesis.
Some common levels of significance α are 0.1, 0.05 and 0.01.
Mathematically, the way in which α reflects the level of evidence required is through the fact that α determines the critical value(s) and the region of rejection.
Once the level of significance α has been chosen, the critical value(s) can be determined. As we've seen in this section, there will either be two critical values or one critical value.
For a two-sided hypothesis test there are two critical values. These two values are the two z-scores: zα/2 and -zα/2.
For a one-sided hypothesis test there is one critical value. It will either be
In any case, critical values can be found by referring to the standard normal table or by using statistical software.
Once the critical value(s) have been determined, the region of rejection can be written down. As with critical values, the region of rejection will depend on whether the test is one-sided or two-sided. However, generally the region of rejection is the area 'outside' the critical values.
If there are two critical values (zα/2 and -zα/2) then the region of rejection is the set of values greater than zα/2 and values less than -zα/2.
If there is one critical value and it is positive (zα) then the region of rejection is the set of values greater than zα.
If there is one critical value and it is negative (-zα) then the region of rejection is the set of values less than -zα.
Of course, the point of all of this is to test a claim! The sample is how we test it. So you need to collect a sample and calculate the sample statistic corresponding to your population parameter. That is, if your test is about a population proportion then you need to calculate a sample proportion. If your test is about a population mean then you need to calculate a sample mean.
The sample statistic will be a value in the sampling distribution of the parameter. This sampling distribution will follow some normal distribution that is known once you assume that the null hypothesis is true. Therefore, the z-score of the sample statistic can be calculated. This z-score is the test statistic for the test.
Once the test statistic has been calculated, you can determine whether or not it is in the region of rejection. If the test statistic is in the region of rejection, the null hypothesis is rejected. If the test statistic is not in the region of rejection, the null hypothesis is not rejected.
A general guide?
The above step-by-step guide is a comprehensive summary of the methodology of hypothesis testing. However, one pragmatic assumption is being made: we are assuming that the sampling distribution is normal. As a consequence, we assume other things, like the fact that critical values are z-scores, and that the test-statistic is a z-score.
Hypothesis testing can be used to test a wide variety of population parameters, and it is true that the sampling distribution does not always follow the normal distribution. However, the above guide is general enough to serve us for this chapter. The only exception that will arise in this chapter is when we are testing the population mean when the population standard deviation is not known. When we get to that test, we will explain exactly how the test differs. (Rest assured that the methodology is exactly the same - only the numbers will change!)