Introduction

At the beginning of this chapter we broke a hypothesis test up into three steps. These were:

Step 1: Specify a value to be tested for a population parameter and define the null hypothesis H₀ and the alternative hypothesis H_A.

Step 2: Collect a sample and use this to test the null hypothesis.

Step 3: Either reject the null hypothesis and conclude that the value specified for the population parameter is wrong, or do not reject the null hypothesis.

The section we've just finished covered the reasoning behind how Step 2 works. In brief, it works like this:

• assume that the claim being tested is true at the beginning of the test
• use sampling distributions to work out what this assumption means for samples that may be collected
• collect a sample and use it as evidence to test this assumption

This results in the decision in the final step, Step 3, above.

However, as with statistical estimation in Chapter 7, the above method cannot be complete without relating it all to the standard normal distribution, Z. For example, in the previous section we had a look at a hypothesis test for the fairness of a coin. At one stage we used the following handy fact:

95% of the normal distribution falls within 1.96 standard deviations of the mean

This enabled us to develop some decision-making power, in that we could collect a sample and then decide whether or not to reject the null hypothesis. But what if we want to run a hypothesis test at some other 'level' than 95%? Just as we did with estimation, we do this by relating the sampling distribution to the standard normal distribution. This is what we'll be doing in this section.

Always Z?

You may recall in Chapter 7 that we didn't always relate the sampling distribution back to the standard normal distribution. In particular, when trying to estimate the mean without knowing the population standard deviation, we used a different kind of standard distribution: the t-distribution. And, sure enough, the same will be true for testing.

So we are slightly over-simplifying the issue when we say that the sampling distribution will always be related to Z. However, for the purposes of demonstrating the general technique of hypothesis testing, for the moment we will always use Z. Later in this chapter, we will cover all of the details and considerations that arise when σ is unknown.