These examples bring us to the concept of the null hypothesis and alternative hypothesis in a hypothesis test. Whenever we conduct a test, these two opposing hypotheses are always involved.
The null hypothesis is the hypothesis that the population parameter assumes some particular value. This hypothesis is typically denoted H0.
In this chapter we will be looking at tests for the population mean of a numerical variable and the population proportion for a categorical variable. So in this chapter, the null hypothesis can really only take on two different forms. It will either look something like this, for a population mean:
H0: μ = 60
or something like this, for a population proportion:
H0: π = 0.5
We often express the null hypothesis like the above. We start with a 'H0:', to let everyone know we are about to announce the null hypothesis, and then we end with a claim like 'μ = 60'. It clearly and succinctly puts everything in the claim we are testing into one equation.
As we will see, once a hypothesis test has been conducted, we will either say that the sample provides enough evidence to reject the null hypothesis (and so we conclude that the claim proposed is not true) or that the sample does not provide enough evidence to reject the null hypothesis. Notice that we never ever say that there is enough evidence to conclude that the null hypothesis is true. (But more on that soon!)
The alternative hypothesis is the hypothesis that the population parameter assumes some set of values other than the one value assigned in the null hypothesis. This hypothesis is typically denoted HA.
The alternative hypothesis is sometimes thought of as the 'opposite' claim to the null hypothesis. And this is sometimes true. However, the alternative hypothesis does not necessarily need to be the precise opposite of the null hypothesis. Its defining feature is that it is the claim that we accept if, after we conduct the hypothesis test, we reject the null hypothesis. That is, it is the statement that may be considered 'proven' if the sample provides enough evidence to reject the null hypothesis.
For this reason, the exact nature of the alternative hypothesis will depend on why the test is being conducted in the first place. The following two examples demonstrate two different kinds of alternative hypotheses.
Fair coin - alternative hypothesis
When Fred is testing the fairness of the coin he is testing the null hypothesis that π = 0.5. For him, the alternative hypothesis is simply that the coin is not fair, that is, that π ≠ 0.5. He would write this as:
HA: π ≠ 0.5
Battery life - alternative hypothesis
The consumer watchdog may have been put on to the case of the battery company because someone suspected that the batteries have a lifetime that is less than the advertised 60 hours. In this case the alternative hypothesis would be written as:
HA: μ < 60
Note that, in each case, the alternative hypothesis is a reflection of the specific assertion that is made if the null hypothesis is found to be 'wrong'. In any test, the null hypothesis is taken to be a default value for a population parameter. In the example of the coin, the null hypothesis is that the coin is fair. In the example of the batteries, the null hypothesis is that the manufacturer is correct in the advertised claim of 60 hours.
Each of these null hypotheses may be found to be wrong. And for the coin example, Fred aims to conclude that the coin simply isn't fair, which is why his alternative hypothesis is that π is not equal to 0.5. For the battery example, the watchdog is interested to see if there is evidence that the average lifetime is lower than the advertised lifetime. So its alternative hypothesis is that μ is less than 60.
The above two examples indicate the two different kinds of alternative hypotheses there are. If the alternative hypothesis asserts a simple inequality, that the population parameter is not equal to some value, like:
HA: π ≠ 0.5
then it is known as a two-sided alternative hypothesis and this would be known as a two-sided hypothesis test. But if the alternative hypothesis asserts that the population parameter is specifically greater than (or specifically less than) some value, like:
HA: μ < 60
or like:
HA: μ > 60
then it is known as a one-sided alternative hypothesis and this would be known as a one-sided hypothesis test.
Whether we are conducting a one-sided or two-sided hypothesis test will have an impact on precisely how we conduct the test, so it is important to work out what sort of test we are using.