## 28.2 About hypotheses and assumptions

### 28.2.1 Null hypotheses

Hypotheses always concern a population parameter. Hypothesising, for example, that the sample mean body temperature is equal to $$37.0^\circ\text{C}$$ is pointless, because it clearly isn’t: the sample mean is $$36.8051^\circ\text{C}$$. Besides, the RQ is about the unknown population: the P in POCI stands for Population.

The null hypothesis $$H_0$$ offers one possible reason why the value of the sample statistic (such as the sample mean) is not the same as the value of the proposed population parameter (such as the population mean): sampling variation. Every sample is different, and so the sample statistic will vary from sample to sample; it may not be equal to the population parameter, just because of the sample used by chance. Null hypotheses always have an ‘equals’ in them (for example, the population mean equals 100, is less than or equal to 100, or is more than or equal to 100), because (as part of the decision making process), something specific must be assumed for the population parameter.

The parameter can take many different forms, depending on the context. The null hypothesis about the parameter is the default value of that parameter; for example,

• there is no difference between the parameter value in two (or more) groups;
• there is no change in the parameter value; or
• there is no relationship as measured by a parameter value.

Hypothesis testing starts by assuming that the null hypothesis is true.

The onus is on the data to provide evidence to refute this default position.

The null hypothesis is always about a population parameter, and always has the form ‘no difference, no change, no relationship.’
Definition 28.1 (Null hypothesis) The null hypothesis proposes that sampling variation explains the difference between the proposed value of the parameter, and the observed value of the statistic.

### 28.2.2 Alternative hypotheses

The other hypothesis is called the alternative hypothesis $$H_1$$. The alternative hypothesis offers another possible reason why the value of the sample statistic (such as the sample mean) is not the same as the value of the proposed population parameter (such as the population mean). The alternative hypothesis proposes that the value of the population parameter really is not the value claimed in the null hypothesis.

Definition 28.2 (Alternative hypothesis) The alternative hypothesis proposes that the difference between the proposed value of the parameter and the observed value of the statistic cannot be explained by sampling variation: the proposed value of the parameter is probably not true.`

Alternative hypotheses can be one-tailed or two-tailed. A two-tailed alternative hypothesis means, for example, that the population mean could be either smaller or larger than what is claimed. A one-tailed alternative hypothesis admits only one of those two possibilities. Most (but not all) hypothesis tests are two-tailed.

The decision about whether the alternative hypothesis is one- or two-tailed is made by reading the RQ (not by looking at the data). Indeed, the RQ and hypotheses should (in principle) be formed before the data are obtained, or at least before looking at the data if the data are already collected.

The ideas are the same whether the alternative hypothesis is one- or two-tailed: based on the data and the sample statistic, a decision is to be made about whether the alternative hypotheses is supported by the data.

Example 28.1 (Alternative hypotheses) For the body-temperature study, the alternative hypothesis is two-tailed: The RQ asks if the population mean is $$37.0^\circ\text{C}$$ or not. That is, two possibilities are considered: that $$\mu$$ could be either larger or smaller than $$37.0^\circ\text{C}$$.

A one-tailed alternative hypothesis would be appropriate if the RQ was: ‘Is the population mean internal body temperature greater than $$37.0^\circ\text{C}$$?’ or Is the population mean internal body temperature smaller than $$37.0^\circ\text{C}$$?.