27.2 Hypotheses and notation: Assumption

The decision making process begins by assuming that the population mean internal body temperature is ${37.0}^{\circ }\text{C}$.

The sample mean $\overline{x}$ is likely to be different for every sample (sampling variation). The sampling distribution of $\overline{x}$ describes how the value of $\overline{x}$ varies from sample to sample. Because $\overline{x}$ varies, the sample mean $\overline{x}$ probably won’t be exactly ${37.0}^{\circ }\text{C}$, even if $\mu$ is ${37.0}^{\circ }\text{C}$.

If $\overline{x}$ is not ${37.0}^{\circ }\text{C}$, two broad reasons could explain why:

1. The population mean body temperature is ${37.0}^{\circ }\text{C}$, but $\overline{x}$ isn’t exactly ${37.0}^{\circ }\text{C}$ due to sampling variation (that is, the sample mean varies and is likely to be different in every sample); or
2. The population mean body temperature is not ${37.0}^{\circ }\text{C}$, and the sample mean body temperature reflects this.

These two possible explanations are called hypotheses. More formally, the two hypotheses above are:

1. The null hypothesis ($H_0$): $\mu ={37.0}^{\circ }\text{C}$; the population mean body temperature is ${37.0}^{\circ }\text{C}$; and
2. The alternative hypothesis ($H_1$): $\mu \ne {37.0}^{\circ }\text{C}$; the population mean body temperature is not ${37.0}^{\circ }\text{C}$.

Since the null hypothesis is assumed true, the evidence is evaluated to determine if it is supported by the data, or not.

Note that the alternative hypothesis asks if $\mu$ is ${37.0}^{\circ }\text{C}$ or not: the value of $\mu$ may be smaller or larger than ${37.0}^{\circ }\text{C}$. Two possibilities are considered: for this reason, this alternative hypothesis is called a two-tailed alternative hypothesis.