## 27.2 Hypotheses and notation: One mean

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

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

If $$\bar{x}$$ 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 $$\bar{x}$$ 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.