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:

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
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:

The null hypothesis (\(H_0\)): \(\mu=37.0^\circ\text{C}\); the population mean body temperature is \(37.0^\circ\text{C}\); and
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.