27.2 Hypotheses and notation: One mean
The decision making process begins by assuming that the population mean internal body temperature is 37.0∘C.
The sample mean ˉx is likely to be different for every sample (sampling variation). The sampling distribution of ˉx describes how the value of ˉx varies from sample to sample. Because ˉx varies, the sample mean ˉx probably won’t be exactly 37.0∘C, even if μ is 37.0∘C.
If ˉx is not 37.0∘C, two broad reasons could explain why:
- The population mean body temperature is 37.0∘C, but ˉx isn’t exactly 37.0∘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∘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 (H0): μ=37.0∘C; the population mean body temperature is 37.0∘C; and
- The alternative hypothesis (H1): μ≠37.0∘C; the population mean body temperature is not 37.0∘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 μ is 37.0∘C or not: the value of μ may be smaller or larger than 37.0∘C. Two possibilities are considered: for this reason, this alternative hypothesis is called a two-tailed alternative hypothesis.