## 29.2 Hypotheses and notation: Mean differences

The RQ asks if the mean energy saving in the population is zero or not. The parameter is the population mean difference. To make things clear, notation is needed (recapping Sect. 23.3):

• $$\mu_d$$: The population mean difference.
• $$\bar{d}$$: The sample mean difference.
• $$s_d$$: The sample standard deviation of the differences.
• $$n$$: The number of differences.
• $$\text{s.e.}(\bar{d})$$: The standard error of the mean differences, where $$\displaystyle \text{s.e.}(\bar{d}) = \frac{s_d}{\sqrt{n}}$$.

The hypotheses, therefore, can be written in terms of the parameter $$\mu_d$$. The null hypothesis is ‘there is no change in the energy consumption, in the population’:

• $$H_0$$: $$\mu_d=0$$.

As noted in Sect. 28.2, the null hypothesis states that there is ‘no difference, no change, no relationship,’ as measured by a parameter value. This hypothesis, the initial assumption, postulates that the mean difference may not be zero in the sample due to sampling variation.

Since the RQ asks specifically if the insulation saves energy, the alternative hypothesis will be a one-tailed hypothesis:

• $$H_1$$: $$\mu_d>0$$ (one-tailed).

This hypothesis says that the mean energy saving in the population is greater than zero. The alternative hypothesis is one-tailed because of the wording of the RQ. Recall that the differences are defined as energy savings.