23.7 Confidence intervals: Mean differences

The CI for the mean difference has the same form as for a single mean (Chap. 22), so an approximate 95% confidence interval (CI) for ${\mu }_d$ is

$$\overline{d}\pm 2\times \text{s.e.}\left(\overline{d}\right).$$ This is the same as the CI for $\overline{x}$ if the differences are considered as the data.

For the insulation data:
$$0.54\pm (2\times 0.3211784),$$ or $0.54\pm 0.642$. This CI is equivalent to $0.54-0.642=-0.102$, up to $0.54+0.642=1.182$. We write:

Based on the sample, an approximate 95% CI for the population mean energy saving after adding the wall cavity insulation is from $-0.10$ to $1.18$MWh.

The negative number is not an energy consumption value; it is a negative mean amount of energy saved. Saving a negative amount is like using more energy. So the 95% CI is saying that we are reasonably confident that, after adding the insulation, the mean energy-use difference is between using $0.10$MWh more energy to using $1.18$MWh less energy. Alternatively, the plausible values for the mean energy savings are between $-0.10$ to $1.18$MWh.