## 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

$\bar{d} \pm 2 \times\text{s.e.}(\bar{d}).$ This is the same as the CI for $$\bar{x}$$ 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.