## 23.6 Means differences: Sampling distribution

The study concerns the mean energy saving (the mean difference). Every sample of $$n = 10$$ houses is likely to comprise different houses, and hence different before and after energy consumptions will be recorded, and hence different energy savings will be recorded. As a result, the sample mean energy differences will vary from sample to sample. That is, the mean differences have a sampling distribution, and a standard error.

Since the differences are like a single sample of data (Chap. 22), the sampling distribution for the differences will have a similar sampling distribution to the mean of a single sample $$\bar{x}$$ (provided the conditions are met; Sect. 23.9).

Definition 23.2 (Sampling distribution of a sample mean difference) The sampling distribution of a sample mean difference is described by:

• an approximate normal distribution;
• centred around $$\mu_d$$ (the population mean difference);
• with a standard deviation of $$\displaystyle\text{s.e.}(\bar{d}) = \frac{s_d}{\sqrt{n_d}}$$,
when certain conditions are met, where $$n$$ is the size of the sample, and $$s_d$$ is the standard deviation of the individual differences in the sample.

For the home insulation data, the variation in the sample mean differences $$\bar{d}$$ can be described by

• approximate normal distribution;
• centred around $$\mu_d$$;
• with a standard deviation of $$\displaystyle\text{s.e.}(\bar{d}) = \frac{1.015655}{\sqrt{10}} = 0.3211784$$, called the standard error of the differences.

Notice that many decimal places are used in the working here; results will be rounded when reported.