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 \(\displaystyle\text{s.e.}(\bar{d}) = \frac{s_d}{\sqrt{n_d}}\),
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.