23.6 Sampling distribution: Means differences

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.