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 ˉx (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 μd (the population mean difference);
  • with a standard deviation of s.e.(ˉd)=sdnd,
when certain conditions are met, where n is the size of the sample, and sd is the standard deviation of the individual differences in the sample.

For the home insulation data, the variation in the sample mean differences ˉd can be described by

  • approximate normal distribution;
  • centred around μd;
  • with a standard deviation of s.e.(ˉd)=1.01565510=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.