29.3 Sampling distribution: Expectation

Initially, assume that μd=0. However, the sample mean energy saving will vary depending on which sample is randomly obtained, even if the mean saving in the population is zero: the sample mean energy saving has sampling variation and hence a standard error. The sampling distribution of ˉd can be described, which describes what values of the statistic might be expected in the sample if the μd=0.

Answering the RQ requires data. The data should be summarised numerically (using your calculator or software, such as jamovi (Fig. 29.1) or SPSS (Fig. 29.2)), and graphically (Fig. 23.1).

jamovi output for the insulation data

FIGURE 29.1: jamovi output for the insulation data

SPSS output for the insulation data

FIGURE 29.2: SPSS output for the insulation data

The sample mean difference is ˉd=0.5400… but this value of ˉd will vary from sample to sample even if μd=0. The amount of variation in ˉd is quantified using the standard error. More precisely, the possible values of the sample mean differences can, under certain conditions, described using

  • an approximate normal distribution; with
  • a mean of μd=0 (taken from H0); and
  • a standard deviation (called the standard error) of s.e.(ˉd)=sd/n=0.3212, where sd is the standard deviation of the differences.

This describes what can be expected from the possible values of ˉd (Fig. 29.3), just through sampling variation (chance) if μd=0.

The sampling distribution of sample means, if the energy saving in the population really was zero

FIGURE 29.3: The sampling distribution of sample means, if the energy saving in the population really was zero