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](jamovi/InsulationBeforeAfter/InsulationNumSummary.png)
FIGURE 29.1: jamovi output for the insulation data
![SPSS output for the insulation data](SPSS/InsulationBeforeAfter/InsulationNumSummary.png)
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](29-Testing-MeanDifference_files/figure-html/EnergySamplingDist-1.png)
FIGURE 29.3: The sampling distribution of sample means, if the energy saving in the population really was zero