23.2 Mean differences: An example
The Electricity Council in Bristol wanted to determine if a certain type of wall-cavity insulation reduced energy consumption in winter (The Open University 1983). Their (Descriptive) RQ was:
What is the mean reduction in energy consumption after adding home insulation?
The parameter is \(\mu_d\), the population mean reduction in energy consumption.
For the collected data (shown below) the same variable (energy consumption) is measured twice for each unit of analysis (the house): energy consumption before adding insulation and after adding insulation.
Finding the difference in energy consumption for each house seems sensible, as the data are paired. Once the differences are computed, the process for computing a CI is the same as in Chap. 22, where these changes (or differences) are used as the data.
Be clear about how the differences are computed. Differences could be computed as Before minus After (the energy consumption saving), or After minus Before (the energy consumption increase).
Either is fine, as long as you are consistent throughout. The meaning of any conclusions will be the same.Here, discussing energy savings seems most natural, so we compute the differences as energy savings: Before minus After.
One energy saving value is negative. This does not mean negative energy usage: the values are differences (more specifically, energy reductions or savings).The differences are computed as Before minus After, so a negative value means that the After value is greater than the Before value: an increase in energy consumption.
As always, begin by understanding the data: producing appropriate graphical and numerical summaries.