## 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 reductionin 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*).

*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.