## 27.6 Making decisions with \(P\)-values

\(P\)-values tells us the likelihood of observing the sample statistic
(or something more extreme),
based on the **assumption** about the population parameter being true.
In this context,
the \(P\)-value tells us the likelihood of
observing the value of \(\bar{x}\)
(or something more extreme), just through sampling variation (chance)
if \(\mu=37\).
The \(P\)-value is a probability,
albeit a probability of something quite specific,
so it is a value between 0 and 1.
Then:

‘Big’ \(P\)-values mean that the sample statistic (i.e., \(\bar{x}\)) could reasonably have occurred through sampling variation, if the assumption about the parameter (stated in \(H_0\)) was true (Fig. 27.7, top panel): The data

**do not contradict**the assumption (\(H_0\)).‘Small’ \(P\)-values mean that the sample statistic (i.e., \(\bar{x}\)) is unlikely to have occurred through sampling variation, if the assumption about the parameter (stated in \(H_0\)) was true: (Fig. 27.7, bottom panel): The data

**contradict**the assumption.

What is meant by ‘small’ and ‘big?’
It is *arbitrary*: no definitive rules exist.
Commonly,
a \(P\)-value smaller than 1% (that is, smaller than 0.01)
is usually considered ‘small,’
and
a \(P\)-value larger than 10% (that is, larger than 0.10)
is usually considered ‘big.’
Between the values of 1% and 10% is often a ‘grey area.’

Traditionally,
a \(P\)-value is ‘small’ if it is less than 5%
(less than 0.05),
and ‘big’ if greater than 5%
(greater than 0.05).
However,
again this is *arbitrary*,
and
binary decision making (*either* big *or* small)
is unreasonable.
More reasonably,
\(P\)-values should be interpreted
as providing varying strength of evidence in support of
the alternative hypothesis \(H_1\)
(Table 28.1).
These are not definitive,
but are only guidelines.
Of course,
conclusions should be written in the context of the problem.

For *one-tailed tests*,
the \(P\)-value is *half* the value of the two-tailed \(P\)-value.

`Sig.`

,
and sometimes explicitly notes that they are two-tailed.
For the body-temperature data then,
where \(P<0.001\),
the \(P\)-value is *very* small,
so there is *very strong evidence*
that the
population mean body temperature is not \(37.0^\circ\text{C}\).