## 27.5 \(P\)-values: One mean

This is the decision-making progress so far:

**Assume**that the population mean is \(37.0^\circ\text{C}\) (this is \(H_0\)).- Based on this assumption,
describe what to
**expect**from the sample means (Fig. 27.2). - The
**observed statistic**is computed, relative to what is expected using a \(t\)-score (Fig. 27.3): \(t=-5.453\).

The value of the \(t\)-score shows that
the value of \(\bar{x}\) is highly unusual.
*How* unusual can be assessed more precisely
using a *\(P\)-value*,
which is used widely in scientific research.
The \(P\)-value is a way of measuring how unusual an observation is
(if \(H_0\) is true).

\(P\) values can be *approximated* using the
68–95–99.7 rule and a diagram
(Sect. 27.5.1),
but more commonly by using software
(Sect. 27.5.2).

### 27.5.1 Approximating \(P\)-values using the 68–95–99.7 rule

The \(P\)-value is
the area *more extreme* than the calculated \(t\)-score.
For example:

*If*the calculated \(t\)-score was \(t=-1\), the*two-tailed*\(P\)-value would be the shaded area in Fig. 27.4 (top panel): About 32%, based on the 68–95–99.7 rule. Because the alternative hypothesis is*two-tailed*,*both*sides of the mean are considered: the \(P\)-value would be the same if \(t=+1\).*If*the calculated \(t\)-score was \(t=-2\), the*two-tailed*\(P\)-value would be the shaded area shown in Fig. 27.4 (bottom panel): About 5%, based on the 68–95–99.7 rule. Because the alternative hypothesis is*two-tailed*,*both*sides of the mean are considered: the \(P\)-value would be the same if \(t=+2\).

Clearly, from what the \(P\)-value means, a \(P\)-value is always between 0 and 1.

**Think 27.2 (\(P\)-values)**What do you think the \(P\)-value will be for \(t=-5.45\) (using Fig. 27.3)?

### 27.5.2 Finding \(P\)-values using sofware

Software computes the
\(t\)-score and a precise \(P\)-value
(jamovi: Fig. 27.5;
SPSS: Fig. 27.6).
The output
(in jamovi, under the heading `p`

;
in SPSS, under the heading `Sig. (2-tailed)`

)
shows that the \(P\)-value is indeed very small.
Although SPSS reports the \(P\)-value as 0.000,
\(P\)-values can never be *exactly* zero,
so we interpret this as
‘zero to three decimal places,’
or that \(P\) is less than 0.001
(written as \(P<0.001\), as jamovi reports).

`0.000`

,
it really means (and we should write) \(P<0.001\):
That is, the \(P\)-value is *smaller*than 0.001.

This \(P\)-value means that,
assumpting \(\mu=37.0^\circ\)C,
observing a sample mean as low as \(36.8051^\circ\)C
just through sampling variation
(from a sample size of \(n=130\))
is almost *impossible*.
And yet, we did…

Using the
decision-making process,
this implies that the initial assumption
(the null hypothesis)
is contradicted by the data:
The evidence suggests that the
*population* mean body temperature is *not* \(37.0^\circ\text{C}\).

SPSS always produces **two-tailed** \(P\)-values,
calls then *Significance values*,
and labels them as `Sig.`