27.5 P-values: One mean
This is the decision-making progress so far:
- Assume that the population mean is 37.0∘C (this is H0).
- 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 ˉx 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 H0 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.

FIGURE 27.4: Computing P-values for the body temperature data
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 μ=37.0∘C, observing a sample mean as low as 36.8051∘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∘C.

FIGURE 27.5: jamovi output for conducting the t-test for the body temperature data

FIGURE 27.6: SPSS output for conducting the t-test for the body temperature data
SPSS always produces two-tailed P-values,
calls then Significance values,
and labels them as Sig.