27.5 P-values: One mean

This is the decision-making progress so far:

  1. Assume that the population mean is 37.0C (this is H0).
  2. Based on this assumption, describe what to expect from the sample means (Fig. 27.2).
  3. 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.

Computing $P$-values for the body temperature data

FIGURE 27.4: Computing P-values for the body temperature data

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).

When software reports a P-value of 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=37.0C, observing a sample mean as low as 36.8051C 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 37.0C.

jamovi output for conducting the $t$-test for the body temperature data

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

SPSS 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.

jamovi can produce one- or two-tailed P-values.