27.3 Sampling distribution: One mean

A RQ is answered using data (this is partly what is meant by evidence-based research). Fortunately, for the body-temperature study, data are available from a comprehensive American study (Shoemaker 1996).

Summarising the data is important, because the data are the means by which the RQ is answered (data below).

A graphical summary (Fig. 27.1) shows that the internal body temperature of individuals varies from person to person: this is natural variation. A numerical summary (from software) shows that:

  • The sample mean is \(\bar{x} = 36.8051^\circ\)C;
  • The sample standard deviation is \(s = 0.40732^\circ\)C;
  • The sample size is \(n=130\).

The sample mean is less than the assumed value of \(\mu=37^\circ\text{C}\)… The question is why: can the difference reasonably be explained by sampling variation, or not?

A 95% CI can also be computed (using software or manually): the 95% CI for \(\mu\) is from \(36.73^\circ\) to \(36.88^\circ\)C. This CI is narrow, implying that \(\mu\) has been estimated with precision, so detecting even small deviations of \(\mu\) from \(37^\circ\) should be possible.

The histogram of the body temperature data

FIGURE 27.1: The histogram of the body temperature data

The decision-making process assumes that the population mean temperature is \(\mu=37.0^\circ\text{C}\), as stated in the null hypothesis. Because of sampling variation, the value of \(\bar{x}\) sometimes would be smaller than \(37.0^\circ\text{C}\) and sometimes greater than \(37.0^\circ\text{C}\).

How much variation in the value of \(\bar{x}\) could be expected, simply due to sampling variation, when \(\mu=37.0^\circ\text{C}\)? This variation is described by the sampling distribution.

The sampling distribution of \(\bar{x}\) was discussed in Sect. 22.2 (and Def. 22.1 specifically). From this, if \(\mu\) really was \(37.0^\circ\)C and if certain conditions are true, the possible values of the sample means can be described using:

  • An approximate normal distribution;
  • With mean \(37.0^\circ\text{C}\) (from \(H_0\));
  • With standard deviation of \(\displaystyle \text{s.e.}(\bar{x}) = \frac{s}{\sqrt{n}} = \frac{0.40732}{\sqrt{130}} = 0.035724\). This is the standard error of the sample means.

A picture of this sampling distribution (Fig. 27.2) shows how the sample mean varies when \(n=130\), simply due to sampling variation, when \(\mu = 37^\circ\text{C}\). This enables questions to be asked about the likely values of \(\bar{x}\) that would be found in the sample, when the population mean is \(\mu = 37^\circ\text{C}\).

The distribution of sample mean body temperatures, if the population mean is $37^\circ$C and $n=130$.  The grey vertical lines are 1, 2 and 3 standard deviations from the mean.

FIGURE 27.2: The distribution of sample mean body temperatures, if the population mean is \(37^\circ\)C and \(n=130\). The grey vertical lines are 1, 2 and 3 standard deviations from the mean.

Think 27.1 (Values of \(\bar{x}\)) Given the sampling distribution shown in Fig. 27.2, use the 68–95–99.7 rule to determine how often will \(\bar{x}\) be larger than 37.036 degrees C just because of sampling variation, if \(\mu\) really is \(37^\circ\)C.
About 16% of the time.

References

Shoemaker AL. What’s normal? – temperature, gender, and heart rate. Journal of Statistics Education [Internet]. 1996;4(2). Available from: http://jse.amstat.org/v4n2/datasets.shoemaker.html.