17.7 Comparing exact and approximate areas

Armed with knowledge of obtaining exact areas, let’s return to Example 17.5:

Example 17.6 (Using normal distributions) Suppose heights of Australian adult males have a mean of \(\mu=175\)cm, and a standard deviation of \(\sigma=7\)cm, and (approximately) follow a normal distribution. Using this model, what proportion are shorter than 160cm?

The general approach to computing probabilities from normal distributions is:

  • Draw a diagram: Mark on 160 cm (Fig. 17.5).
  • Shade the required region of interest: ‘less than 160 cm tall’ (Fig. 17.5).
  • Compute the \(z\)-score using Equation (17.1).
  • Use the \(z\) tables in Appendix B.2.
  • Compute the answer.

The number of standard deviations that 160cm is from the mean is using Equation (17.1):

\[\begin{align*} z &= \frac{x-\mu}{\sigma} \\[3pt] &= \frac{160-175}{7} = \frac{-15}{7} = -2.14. \end{align*}\] That is, 160cm is 2.14 standard deviations below the mean, so use \(z=-2.14\) in the tables. The diagram at the top of the tables reminds us that this is the probability (area) that the value of \(z\) is less than \(z=-2.14\) (Fig. 17.5). The probability of finding an Australian man less than 160cm tall is about 1.6%.

More complicated questions can be asked too, as shown in the next section.