## 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.
\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%.