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):

$$\begin{array}{cc}z& =\frac{x-\mu }{\sigma }\\ & =\frac{160-175}{7}=\frac{-15}{7}=-\mathrm{2.14.}\end{array}$$ 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.