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 μ=175cm,
and a standard deviation of σ=7cm,
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):
z=x−μσ=160−1757=−157=−2.14. 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.