## 6.3 Normal distributions and heights

In the (NNS) (conducted with the 1995 *National Health Survey* (NHS)), trained nutritionists measured the heights of the respondents (Australian Bureau of Statistics 1995).

The values quoted are used for modelling purposes, and are essentially the population means \(\mu\) and standard deviations \(\sigma\).

- Take a wild guess: What percentage of your peers (people your age and gender) do you think would be
*shorter*than you? _______ - Write down your own height: ________
- From Fig. 6.1, write down the mean and standard deviation for people in your
*age group*and*gender*.

Using this information, compute the \(z\)-score for your height, relative to people in your age group and gender. Explain what this means.

- Heights have an approximate normal (bell-shape) distribution, so the \(68\)--\(95\)--\(99.7\) rule applies.

**Using the \(68\)--\(95\)--\(99.7\) rule**,*roughly*estimate the percentage of people of your age and gender that are*shorter*than you. (Hint: Draw a picture!) - Using
**the normal distribution tables**, together with the information from**Fig. 6.1**, compute the percentage of people of your age and gender that are*shorter*than you. - Using the \(68\)--\(95\)--\(99.7\) rule, the middle \(95\)% of females \(18\) and over (using the
**Fig. 6.2**) are within what height range?

For this part, we want to just look at

**females over \(18\)**; from**Fig. 6.2**, for this group, the mean is \(\mu = 161.4\) cm, and the standard deviation is \(\sigma = 6.7\) cm. Use this information to answer the following questions.- Compute the
*percentage*of females \(18\) and over that are*shorter*than \(171\) cm. - What are the
*odds*that a female \(18\) and over is*shorter*than \(171\) cm? - Compute the
*percentage*of females \(18\) and over that are*taller*than \(171\) cm? - What are the
*odds*that a female \(18\) and over is*taller*than \(171\) cm? - Compute the
*percentage*of females \(18\) and over that are*between*170 and \(180\) cm? - What are the
*odds*that a female \(18\) and over is*between*than \(170\) and \(180\) cm?

- Compute the
For this part, use the data in

**Fig. 6.2**. Approximately \(20\)% of females \(18\) and over are shorter than what height?

### References

Australian Bureau of Statistics. How Australians measure up [Internet]. Australian Bureau of Statistics; 1995. Report No.: 4359.0. Available from: http://www.ausstats.abs.gov.au/.