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.

FIGURE 6.1: Part of the ABS report on heights of Australians

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?

FIGURE 6.2: Part of the ABS report on heights of females Australians

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.
1. Compute the percentage of females \(18\) and over that are shorter than \(171\) cm.
2. What are the odds that a female \(18\) and over is shorter than \(171\) cm?
3. Compute the percentage of females \(18\) and over that are taller than \(171\) cm?
4. What are the odds that a female \(18\) and over is taller than \(171\) cm?
5. Compute the percentage of females \(18\) and over that are between 170 and \(180\) cm?
6. What are the odds that a female \(18\) and over is between than \(170\) and \(180\) cm?
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/.