6.2 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 \(\sigma\).

  1. Take a wild guess: What percentage of your peers (your age and gender) do you think would be shorter than you?
  2. Write down your own height.
  3. From Fig. 6.1, write down the mean and standard deviation for people in your age group and gender.
Part of the ABS report on heights of Australians

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

  1. Using this information, compute the \(z\)-score for your height, relative to people in your age group and gender. Explain what this means.
  2. 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!)
  3. 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.
  4. 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?
Part of the ABS report on heights of females Australians

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

  1. 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 171cm.
    • What are the odds that a female 18 and over is shorter than 171cm?
    • Compute the percentage of females 18 and over that are taller than 171cm?
    • What are the odds that a female 18 and over is taller than 171cm?
    • Compute the percentage of females 18 and over that are between 170 and 180cm?
    • What are the odds that a female 18 and over is between than 170 and 180cm?
  2. 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. 1995. “How Australians Measure up.” 4359.0. Australian Bureau of Statistics. http://www.ausstats.abs.gov.au/Ausstats/subscriber.nsf/Lookup/CA25687100069892CA256889001F4A36/$File/43590_1995.pdf.