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 .

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 (people 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.
Using this information, compute the $$z$$-score for your height, relative to people in your age group and gender. Explain what this means.
1. 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!)
2. 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.
3. 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?
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.

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