Question

Researchers studying anthropometry collected body girth measurements

and skeletal diameter measurements, as well as age, weight, height and gender, for 507 physically

active individuals. The histogram below shows the sample distribution of heights in centimeters.

(a) What is the point estimate for the average height of active individuals? What about the

median?

(b) What is the point estimate for the standard deviation of the heights of active individuals?

What about the IQR?

(c) Is a person who is 1m 80cm (180 cm) tall considered unusually tall? And is a person who is

1m 55cm (155cm) considered unusually short? Explain your reasoning.

(d) The researchers take another random sample of physically active individuals. Would you

expect the mean and the standard deviation of this new sample to be the ones given above.

Explain your reasoning.

(e) The samples means obtained are point estimates for the mean height of all active individuals,

if the sample of individuals is equivalent to a simple random sample. What measure do we use

to quantify the variability of such an estimate? Compute this quantity using the data from

the original sample under the condition that the data are a simple random sample.

Answer 1

(a)

The point estimate for the average height of active individuals is,

$$ \text { mean, } \bar{x}=171.1 $$

The median of active individuals is,

$$ \text { Median }=170.3 $$

(b)

The point estimate for the standard deviation of the heights of active individuals is,

$$ S D=9.4 $$

The Inter quartile range of the heights is,

$$ \begin{aligned} I Q R &=Q_{3}-Q_{1} \\ &=177.8-163.8 \\ &=14 \end{aligned} $$

(c)

Is a person who is \(1 \mathrm{~m} 80 \mathrm{~cm}(180 \mathrm{~cm})\) tall considered unusually tall?

Data beyond two standard deviations away from the mean is considered "unusual" data.

$$ \begin{aligned} (\text { Mean } \pm 2 S D) &=(171.1-(2 \times 9.4), 171.1+(2 \times 9.4)) \\ &=(171.1-18.8,171.1+18.8) \\ &=(152.3,189.9) \end{aligned} $$

\(180 \mathrm{~cm}\) is within the 2 standard deviation limits, so the height \(180 \mathrm{~cm}\) is not considered as unusually tall.

\(155 \mathrm{~cm}\) is within the 2 standard deviation limits, so the height \(155 \mathrm{~cm}\) is not considered as unusually short.

(d)

The mean and standard deviation are not exactly the same. There is a very high probability that another sample would yield different, but similar results, given a sample size of 507 persons. This is inherent to random sampling and random distributions.

(e)

The standard error of the height is,

$$ \begin{aligned} S E &=\frac{9.4}{\sqrt{507}} \\ &=0.417 \end{aligned} $$