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124 CHAPTER 4 PROBABILITY DISTRIBUTIONS
approximately normally distributed. Consequently, even though no variable encountered
in practice is precisely normally distributed, the normal distribution can be used
to model the distribution of many variables that are of interest. Using the normal distribution
as a model allows us to make useful probability statements about some variables
much more conveniently than would be the case if some more complicated model
had to be used.
Human stature and human intelligence are frequently cited as examples of variables
that are approximately normally distributed. On the other hand, many distributions
relevant to the health field cannot be described adequately by a normal distribution.
Whenever it is known that a random variable is approximately normally distributed, or
when, in the absence of complete knowledge, it is considered reasonable to make this
assumption, the statistician is aided tremendously in his or her efforts to solve practical
problems relative to this variable. Bear in mind, however, that “normal” in this context
refers to the statistical properties of a set of data and in no way connotes normality in
the sense of health or medical condition.
There are several other reasons why the normal distribution is so important in statistics,
and these will be considered in due time. For now, let us see how we may answer
simple probability questions about random variables when we know, or are willing to
assume, that they are, at least, approximately normally distributed.
EXAMPLE 4.7.1
The Uptimer is a custom-made lightweight battery-operated activity monitor that records
the amount of time an individual spends in the upright position. In a study of children
ages 8 to 15 years, Eldridge et al. (A-10) studied 529 normally developing children who
each wore the Uptimer continuously for a 24-hour period that included a typical school
day. The researchers found that the amount of time children spent in the upright position
followed a normal distribution with a mean of 5.4 hours and standard deviation of 1.3
hours. Assume that this finding applies to all children 8 to 15 years of age. Find the probability
that a child selected at random spends less than 3 hours in the upright position in
a 24-hour period.
Solution:
First let us draw a picture of the distribution and shade the area corresponding
to the probability of interest. This has been done in Figure 4.7.1.
s = 1.3
3.0 m = 5.4
FIGURE 4.7.1 Normal distribution to approximate
distribution of amount of time children spent in upright
position (mean and standard deviation estimated).
x