Introductory Biostatistics

approximation; in practice,n¼25 or more could be considered adequately
large). This means that we have the two properties

mx¼m

s^2 x¼

s^2

n

as seen in Example 4.1.
The following example shows how goodxis as an estimate for the popula-
tionmeven if the sample size is as small as 25. (Of course, it is used only as an
illustration; in practice,mands^2 are unknown.)

Example 4.2 Birth weights obtained from deliveries over a long period of time
at a certain hospital show a meanmof 112 oz and a standard deviationsof
20.6 oz. Let us suppose that we want to compute the probability that the mean
birth weight from a sample of 25 infants will fall between 107 and 117 oz (i.e.,
the estimate is o¤ the mark by no more than 5 oz). The central limit theorem is
applied and it indicates thatxfollows a normal distribution with mean

mx¼ 112

and variance

s^2 x¼

ð 20 : 6 Þ^2

25

or standard error

sx¼ 4 : 12

It follows that

Prð 107 axa 117 Þ¼Pr

107
112

4 : 12

aza

117
112

4 : 12



¼Prð
 1 : 21 aza 1 : 21 Þ

¼ð 2 Þð 0 : 3869 Þ

¼ 0 : 7738

In other words, if we use the mean of a sample of sizen¼25 to estimate the
population mean, about 80% of the time we are correct within 5 oz; this figure
would be 98.5% if the sample size were 100.

ESTIMATION OF MEANS 153