Introductory Biostatistics

4.2.1 Confidence Intervals for a Mean

Similar to what was done in Example 4.2, we can write, for example,

Pr
 1 : 96 a

x
m

s=

ffiffiffi

n

p a 1 : 96



¼ð 2 Þð 0 : 475 Þ

¼ 0 : 95

This statement is a consequence of the central limit theorem, which indicates
that for a large sample sizen,xis a random variable (in the context of repeated
sampling) with a normal sampling distribution in which

mx¼m

sx^2 ¼s^2 =n

The quantity inside brackets in the equation above is equivalent to

x
 1 : 96 s=

ffiffiffi

n

p

amaxþ 1 : 96 s=

ffiffiffi

n

p

All we need to do now is to select a random sample, calculate the numerical
value ofxand its standard error withsreplaced by sample variances,s=

ffiffiffi

n

p
,
and substitute these values to form the endpoints of the interval

xG 1 : 96 s=

ffiffiffi

n

p

In a specific numerical case this will produce two numbers,

a¼x
 1 : 96 s=

ffiffiffi

n

p

and

b¼xþ 1 : 96 s=

ffiffiffi

n

p

and we have the interval

aamab

But here we run into a logical problem. We are sampling from a fixed popula-
tion. We are examining values of a random variable obtained by selecting a
random sample from that fixed population. The random variable has a distri-
bution with meanmthat we wish to estimate. Since the population and the
distribution of the random variable we are investigating are fixed, it follows
that the parametermis fixed. The quantitiesm,a, andbare all fixed (after the
sample has been obtained); then we cannot assert that the probability thatmlies

154 ESTIMATION OF PARAMETERS