The Essentials of Biostatistics for Physicians, Nurses, and Clinicians

5.3 Confi dence Intervals 63

construct the interval. Since ZX=−()/(/)()/ˆˆμσn nX= −μσ, we

have PnX[ 1.96−≤ − ≤ =(ˆ μσ)/ 1.96] 0.95. We invert this probability

statement about Z into a probability statement about μ falling inside an

interval as follows:

PnX P nX

nPX

[ 1.96 ( )/ 1.96] [ 1.96 / ( )

1.96 / ] [ 1

−≤ − ≤ =− ≤−

≤=−−

ˆˆ

ˆ

μσ σ μ

σ ..96 /σμnX n≤− ≤− +ˆ 1.96 /σ].

Then multiplying all three sides of the inequality in the probability
statement by − 1, we have PX[/ˆ+≥≥− = 196. σμn Xˆ 196. σ/]n 095..
This probability statement can be interpreted as the interval
[/ XnXnˆ−+ 196 .,σσˆ 196. /] is a two - sided 95% confi dence interval
for the unknown parameter μ. We can calculate the endpoints of this
interval since σ is known. However, in most practical problems σ is an
unknown nuisance parameter. For n very large, we can use the sample
estimate S for the standard deviation in place of σ and calculate the
endpoints of the interval in the same way.
If the sample size is small, then Z is replaced by TnX S=−()/ˆ μ.
This statistic T has a Student t - distribution with n − 1 degrees of
freedom. But then to make the same statement with 95% confi dence
the normal percentile value of 1.96 must be replaced by the correspond-
ing value from the t distribution with n − 1 degrees of freedom. From
a table for the central t - distribution that can be found in many
text books (Chernick and Friis ( 2003 , p. 371), we see for n − 1 = 4, 9,
14, 19, 29, 40, 60, 120, we have the comparable t - percentile C = 2.776,
2.262, 2.145, 2.093, 2.045, 2.021, 2.000, 1.980. As the degrees
of freedom get larger, C approaches the normal percentile of 1.960.
So between 40 and 60, the approximation by the normal is pretty
good.
The cartoon in Figure 5.3 illustrates the concept visually. In an
experiment like the one shown there, since the confi dence interval is a
95% two - sided interval, and the true parameter value is 0.5, we would
expect 19 intervals to include 0.5 and 1 to miss. But this too is subject
to variability. In the example above, all 20 intervals included 0.5,
although one almost missed. If we repeated this experiment indepen-
dently, we could get 19, 18, 17, or all 20 intervals containing 0.5. It is
theoretically possible for a number smaller than 17 to include 0.5 but
that would be highly unlikely.