For example, forg¼ 0 :05, then Z 1 ð 0 : 05 Þ¼ 1 : 96
and Z 2 ð 0 : 05 Þ¼ 1 :96.
As long asg< 0 :5, we can always find a value
ZðgÞ>0 such thatZ 1 ðgÞ¼ZðgÞandZ 2 ðgÞ¼
ZðgÞso that Equation (3) holds.
Now, by substituting the definition of Z in
expression (2) withZðgÞ¼Z 2 ðgÞ¼Z 1 ðgÞand
rearranging terms, the inequalityZ 1 ðgÞZZ 2
ðgÞcan be re-written as
Lg¼XnZðgÞs=
ffiffiffi
np
XnþZðgÞs=ffiffiffi
np
¼Ug
ð 4 ÞNow, let us take a closer look at expression (4). The
value at the center,m, is the population mean, which
is the unknown quantity we are estimating. The two
expressions on the right-hand and the left-hand
sides of (4) are variables calculated from the data.
Thus, expression (4) represents a random interval
containing the population meanm. Expression (3)
assigns a probability 1g that (4) holds. The
interpretation of this is that if we conduct an experi-
ment and calculate the lower and upper limits of the
interval,LgandUg, respectively, then the interval
(Lg,Ug) will contain the true (and unknown!) popu-
lation mean with probability 1g. The interval (4)
is called aconfidence intervalfor the population
mean, and 1gis called theconfidence levelof the
interval, often expressed as a percent.
Let us illustrate these ideas using the data of
Table 25.4. Suppose we wish to estimate the dif-
ferencebetween the population means of the
non-exercising and the exercising students by con-
structing a confidence interval with confidence
level 95%. Then substitutingDforXnand SED
fors=
ffiffiffi
np
in (4), and recalling that Zð 0 : 05 Þ¼
1 :96, we obtain the confidence limits
L 0 : 05 ¼DSEDZð 0 : 05 Þ¼ 5 : 2 1 : 105 1 : 96 ¼ 3 : 03 ;
and
U 0 : 05 ¼DþSEDZð 0 : 05 Þ
¼ 5 : 2 þ 1 : 105 1 : 96 ¼ 7 : 26 :Thus, the interval (3.03, 7.26) is a 95% confidence
interval for the effect. It should be emphasized
that the probability statement about the confidence
level of 0.95 does not relate to the specific interval
(3.03, 7.26), as this specific interval is an outcome
of the specific sample used for the calculation and
either contains the parameteror does not. It is a
theoretical probability pertaining to a generic inter-
val calculated from asample following the stepswe
described above. Thus, if we could repeat the
experiment many times, each time calculating a
confidence interval in the way we have just done,
we should expect approximately 95% of these
intervals to contain the true mean effect.Of
course, when calculating a confidence interval
from a sample, there is no way to tell whether or
not the interval contains the parameter it is estimat-
ing. The confidence level provides us with a certain
level of assurance that it is so, in the sense we just
described. One might ask, why not choosegto be a
very small number such as 0.01 or 0.001 and thus
obtain an arbitrarily large confidence level? One
can see from the way Z(g) is defined that it
increases asgdecreases. For example,Zð 0 : 01 Þ¼
2 :58 and Zð 0 : 001 Þ¼ 3 :25 which would corre-
spond to the confidence intervals (2.35, 8.05) and
(1.54, 8.86), respectively. So the answer becomes
self-evident: Yes, one can choose an arbitrarily
high confidence level but this will come at the
price that the resulting confidence interval will be
so wide that it becomes meaningless. In other
words, there is a tradeoff between confidence and
accuracy. It seems that 95% confidence achieves a
satisfactory balance between the two in most cases.
Confidence intervals are often calculated after
performing a statistical test. When the test is sta-
tistically significant, we have reason to believe that
the effect is real. The confidence interval gives us
additional information as to the size of the effect.
Confidence intervals are also calculated during
exploratory analyses. The purpose of such analyses
is to explore the data, identify possible effects and
generate hypotheses for future studies rather than
make specific inferences. Confidence intervals are
extremely useful tools toward this goal.
Another common use of confidence intervals is
in the establishment of equivalence between two
treatments. Here ‘equivalence’ is not synonymous
with ‘equality’. It means that the difference, if
any, between the effects of the two treatments is330 CH25 STATISTICAL PRINCIPLES AND APPLICATION IN BIOPHARMACEUTICAL RESEARCH