betweenaandbis 0.95. In fact, eithermlies inða;bÞor it does not, and it is
not correct to assign a probability to the statement (even the truth remains
unknown).
The di‰culty here arises at the point of substitution of the numerical values
observed forxand its standard error. The random variation inxis variation
from sample to sample in the context of repeated sampling. When we substitute
xand its standard errors=
ffiffiffi
np
by their numerical values resulting in interval
ða;bÞ, it is understood that the repeated sampling process could produce many
di¤erent intervals of the same form:
xG 1 :96SEðxÞAbout 95% of these intervals would actually includem. Since we have only one
of these possible intervals, the intervalða;bÞfrom our sample, we say we are
95% confident thatmlies between these limits. The intervalða;bÞis called a
95% confidence intervalform, and the figure ‘‘95’’ is called thedegree of confi-
denceorconfidence level.
In forming confidence intervals, the degree of confidence is determined by
the investigator of a research project. Di¤erent investigators may prefer di¤er-
ent confidence intervals; the coe‰cient to be multiplied with the standard error
of the mean should be determined accordingly. A few typical choices are given
in Table 4.4; 95% is the most conventional.
Finally, it should be noted that since the standard error is
SEðxÞ¼s=ffiffiffi
npthe width of a confidence interval becomes narrower as sample size increases,
and the process above is applicable only to large samples (n>25, say). In the
next section we show how to handle smaller samples (there is nothing magic
about ‘‘25’’; see the note at the end of Section 4.2.2).
Example 4.3 For the data on percentage saturation of bile for 31 male
patients of Example 2.4:
40 ; 86 ; 111 ; 86 ; 106 ; 66 ; 123 ; 90 ; 112 ; 52 ; 88 ; 137 ; 88 ; 88 ;
65 ; 79 ; 87 ; 56 ; 110 ; 106 ; 110 ; 78 ; 80 ; 47 ; 74 ; 58 ; 88 ; 73 ; 118 ;
67 ; 57TABLE 4.4
Degree of Confidence Coe‰cient
99% 2.576
!95% 1.960
90% 1.645
80% 1.282ESTIMATION OF MEANS 155