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8-2 CONFIDENCE INTERVAL ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE KNOWN 251

this interval with probability 0.95. However, with a little reflection, it’s easy to see that this can-
not be correct; the true value of is unknown and the statement 63.84 65.08 is either
correct (true with probability 1) or incorrect (false with probability 1). The correct interpretation
lies in the realization that a CI is a random interval because in the probability statement defin-
ing the end-points of the interval (Equation 8-4), L and U are random variables. Consequently,
the correct interpretation of a 100(1 )% CI depends on the relative frequency view of prob-
ability. Specifically, if an infinite number of random samples are collected and a 100(1 )%
confidence interval for is computed from each sample, 100(1 )% of these intervals will
contain the true value of .
The situation is illustrated in Fig. 8-1, which shows several 100(1 )% confidence
intervals for the mean of a normal distribution. The dots at the center of the intervals indi-
cate the point estimate of (that is, ). Notice that one of the intervals fails to contain the true
x
value of . If this were a 95% confidence interval, in the long run only 5% of the intervals
would fail to contain .
Now in practice, we obtain only one random sample and calculate one confidence interval.
Since this interval either will or will not contain the true value of , it is not reasonable to attach
a probability level to this specific event. The appropriate statement is the observed interval [l, u]
brackets the true value of with confidence 100(1 ). This statement has a frequency inter-
pretation; that is, we don’t know if the statement is true for this specific sample, but the method
used to obtain the interval [l, u] yields correct statements 100(1 )% of the time.

Confidence Level and Precision of Estimation
Notice in Example 8-1 that our choice of the 95% level of confidence was essentially
arbitrary. What would have happened if we had chosen a higher level of confidence, say, 99%?
In fact, doesn’t it seem reasonable that we would want the higher level of confidence? At
0.01, we find z 2 z 0.01 2 z 0.005 2.58, while for 0.05, z 0.025 1.96 . Thus, the
length of the 95% confidence interval is

211.96 2n2 3.92 2n

whereas the length of the 99% CI is

212.58 2n2 5.16 2n

Thus, the 99% CI is longer than the 95% CI. This is why we have a higher level of confidence
in the 99% confidence interval. Generally, for a fixed sample size n and standard deviation ,
the higher the confidence level, the longer the resulting CI.

µ

Figure 8-1 Repeated
construction of a con- 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
fidence interval for . Interval number