SECTION 6.4 Confidence Intervals 383
could be used. To form a 90% confidence interval from a measured
meanx, we would replace the number 1.96 used above with the value
ofzfor which random samples from a normal population with mean 0
and standard deviation 1 would lie between±z99% of the time. Here,
it turns out thatz≈ 2 .58:
In general, the (1−α)×100% confidence interval for the mean is
obtained by determining the valuezα/ 2 such that a normally-distributed
random variableZof mean 0 and standard deviation 1 will satisfy
P(−zα/ 2 ≤Z≤zα/ 2 ) = 1−α.Below are tabulated some of the more traditional values:
Confidence Relevant
Level z-value
( 1 −α) α zα/ 2
0.90 0.10 1.645
0.95 0.05 1.960
0.98 0.02 2.326
0.99 0.01 2.576In summary, the (1−α)×100% confidence interval for the mean is
formed from the sample meanxby constructing