Figure 11-13. Raw scores scaled to normal probability density function.
Normal distribution with known variance σ^2
If the variance of the population is known then use the central limit theorem to
construct the following confidence interval for the mean μof the population based
upon a 1 −α=γlevel of confidence
CON F {x−
√σ
nzα/^2 ≤μ≤x+
√σ
nzα/^2 } (11 .81)
Normal distribution with unknown variance σ^2
In the case where the population variance is unknown,
then make use of the fact that t=|x−μ|
s/
√
n
follows a stu-
dent t-distribution to construct a confidence interval.
From the student t-distribution determine the value
tα/ 2 ,n − 1 based upon n− 1 degrees of freedom, nbeing
the sample size, such that the right tailed area under
equals α/ 2 as illustrated in the accompanying figure.
Some examples for a sample size of n= 11 and degrees of freedom n−1 = 10 are given
in the following table.
1 −α .90 .95 .99 .999
α .10 .05 .01 .001
α/ 2 .05 .025 .005 .0005
tα/ 2 , 10 1.812 2.228 3.169 4.144
