Advanced High-School Mathematics

382 CHAPTER 6 Inferential Statistics

therefore the random variable Z = Xσ/−√μn is normally distributed with
mean 0 and standard deviation 1. The valuesz≈± 1 .96 are the values
such that a normally-distributed random variableZ with mean 0 and
variance 1 will satisfyP(− 1. 96 ≤Z≤ 1 .96) = 0.95; see figure below

In other words, we have

P(− 1. 96 ≤

X−μ

σ/

√

n

≤ 1 .96) = 0. 95.

We may rearrange this and write

P(X− 1. 96

σ

√

n

≤μ≤X+ 1. 96

σ

√

n

).

Once we have calculated the meanxof nindependent samples, we

obtain a specific interval



x− 1. 96 √σ

n

,x− 1. 96

σ

√

n



which we call the

95% confidence interval for the mean μ of the given population.
Again, it’s important to realize that once the sample has been taken
and x has been calculated, there’s nothing random at all about the
above confidence interval: it’s not correct that it contains the true
meanμwith probability 95%, it either does or it doesn’t!

Of course, there’s nothing really special about the confidence level
95%—it’s just a traditionally used one. Other confidence levels fre-
quently used are 90% and 99%, but, of course, any confidence level