226 Fundamentals of Statistics
standard error. The calculated value will then follow a standard normal dis-
tribution; that is,
x2m
s@!n,N^1 0, 1^2
This value is called a z test statistic. We can then compare the z test sta-
tistic to z values taken from a standard normal distribution. A z value, usu-
ally written as zP, is the point z on a standard normal curve such that for
random variable Z, P 1 Z#zP 25 p. For example, z0.95 5 1. 6 45 because 95%
of the area under the curve is to the left of 1.645. See Figure 6-1.Figure 6-1
The z valuez0.95 = 1.645Figure 6-1 shows a one-sided z value, but for confi dence intervals, we’re
more interested in a two-sided z value, where p is the probability of the
value falling in the center of the distribution and a (which equals 1 2 p)
is the probability of its falling in one of the two tails. For a two-sided range
of size p, these z values are 2 z 12 a/ 2 andz 12 a/ 2. In other words, for a ran-
dom variable Z, P 12 z 1 2a/ 2 ,Z,z 1 2a/ 2251 2a5p. If we want to find
the central 95% of the standard normal curve, p 50. 9 5, a5 0. 0 5, and
z 12 0.05/ 25 z0.9 755 1. 96. This means that 95% of the values on a standard
normal curve lie between 2 1.96 and 1.96. See Figure 6-2.