Worksheet on The Standard Normal Distribution
(Also known as the z-distribution)
The standard normal distribution is the normal distribution whose mean is 0 and the
standard deviation is 1. The total area under the curve is 1, representing the fact that the
sum of all probabilities is 1. The probability that a randomly selected number x is
between two values (say a and b) is given by the area under the curve between a and b.
With this worksheet, we will learn how to find the probability that a selected number x is
between two values (i.e., finding p(a<x<b)) using a table.
A. Identifying the Parts of the Curve: The Body and The Tail
B = body region = 0 < z < 1.3
An interval from 0 to z is called
a body region.
if z is positive
T = tail region = z > 1.3 region = [-z,0] if
z is negative
The part that is leftover on the end is
called a tail region.
One body region and its tail represent half of the area under the normal curve:
Body + Tail = z > 0 = 50%
PROBLEMS
1. Sketch the region given by the interval 0 < z <1. Is it a body region or a tail region?
(0 < z < 1) =
2. Sketch the region given by the interval z > 2. Is it a body region or a tail region?
(z > 2) =
B. Identifying the Probability that x is in a Body Region
The probability that x is in a Body region
0< z < z* is given by z* = p(0<z<z*) that
can be read from the body table for the
standard normal distribution. This is
located in Appendix F of your book.
PROBLEMS
1. What percent of the standard normal z-distribution lies between z=0.00 and z=1.10?
Body Region = 0.00 < z < 1.10
p(0.00 < z < 1.10) =
2. What percent of the standard normal z-distribution lies between z=0 and z=1.12?
Body Region =
p(0.00 < z < 1.12) =
3. What percent of the standard normal z-distribution lies between z=0 and z=1.15?
Body Region =
p(0.00 < z < 1.15) =
4. What percent of the standard normal z-distribution lies between z=0 and z=1.21?
Body Region =
p(0.00 < z < 1.21) =
5. What percent of the standard normal z-distribution lies between z=0 and z=2.00?
Body Region =
p(0.00 < z < 2.00) =
C. Identifying the Probability that x is in a Tail Region
The probability that x is in a tail region z
> z* is (.5 - z*) where z* is the
probability of the body region p(0<z<z*).
The probability of the body region is
given in the body table for the standard
normal distribution.
PROBLEMS
1. What percent of the standard normal z-distribution lies in the body region between
z=0.00 and z=1.00?
Body Region =
p(0.00 < z < 1.00) =
2. What percent of the standard normal z-distribution lies in the tail region z > 1.00?
Tail Region =
p(z > 1.00) =
3. What percent of the standard normal z-distribution lies in the tail region between
z=1.18 and z=?
Tail Region =
p(z > 1.18) =
Body region =
D. Identifying the Probability that x is in a Composite Region
The probability that x is in an interval [a,b] will be a sum of the body and tail regions
contained in the interval [a,b].
Step 1: Shade in the interval on a sketch of the normal distribution curve.
Step 2: Identify the body and tail regions contained in the interval.
Step 3: Add the probabilities for each body and tail region contained in the interval,
keeping in mind two facts:
a) a region to the left of the mean will have the same probability as the region of the
same width on the right hand side due to symmetry.
b) A body and tail region together on one side have a probability of .5.
PROBLEMS
1. What percent of the standard normal z-distribution lies in the composite region
between z=-1 and z=1?
p(-1<z<1) =
2. What percent of the standard normal z-distribution lies in the composite region
defined by z < 2?
p(z<2) =