Chapter 6: Normal Distributions
Since normal distributions play a large role in statistics, we will talk mainly
about normal distributions in this chapter.
The normal distributions are symmetric, single peaked, bell-shaped density
curves. All normal distributions have the same overall shape. The exact
density curve for a particular normal distribution is described by giving its
mean and its standard deviatin .
The mean is located at the center of the symmetric curve and is the same as
median. Changing without changing moves the normal curve to a new
location without altering its spread.
The standard deviation controls the spread of a normal curve. Example 6.1: If the random variable Z has the standard normal
() 7 5P.(Z1
() 7 5P.(Z1
Top-Hat Question (Review 6-1):
(c) P(1.43 Z 1.37) Example 6.2: If the random variable Z has the standard normal
distribution, locate the value of z that satisfies
(a) P(Z z ) 0.1736
(b) P(Z z) 0.025
Top-Hat Question (Review 6-2):
(c) P(z Z z) 0.90 Y YMean
The standardized variable Z has mean 0 and
standard deviation 1. It does not depend on the unit of measurement.
IfY has a normal distribution with mean of and standard deviation of ,
then Z has the standard normal distribution.
Example 6.3: If the random variable Y has a normal distribution
of 60 and standard deviation of 4 , finP(55 63). Example 6.4: Replacement times for CD players are normally distributed
with a mean of 7.1 years and a standard deviation of 1.4 years. Find the