Chapter 5  Normal Probability Distributions
Section 5.1  Introduction to Normal Distributions
Objectives
•
Interpret graphs of normal probability distributions
•
Find areas under the standard normal curve
1)
Continuous random variable 
Has an infinite number of possible values that can be represented by an
interval on the number line.
2)
Continuous probability distribution 
The probability distribution of a continuous random variable.
3)
Normal distribution  (a)
A continuous probability distribution for a random variable,
x
.
(b)
The most
important continuous probability distribution in statistics.
(c)
The graph of a normal distribution is called the
normal curve
.
4)
Properties of Normal Distributions

1.
The mean, median, and mode are equal.
2.
The normal curve is bellshaped and symmetric about the mean.
3.
The total area under the curve is equal to one.
4.
The normal curve approaches, but never touches the
x
axis as it extends farther and farther away from the
mean.
5.
Between μ – σ and μ + σ (in the center of the curve), the graph curves downward.
The graph curves upward
to the left of μ – σ and to the right of μ + σ.
The points at which the curve changes from curving upward to
curving downward are called the
inflection points
.
5)
Probability density function (pdf)
is a function that graphs the probability distribution of a continuous
random variable.
Pdf for an normal curve is:
6)
Means and Standard Deviations for Normal Distributions 
(a)
A normal distribution can have any
mean and any positive standard deviation.
(b)
The mean gives the location of the line of symmetry.
(c)
The
standard deviation describes the spread of the data.
Example#1 & Try It Yourself
1  (p. 241)
Look at the normal curves like in the
margin of page 241 & ex. #1.
Which curve has the greatest mean?
Which curve has the greatest standard deviation?
{Look at normal curves
in Excel spread sheet 
Normal Graph
Illustration
}.
a.
Where is the line of symmetry in the graphs?
b.
Which graph is more spread out and why?
Example#2 & Try It Yourself
2  (p. 242)
Look at the normal curves like in the
examples on page 242.
Note
line of symmetry, inflection points and estimate the standard deviation (and mean).
{look at normal curves in
excel}.
a.
Where is the line of symmetry in the graphs?
b.
Estimate the inflection points and the distance from the mean to where the curve is at the xaxis to
estimate the standard deviation.
7)
Standard normal distribution

A normal distribution with a mean of 0 and a standard deviation of 1.
8)
Properties of the Standard Normal Distribution  1.
The cumulative area is close to 0 for
z
scores close
to
z
=

3.49.
2.
The cumulative area increases as the
z
scores increase.
3.
The cumulative area for
z
= 0 is
0.5000.
4.
The cumulative area is close to 1 for
z
scores close to
z
= 3.49.
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Use zscore to transform
Normal Distribution to Standard Normal distributions.
Area under the Standard
Normal Curve gives you probabilities of zvalues or cumulative probabilities of the zvalues.
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