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Chapter 9
Statistics
Section 9.3
The Normal Distribution
Continuous Distributions

Recall that a histogram and its
corresponding frequency polygon can be
constructed from information obtained from
a frequency distribution or a probability
distribution.
Continuous Distributions

A bank manager collects data to determine the
amount of time to the nearest minute tellers
spend on each transaction.
Continuous Distributions

A distribution in which the outcomes can
take any real number value within some
interval is a continuous distribution.

The graph of a continuous distribution is a
curve.

Distributions whose peak is not at the
center are called skewed.
Normal Distributions
Many natural and social phenomena produce
continuous probability distributions whose
graphs can be approximated by bell-shaped
curves.
 These kinds of distributions are called normal
distributions and their graphs are called normal
curves.
 For a normal distribution, the Greek letter µ
(mu) is used to denote the mean, and σ
(sigma) is used to denote the standard
deviation.

Examples of Normal Distributions
The smaller the standard deviation, the taller and narrower
the curve will be.
The larger the standard deviation, the wider and more flat
the curve will be.
Properties of Normal Distributions

The peak occurs directly above the mean µ.

The curve is symmetric about the vertical line through
the mean.

The curve never touches the x-axis – it extends
indefinitely in both directions.

The area under the curve (and above the horizontal
axis) is always 1. (Sum of the probabilities in a
probability distribution is always 1.)
Determining Probabilities of a
Normal Distribution
Determining Probabilities of a
Normal Distribution
To use normal curves effectively, we must be
able to calculate areas under portions of these
curves.
 These calculations have already been done for
the normal curve with mean µ = 0 and
standard deviation σ = 1. (This is the
standard normal curve.
 The table of these calculations is found in the
Appendix of your textbook.

Area Under a Normal Curve Table
Standard Normal Curve

The horizontal axis of the standard normal
curve is usually labeled z.

When calculating normal probability,
always draw a normal curve with the
mean and z-scores every time.
Example 1

Find the percent of the area under a normal
curve between the mean and the given
number of standard deviations from the
mean.
a.) 1.87
b.) -0.95
Example 2

Find the percent of the total area under the
standard normal curve between each pair of
z-scores.
a.) 1.05 and 2.46
b.) -2.15 and 1.17
Example 3

Find a z-score satisfying the following
conditions.
a.) 45% of the total area is to the left of z.
b.) 20% of the total area is to the right of z.
Important!!
The key to finding areas under any normal
curve is to express each number x on the
horizontal axis in terms of a standard
deviation above or below the mean.
 The z-score for x is the number of
standard deviations that x lies from the
mean (positive if x is above the mean,
negative if x is below the mean).

Converting a Data Value X
to a Z-score
Importance of Z-scores
By converting data values to z-scores and using the table
for the standard normal curve, we can find areas under any
normal curve.
Since the areas are probabilities, we can now how handle
a variety of a applications.
FUN!! 
Applications of the Standard
Normal Curve
Example 4

A certain type of light bulb has an average life
of 500 hours, with a standard deviation of
100 hours. The length of life of the bulb can
be closely approximated by a normal curve.
An amusement park buys and installs 10,000
such bulbs. Find the total number that can
be expected to last for the following periods
of time
a.) at least 500 hours
b.) between 650 and 780 hours
Example 5

A machine that fills quart orange juice
cartons is set to fill them with 32.1 oz. If the
actual contents of the cartons vary normally,
with a standard deviation of 0.1 oz, what
percent of the cartons contains less than a
quart (32 oz)?
Example 6

On standard IQ tests, the mean is 100, with
a standard deviation of 15. The results are
very close to fitting a normal curve. Suppose
an IQ test is given to a very large group of
people. Find the percent of those people
whose IQ scores are more than 130.