Download Section 8.5 Normal Distributions

Section 8.5 Normal Distributions
Density curve: the smooth curve identifying the shape of a
distribution.
Basic Properties of Density Curves
Property 1: A density curve is always on or above the
horizontal axis.
Property 2: The total area under a density curve (and above
the horizontal axis) equals 1
Variables and Their Density Curves
For a variable with a density curve, the percentage of all
possible observations of the variable that lie within any
specified range equals (at least approximately) the
corresponding area under the density curve, expressed as a
percentage.
60% of the data is
within the values of
A and B
0.2
1
0.1
A
B
This section will focus on the most important density curve-the
normal density curve. The normal density curve has a
symmetric “bell-shaped” curve
Normal distribution: is a distribution represented by a normal
curve.
What is important about the normal distribution?
It can be used as a more convenient approximation for a
variable X which we will be exploring in this section
How can we express the normal curve as a distribution for X?
1. As a smooth approximation to a histogram that is based
upon a sample of X.
2. As an idealized representation for the population
distribution of X by knowing µ-the population mean and σthe population standard deviation.
If a variable X follows normal distribution, then it can be
represented by X ~ N   ,   . N represents that the distribution
is normally distributed,  represents the population mean, and
 represents the population standard deviation. All normal
curves can be described by this single formula.
If a variable X follows a normal distribution with mean  and
standard deviation  , then the density curve of the distribution
of X is given by the formula
f (x) 
1

2
e
1 x 
 

2  
2
Where
 f ( x ) is called the density function


e

is a constant about 2.71
is a constant about 3.14
µ-3σ
µ-2σ µ-σ
µ
µ+σ
µ+2σ µ+3σ
Example 1
Sketch the normal distribution with
a. µ = -2 and σ = 2
b. µ = -2 and σ = 1/2
c. µ = 0 and σ = 2
Can we look at these distributions in our calculator?
Yes
Graphing Procedure 1
1. In your graphing mode go to
2. Y1=
3. Press 2 VARS →DISTR
4. You will see
DISTR
DRAW
5. Press 1:normalpdf(
6. Press x,-2, 2 )
You will see the first distribution.
You may need to adjust your window the y-values will not be
negative and the maximum y-value will not be larger than 1 for
these graphs
We looked at relative frequency histograms and density curves
to find the probabilities that our X-values were between a
certain range. Now we will be applying that same sort of idea
with our normal curve.
Normally Distributed Variables and Normal-Curves Areas
For a normally distributed variable, the percentage of all
possible observations that lie within any specified range equals
the corresponding area under its associated normal curve,
expressed as a percentage. This result holds approximately for
a variable that is approximately normally distributed.
Example 2
The area under a particular normal curve between 10 and 15 is
0.6874. A normally distributed variable has the same mean
and standard deviation as the parameters for this normal
curve. What percentage of all possible observations of the
variable lie between 10 and 15? Explain your answer.
The most common normal curve X ~ N   ,   is the standard
normal curve Z ~ N  0 ,1  There are tables made up so we can
find the area under curve if we have a z-table.
Standard Normal distribution: Standard Normal Curve
A normally distributed variable having mean 0 and standard
deviation 1 is said to have the standard normal distribution.
Its associated normal curve is called the standard normal
curve.
Basic Properties of the Standard Normal Curve
Property 1: The total area under the standard normal curve is
1.
Property 2: The standard normal curve extends indefinitely in
both directions, approaching but never touching, the horizontal
axis as it does so.
Property 3: The standard normal curve is symmetric about 0;
that is, the part of the curve to the left of the dashed line is the
mirror image of the part of the curve to the right of it.
Property 4: Almost all the area under the standard normal
curve lies between -3 and 3.
-3
-2
-1
0
1
2
3
Procedure 1: Expressing the range in terms of z-scores
and finding the corresponding area.
Step 1: Draw a standard normal curve.
Step 2: Label the z-score(s) on the curve.
Step 3: Shade in the region of interest.
Step 4: Determining the corresponding area under the
standard normal curve using table in Appendix B (pg A18).
-3
-2 -1
0
1
2
3
Example 3
is the standard normal distribution. Find the indicated
probabilities.
Z
a. P  0
 Z  1 .5 
b. P   1 . 3 
c. P   1 . 64
Z  0
 Z  2 . 35
d. P  Z
 1 . 82

e. P  
2 .6  Z


Can we do this on our calculator?
Sort of, but we need to draw our picture(s) so we make sure
that we know what area we are interested in
Graphing Procedure 2
Method to find some of the areas in our calculator
1. Press 2 VARS →DISTR
2. You will see
DISTR
DRAW
3. Press 2:normalcdf(
4. Enter in first value , Enter in second value )
5. Press Enter
We can standardize any normally distributed variable(s) so we
can find the area under the curve
Standardized Normally Distributed Variable Z
The standardized version of a normally distributed variable
Z 
x  

has the standard normal distribution.
X
µ-3σ
-3
µ-2σ µ-σ
-2
-1
µ
0
µ+
σ
1
µ+2σ µ+3σ
Z
2
3
X
,
Procedure 2: To Determine a Percentage or Probability for
a Normally Distributed Variable
Step 1: Sketch the normal curve associated with the variable
Step 2: Shade the region of interest and mark its delimiting xvalue(s).
Step 3: Find the z-score(s) for the delimiting x-value(s) found
in Step 2.
Step 4: Use the standard normal curve in Appendix B to find
the area under the standard normal curve delimited by the zscore(s) found in Step 3.
Example 4
X has a normal distribution with the given mean and standard
deviation. Find the indicated probabilities.
9.   50 ,   10 , find P ( 35  X  65 )
14. 
 100
,
 15
, find
P ( 70  X  80 )
Can we do this on our calculator?
Sort of, but we need to draw our picture(s) so we make sure
that we know what area we are interested in
Graphing Procedure 3
Method to find some of the probabilities in our calculator if
we know the X values,  , and  .
1. Press 2 VARS →DISTR
2. You will see
DISTR
DRAW
3. Press 2:normalcdf(
4. Enter in first value , Enter in second value, type in  ,  )
5. Press Enter
Example 5
20. X has a normal random variable with mean   10 and
standard deviation   5 . Find b such that P (10  X  b )  0 . 4
27. SAT scores are normally distributed with a mean 500 and a
standard deviation of 100. Find the probability that a randomly
chosen test-taker will score between 450 and 550.
30. LSAT test scores are normally distributed with a mean of
151 and a standard deviation of 7. Find the probability that a
randomly chosen test taker will score 144 or lower.
34. If the mean IQ score is 100 and standard deviation is 16
find the number of people in the U.S. if the population is
313,000,000 with an IQ of 140 or higher.
36. LSAT test scores are normally distributed with a mean of
151 and a standard deviation of 7.What score would place you
in the top 2 of test takers?
Empirical Rule
For any data set having roughly a bell-shaped distribution.
 Approximately 68% of the observations lie within one
standard deviation to either side of the mean. This would
    ,     for populations.
 Approximately 95% of the observations lie within two
standard deviations to either side of the mean. This would
   2  ,   2   for populations.
 Approximately 99.7% of the observations lie within three
standard deviations to either side of the mean. This would
   3 ,   3  for populations.