Download PPT

Chapter 3 ~ Normal Probability Distributions
P(a  x  b)
a

b
x
1
Chapter Goals
• Learn about the normal, bell-shaped, or
Gaussian distribution
• How probabilities are found
• How probabilities are represented
• How normal distributions are used in the real
world
2
Normal Probability Distributions
• The normal probability distribution is the most
important distribution in all of statistics
• Many continuous random variables have normal
or approximately normal distributions
• Need to learn how to describe a normal
probability distribution
3
Normal Probability Distribution
1. A continuous random variable
2. Description involves two functions:
a. A function to determine the ordinates of the graph
picturing the distribution
b. A function to determine probabilities
3. Normal probability distribution function:
1 ( x-)
e 2 s
2
1
s 2p
This is the function for the normal (bell-shaped) curve
f ( x) =
4. The probability that x lies in some interval is the area
under the curve
4
The Normal Probability Distribution
s
 - 3s  - 2s  - s

 s
  2s   3s
5
Probabilities for a Normal Distribution
• Illustration
b
P(a  x  b) =  f ( x )dx
a
a
b
x
6
Notes

The definite integral is a calculus topic

We will use a table to find probabilities for normal
distributions

We will learn how to compute probabilities for one special
normal distribution: the standard normal distribution

Transform all other normal probability questions to this
special distribution

Recall the empirical rule: the percentages that lie within
certain intervals about the mean come from the normal
probability distribution

We need to refine the empirical rule to be able to find the
percentage that lies between any two numbers
7
Percentage, Proportion & Probability
• Basically the same concepts
• Percentage (30%) is usually used when talking
about a proportion (3/10) of a population
• Probability is usually used when talking about
the chance that the next individual item will
possess a certain property
• Area is the graphic representation of all three
when we draw a picture to illustrate the situation
8
The Standard Normal Distribution
• There are infinitely many normal probability
distributions
• They are all related to the standard normal
distribution
• The standard normal distribution is the
normal distribution of the standard variable z
(the z-score)
9
Standard Normal Distribution
Properties:
• The total area under the normal curve is equal to 1
• The distribution is mounded and symmetric; it extends indefinitely in
both directions, approaching but never touching the horizontal axis
• The distribution has a mean of 0 and a standard deviation of 1
• The mean divides the area in half, 0.50 on each side
• Nearly all the area is between z = -3.00 and z = 3.00
Notes:

Appendix, Table A lists the probabilities associated with the intervals
from the mean (0) to a specific value of z

Probabilities of other intervals are found using the table
entries, addition, subtraction, and the properties above
10
Which Table to Use?
An infinite number of normal distributions
means an infinite number of tables to look up!
11
Solution: The Cumulative
Standardized Normal Distribution
Cumulative Standardized Normal
Distribution Table (Portion)
Z = 0
Z
.00
.01
sZ =1
.02
.5478
0.0 .5000 .5040 .5080
Shaded Area
Exaggerated
0.1 .5398 .5438 .5478
0.2 .5793 .5832 .5871
Probabilities
0.3 .6179 .6217 .6255
0
Z = 0.12
Only One Table is Needed
12
Standardizing Example
Z=
X -
s
6.2 - 5
=
= 0.12
10
Standardized
Normal Distribution
Normal Distribution
s = 10
 =5
sZ =1
6.2
X
Shaded Area Exaggerated
Z = 0
0.12
Z
13
Example:
P  2.9  X  7.1 = .1664
Z=
X -
s
2.9 - 5
=
= -.21
10
Z=
X -
s
7.1 - 5
=
= .21
10
Standardized
Normal Distribution
Normal Distribution
s = 10
.0832
sZ =1
.0832
2.9
7.1
X
-0.21
Shaded Area Exaggerated
Z = 0
0.21
Z
14
Example:
P  2.9  X  7.1 = .1664
Cumulative Standardized Normal
Distribution Table (Portion)
Z
.00
.01
Z = 0
sZ =1
.02
.5832
0.0 .5000 .5040 .5080
Shaded Area
Exaggerated
0.1 .5398 .5438 .5478
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
(continued)
0
Z = 0.21
15
Example:
P  2.9  X  7.1 = .1664
Cumulative Standardized Normal
Distribution Table (Portion)
Z
.00
.01
.02
Z = 0
sZ =1
.4168
-03 .3821 .3783 .3745
Shaded Area
Exaggerated
-02 .4207 .4168 .4129
-0.1 .4602 .4562 .4522
0.0 .5000 .4960 .4920
(continued)
0
Z = -0.21
16
Notes

The symmetry of the normal distribution is a key factor
in determining probabilities associated with values
below (to the left of) the mean. For example: the area
between the mean and z = -1.37 is exactly the same as
the area between the mean and z = +1.37.

When finding normal distribution probabilities, a sketch
is always helpful. See course worksheet.
17
Example:
P  X  8 = .3821
Z=
X -
s
8-5
=
= .30
10
Standardized
Normal Distribution
Normal Distribution
s = 10
sZ =1
.3821
 =5
8
X
Shaded Area Exaggerated
Z = 0
0.30
Z
18
Example:
P  X  8 = .3821
Cumulative Standardized Normal
Distribution Table (Portion)
Z
.00
.01
Z = 0
(continued)
sZ =1
.02
.6179
0.0 .5000 .5040 .5080
Shaded Area
Exaggerated
0.1 .5398 .5438 .5478
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
0
Z = 0.30
19
Recovering X Values for Known
Probabilities
Standardized
Normal Distribution
Normal Distribution
s = 10
sZ =1
.6179
.3821
 =5
?
X
Z = 0
0.30
Z
X =   Zs = 5  .3010 = 8
20
Assessing Normality
• Not all continuous random variables are normally
distributed
• It is important to evaluate how well the data set
seems to be adequately approximated by a normal
distribution
21
Assessing Normality
(continued)
• Construct charts
– For small- or moderate-sized data sets, do stemand-leaf display and box-and-whisker plot look
symmetric?
– For large data sets, does the histogram or polygon
appear bell-shaped?
• Compute descriptive summary measures
– Do the mean, median and mode have similar
values?
– Is the interquartile range approximately 1.33 s?
– Is the range approximately 6 s?
22
Assessing Normality
• Observe the distribution of the data set
(continued)
– Do approximately 2/3 of the observations lie
between mean  1 standard deviation?
– Do approximately 4/5 of the observations lie
between mean  1.28 standard deviations?
– Do approximately 19/20 of the observations lie
between mean  2 standard deviations?
23
Assessing Normality
• With Minitab
24
Applications of Normal Distributions
• Apply the techniques learned for the z distribution
to all normal distributions
• Start with a probability question in terms of
x-values
• Convert, or transform, the question into an
equivalent probability statement involving
z-values
25
Standardization
• Suppose x is a normal random variable with mean  and
standard deviation s
x-
• The random variable z =
s
distribution

0
has a standard normal
c
c-
s
x
z
26
Notes
• The normal table may be used to answer many kinds of questions
involving a normal distribution
• Often we need to find a cutoff point: a value of x such that there is
a certain probability in a specified interval defined by x
 Example: The waiting time x at a certain bank is approximately
normally distributed with a mean of 3.7 minutes and a
standard deviation of 1.4 minutes. The bank would
like to claim that 95% of all customers are waited on
by a teller within c minutes. Find the value of c that
makes this statement true.
27
Solution
0.0500
0.5000 0.4500
3.7
0
P ( x  c) = 0.95
 x - 3.7  c - 3.7  =
 0.95
P
 14
.
1.4 
  c - 3.7  =
 0.95
P z

14
. 
c
1645
.
x
z
c - 3.7
= 1645
.
14
.
c = (1645
. )(14
. )  3.7 = 6.003
c  6 minutes
28
Notation
• If x is a normal random variable with mean  and
standard deviation s, this is often denoted:
x ~ N(, s)

Example: Suppose x is a normal random variable
with  = 35 and s = 6. A convenient notation to
identify this random variable is: x ~ N(35, 6).
29
Notation
• z-score used throughout statistics in a variety of
ways
• Need convenient notation to indicate the area
under the standard normal distribution
• z(a) is the algebraic name, for the z-score (point on
the z axis) such that there is a of the area
(probability) to the right or left of z(a)
30
Notes
• The values of z that will be used regularly come from
one of the following situations:
1. The z-score such that there is a specified area in one
tail of the normal distribution
2. The z-scores that bound a specified middle
proportion of the normal distribution
31
Example
 Example: Find the numerical value of z(0.99):
0.01
z(0.01)
0
z
• Because of the symmetrical nature of the normal distribution,
z(0.99) = -z(0.01)
• Using Table A: z(0.01) = -2.33
32
Example
 Example: Find the z-scores that bound the middle 0.99 of the
normal distribution:
0.005
0.005
0.495
z(0.005)
0.495
0
z(0.995)
• Use Table A:
z(0.005) = -2.575 and z(0.995) = 2.575
33
Chapter Summary
• Discussed the normal distribution
• Described the standard normal distribution
• Evaluated the normality assumption
34