Download Business Statistics: A Decision

Introduction to Statistics
Chapter 6
Continuous
Probability Distributions
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 5-1
Chapter Goals
After completing this chapter, you should be
able to:

Understand the continuous random variables
and there probability distribution

Recognize when to apply the uniform

Find probabilities using a normal distribution table
and apply the normal distribution
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 5-2
Continuous Probability Distributions

A continuous random variable is a variable that
can assume any value on a continuum (can
assume an uncountable number of values)





thickness of an item
time required to complete a task
temperature of a solution
height, in inches
These can potentially take on any value,
depending only on the ability to measure
accurately.
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 5-3
The Uniform Distribution

The uniform distribution is a
probability distribution that has
equal probabilities for all possible
outcomes of the random variable
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 5-4
The Uniform Distribution
(continued)
The Continuous Uniform Distribution:
f(x) =
1
ba
if a  x  b
0
otherwise
where
f(x) = value of the density function at any x value
a = lower limit of the interval
b = upper limit of the interval
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 5-5
Uniform Distribution
Example: Uniform Probability Distribution
Over the range 2 ≤ x ≤ 6:
1
f(x) = 6 - 2 = .25 for 2 ≤ x ≤ 6
f(x)
.25
2
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
6
x
Chap 5-6
The Normal Distribution
‘Bell Shaped’
 Symmetrical
 Mean, Median and Mode
are Equal
Location is determined by the
mean, μ

Spread is determined by the
standard deviation, σ
The random variable has an
infinite theoretical range:
+  to  
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
f(x)
σ
x
μ
Mean
= Median
= Mode
Chap 5-7
Many Normal Distributions
By varying the parameters μ and σ, we obtain
different normal distributions
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 5-8
The Normal Distribution Shape
f(x)
Changing μ shifts the
distribution left or right.
σ
μ
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Changing σ increases
or decreases the
spread.
x
Chap 5-9
Finding Normal Probabilities
Probability is the
Probability is measured
area under the
curve! under the curve
f(x)
by the area
P (a  x  b)
a
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
b
x
Chap 5-10
Probability as
Area Under the Curve
The total area under the curve is 1.0, and the curve is
symmetric, so half is above the mean, half is below
f(x) P(  x  μ)  0.5
0.5
P(μ  x  )  0.5
0.5
μ
x
P(  x  )  1.0
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 5-11
Empirical Rules
What can we say about the distribution of values
around the mean? There are some general rules:
f(x)
μ ± 1σ encloses about
68% of x’s
σ
μ1σ
σ
μ
μ+1σ
x
68.26%
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 5-12
The Empirical Rule
(continued)

μ ± 2σ covers about 95% of x’s

μ ± 3σ covers about 99.7% of x’s
2σ
3σ
2σ
μ
x
95.44%
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
3σ
μ
x
99.72%
Chap 5-13
Importance of the Rule

If a value is about 2 or more standard
deviations away from the mean in a normal
distribution, then it is far from the mean

The chance that a value that far or farther
away from the mean is highly unlikely, given
that particular mean and standard deviation
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 5-14
The Standard Normal Distribution



Also known as the “z” distribution
Mean is defined to be 0
Standard Deviation is 1
f(z)
1
0
z
Values above the mean have positive z-values,
values below the mean have negative z-values
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 5-15
The Standard Normal

Any normal distribution (with any mean and
standard deviation combination) can be
transformed into the standard normal
distribution (z)

Need to transform x units into z units
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 5-16
Translation to the Standard
Normal Distribution

Translate from x to the standard normal (the
“z” distribution) by subtracting the mean of x
and dividing by its standard deviation:
x μ
z
σ
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 5-17
Example

If x is distributed normally with mean of 100
and standard deviation of 50, the z value for
x = 250 is
x  μ 250  100
z

 3.0
σ
50

This says that x = 250 is three standard
deviations (3 increments of 50 units) above
the mean of 100.
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 5-18
Comparing x and z units
μ = 100
σ = 50
100
0
250
3.0
x
z
Note that the distribution is the same, only the
scale has changed. We can express the problem in
original units (x) or in standardized units (z)
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 5-19
The Standard Normal Table

The Standard Normal table in the textbook
(Appendix D)
gives the probability from the mean (zero)
up to a desired value for z
.4772
Example:
P(0 < z < 2.00) = .4772
0
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
2.00
z
Chap 5-20
The Standard Normal Table
(continued)
The column gives the value of
z to the second decimal point
z
The row shows
the value of z
to the first
decimal point
0.00
0.01
0.02
…
0.1
0.2
.
.
.
2.0
.4772
P(0 < z < 2.00)2.0
= .4772
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
The value within the
table gives the
probability from z = 0
up to the desired z
value
Chap 5-21
General Procedure for
Finding Probabilities
To find P(a < x < b) when x is distributed
normally:

Draw the normal curve for the problem in
terms of x

Translate x-values to z-values

Use the Standard Normal Table
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 5-22
Z Table example

Suppose x is normal with mean 8.0 and
standard deviation 5.0. Find P(8 < x < 8.6)
Calculate z-values:
x μ 8 8
z

0
σ
5
x  μ 8.6  8
z

 0.12
σ
5
8 8.6
x
0 0.12
Z
P(8 < x < 8.6)
= P(0 < z < 0.12)
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 5-23
Z Table example

(continued)
Suppose x is normal with mean 8.0 and
standard deviation 5.0. Find P(8 < x < 8.6)
=8
=5
8 8.6
=0
=1
x
P(8 < x < 8.6)
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
0 0.12
z
P(0 < z < 0.12)
Chap 5-24
Solution: Finding P(0 < z < 0.12)
Standard Normal Probability
Table (Portion)
z
.00
.01
P(8 < x < 8.6)
= P(0 < z < 0.12)
.02
.0478
0.0 .0000 .0040 .0080
0.1 .0398 .0438 .0478
0.2 .0793 .0832 .0871
Z
0.3 .1179 .1217 .1255
0.00
0.12
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 5-25
Finding Normal Probabilities
Suppose x is normal with mean 8.0
and standard deviation 5.0.
 Now Find P(x < 8.6)

Z
8.0
8.6
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 5-26
Finding Normal Probabilities
(continued)
Suppose x is normal with mean 8.0
and standard deviation 5.0.
 Now Find P(x < 8.6)

P(x < 8.6)
.0478
.5000
= P(z < 0.12)
= P(z < 0) + P(0 < z < 0.12)
= .5 + .0478 = .5478
Z
0.00
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
0.12
Chap 5-27
Upper Tail Probabilities
Suppose x is normal with mean 8.0
and standard deviation 5.0.
 Now Find P(x > 8.6)

Z
8.0
8.6
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 5-28
Upper Tail Probabilities
(continued)

Now Find P(x > 8.6)…
P(x > 8.6) = P(z > 0.12) = P(z > 0) - P(0 < z < 0.12)
= .5 - .0478 = .4522
.0478
.5000
.50 - .0478
= .4522
Z
0
0.12
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Z
0
0.12
Chap 5-29
Lower Tail Probabilities
Suppose x is normal with mean 8.0
and standard deviation 5.0.
 Now Find P(7.4 < x < 8)

Z
8.0
7.4
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 5-30
Lower Tail Probabilities
(continued)
Now Find P(7.4 < x < 8)…
The Normal distribution is
symmetric, so we use the
same table even if z-values
are negative:
.0478
P(7.4 < x < 8)
= P(-0.12 < z < 0)
Z
= .0478
8.0
7.4
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 5-31
Chapter Summary

Reviewed key continuous distributions

uniform and normal,

Found probabilities using formulas and tables

Recognized when to apply different distributions

Applied distributions to decision problems
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 5-32