Chap006.ppt

Copyright © 2015, 2012, 2009 Pearson Education, Inc. Chapter 06, Slide 1
The Normal Distribution
and Other Continuous
Distributions
Chapter 6

Learning Objectives
In this chapter, you learn:
 To compute probabilities from the normal distribution
 How to use the normal distribution to solve business
problems
 To use the normal probability plot to determine whether
a set of data is approximately normally distributed
 To compute probabilities from the uniform distribution
 To compute probabilities from the exponential
distribution

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 precisely and
accurately measure

 ‘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  
Mean
= Median
= Mode
X
f(X)
μ
σ

Density Function
2
μ)
(X
2
1
e
2π
1
f(X)





 

 

 The formula for the normal probability density function is
Where e = the mathematical constant approximated by 2.71828
π = the mathematical constant approximated by 3.14159
μ = the population mean
σ = the population standard deviation
X = any value of the continuous variable

By varying the parameters μ and σ, we
obtain different normal distributions
A
B
C
A and B have the same mean but different standard deviations.
B and C have different means and different standard deviations.

Shape
X
f(X)
μ
σ
Changing μ shifts the
distribution left or right.
Changing σ increases
or decreases the
spread.

The Standardized Normal
 Any normal distribution (with any mean and
standard deviation combination) can be
transformed into the standardized normal
distribution (Z)
 To compute normal probabilities need to
transform X units into Z units
 The standardized normal distribution (Z) has a
mean of 0 and a standard deviation of 1

Translation to the Standardized
Normal Distribution
 Translate from X to the standardized normal
(the “Z” distribution) by subtracting the mean
of X and dividing by its standard deviation:
σ
μ
X
Z


The Z distribution always has mean = 0 and
standard deviation = 1

The Standardized Normal
Probability Density Function
 The formula for the standardized normal
probability density function is
Where e = the mathematical constant approximated by 2.71828
π = the mathematical constant approximated by 3.14159
Z = any value of the standardized normal distribution
2
(1/2)Z
e
2π
1
f(Z) 


The Standardized
Normal Distribution
 Also known as the “Z” distribution
 Mean is 0
 Standard Deviation is 1
Z
f(Z)
0
1
Values above the mean have positive Z-values.
Values below the mean have negative Z-values.

Example
 If X is distributed normally with mean of $100
and standard deviation of $50, the Z value
for X = $200 is
 This says that X = $200 is two standard
deviations (2 increments of $50 units) above
the mean of $100.
2.0
$50
100
$
$200
σ
μ
X
Z 





Comparing X and Z units
Note that the shape of the distribution is the same,
only the scale has changed. We can express the
problem in the original units (X in dollars) or in
standardized units (Z)
Z
$100
2.0
0
$200 $X(μ = $100, σ = $50)
(μ = 0, σ = 1)

Finding Normal Probabilities
a b X
f(X) P a X b
( )
≤
Probability is measured by the area
under the curve
≤
P a X b
( )
<
<
=
(Note that the
probability of any
individual value is zero)

f(X)
X
μ
Probability as
Area Under the Curve
0.5
0.5
The total area under the curve is 1.0, and the curve is
symmetric, so half is above the mean, half is below
1.0
)
X
P( 




0.5
)
X
P(μ 



0.5
μ)
X
P( 




The Standardized Normal Table
 The Cumulative Standardized Normal table
in the textbook (Appendix table E.2) gives the
probability less than a desired value of Z (i.e.,
from negative infinity to Z)
Z
0 2.00
0.9772
Example:
P(Z < 2.00) = 0.9772

The Standardized Normal Table
The value within the
table gives the
probability from Z =  
up to the desired Z
value
.9772
2.0
P(Z < 2.00) = 0.9772
The row shows
the value of Z
to the first
decimal point
The column gives the value of
Z to the second decimal point
2.0
.
.
.
(continued)
Z 0.00 0.01 0.02 …
0.0
0.1

General Procedure for
 Draw the normal curve for the problem in
terms of X
 Translate X-values to Z-values
 Use the Standardized Normal Table
To find P(a < X < b) when X is
distributed normally:

 Let X represent the time it takes (in seconds)
to download an image file from the internet.
 Suppose X is normal with a mean of18.0
seconds and a standard deviation of 5.0
seconds. Find P(X < 18.6)
18.6
X
18.0

 Let X represent the time it takes, in seconds to download an image file
from the internet.
 Suppose X is normal with a mean of 18.0 seconds and a standard
deviation of 5.0 seconds. Find P(X < 18.6)
Z
0.12
0
X
18.6
18
μ = 18
σ = 5
μ = 0
σ = 1
(continued)
0.12
5.0
8.0
1
18.6
σ
μ
X
Z 




P(X < 18.6) P(Z < 0.12)

Z
0.12
Z .00 .01
0.0 .5000 .5040 .5080
.5398 .5438
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
Solution: Finding P(Z < 0.12)
0.5478
.02
0.1 .5478
Standardized Normal Probability
Table (Portion)
0.00
= P(Z < 0.12)
P(X < 18.6)

Finding Normal
Upper Tail Probabilities
 Suppose X is normal with mean 18.0
and standard deviation 5.0.
 Now Find P(X > 18.6)
X
18.6
18.0

 Now Find P(X > 18.6)…
(continued)
Z
0.12
0
Z
0.12
0.5478
0
1.000 1.0 - 0.5478
= 0.4522
P(X > 18.6) = P(Z > 0.12) = 1.0 - P(Z ≤ 0.12)
= 1.0 - 0.5478 = 0.4522
Finding Normal
Upper Tail Probabilities

Finding a Normal Probability
Between Two Values
 Suppose X is normal with mean 18.0 and
standard deviation 5.0. Find P(18 < X < 18.6)
P(18 < X < 18.6)
= P(0 < Z < 0.12)
Z
0.12
0
X
18.6
18
0
5
8
1
18
σ
μ
X
Z 




0.12
5
8
1
18.6
σ
μ
X
Z 




Calculate Z-values:

Z
0.12
Solution: Finding P(0 < Z < 0.12)
0.0478
0.00
= P(0 < Z < 0.12)
P(18 < X < 18.6)
= P(Z < 0.12) – P(Z ≤ 0)
= 0.5478 - 0.5000 = 0.0478
0.5000
Z .00 .01
0.0 .5000 .5040 .5080
.5398 .5438
0.2 .5793 .5832 .5871
0.3 .6179 .6217 .6255
.02
0.1 .5478
Table (Portion)

 Suppose X is normal with mean 18.0
and standard deviation 5.0.
 Now Find P(17.4 < X < 18)
X
17.4
18.0
Probabilities in the Lower Tail

Probabilities in the Lower Tail
Now Find P(17.4 < X < 18)…
X
17.4 18.0
P(17.4 < X < 18)
= P(-0.12 < Z < 0)
= P(Z < 0) – P(Z ≤ -0.12)
= 0.5000 - 0.4522 = 0.0478
(continued)
0.0478
0.4522
Z
-0.12 0
The Normal distribution is
symmetric, so this probability
is the same as P(0 < Z < 0.12)

Empirical Rule
μ ± 1σ encloses about
68.26% of X’s
f(X)
X
μ μ+1σ
μ-1σ
What can we say about the distribution of values
around the mean? For any normal distribution:
σ
σ
68.26%

The Empirical Rule
 μ ± 2σ covers about 95.44% of X’s
 μ ± 3σ covers about 99.73% of X’s
x
μ
2σ 2σ
x
μ
3σ 3σ
95.44% 99.73%
(continued)

 Steps to find the X value for a known
probability:
1. Find the Z value for the known probability
2. Convert to X units using the formula:
Given a Normal Probability
Find the X Value
Zσ
μ
X 


Finding the X value for a
Known Probability
Example:
 Let X represent the time it takes (in seconds) to
download an image file from the internet.
 Suppose X is normal with mean 18.0 and standard
deviation 5.0
 Find X such that 20% of download times are less than
X.
X
? 18.0
0.2000
Z
? 0
(continued)

Find the Z value for
20% in the Lower Tail
 20% area in the lower
tail is consistent with a
Z value of -0.84
Z .03
-0.9 .1762 .1736
.2033
-0.7 .2327 .2296
.04
-0.8 .2005
Table (Portion)
.05
.1711
.1977
.2266
…
…
…
…
X
? 18.0
0.2000
Z
-0.84 0
1. Find the Z value for the known probability

2. Convert to X units using the formula:
Finding the X value
8
.
13
0
.
5
)
84
.
0
(
0
.
18
Zσ
μ
X






So 20% of the values from a distribution
with mean 18.0 and standard deviation
5.0 are less than 13.80

Both Minitab & Excel Can Be Used
To Find Normal Probabilities
Find P(X < 9) where X is normal with a mean of 7
and a standard deviation of 2
Excel Minitab

Evaluating Normality
 Not all continuous distributions are normal
 It is important to evaluate how well the data set is
approximated by a normal distribution.
 Normally distributed data should approximate the
theoretical normal distribution:
 The normal distribution is bell shaped (symmetrical)
where the mean is equal to the median.
 The empirical rule applies to the normal distribution.
 The interquartile range of a normal distribution is 1.33
standard deviations.

Comparing data characteristics to theoretical
properties
Construct charts or graphs
 For small- or moderate-sized data sets, construct a stem-and-leaf
display or a boxplot to check for symmetry
 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 1.33σ?
 Is the range approximately 6σ?
(continued)

Comparing data characteristics to theoretical
properties
 Observe the distribution of the data set
 Do approximately 2/3 of the observations lie within mean ±1
standard deviation?
 Do approximately 80% of the observations lie within mean
±1.28 standard deviations?
 Do approximately 95% of the observations lie within mean ±2
standard deviations?
 Evaluate normal probability plot
 Is the normal probability plot approximately linear (i.e. a straight
line) with positive slope?
(continued)

Constructing
A Normal Probability Plot
 Normal probability plot
 Arrange data into ordered array
 Find corresponding standardized normal quantile
values (Z)
 Plot the pairs of points with observed data values (X)
on the vertical axis and the standardized normal
quantile values (Z) on the horizontal axis
 Evaluate the plot for evidence of linearity

A normal probability plot for data
from a normal distribution will be
approximately linear:
30
60
90
-2 -1 0 1 2 Z
X
The Normal Probability Plot
Interpretation

Normal Probability Plot
Interpretation
Left-Skewed Right-Skewed
Rectangular
30
60
90
-2 -1 0 1 2 Z
X
(continued)
30
60
90
-2 -1 0 1 2 Z
X
30
60
90
-2 -1 0 1 2 Z
X Nonlinear plots
indicate a deviation
from normality

An Example: Mutual Fund Returns
The boxplot is skewed to
the right. (The normal
distribution is symmetric.)

Descriptive Statistics
(continued)
• The mean (21.84) is approximately the same as the median
(21.65). (In a normal distribution the mean and median are
equal.)
• The interquartile range of 6.98 is approximately 1.09 standard
deviations. (In a normal distribution the interquartile range is
1.33 standard deviations.)
• The range of 59.52 is equal to 9.26 standard deviations. (In a
normal distribution the range is 6 standard deviations.)
• 77.04% of the observations are within 1 standard deviation of
the mean. (In a normal distribution this percentage is 68.26%.
• 86.79% of the observations are within 1.28 standard
deviations of the mean. (In a normal distribution this
percentage is 80%.)
• 96.86% of the observations are within 2 standard deviations
of the mean. (In a normal distribution this percentage is
95.44%.)
• The skewness statistic is 1.698 and the kurtosis statistic is
8.467. (In a normal distribution, each of these statistics equals
zero.)

Evaluating Normality Via Excel
(continued)
Plot is not a straight
line and shows the
distribution is skewed
to the right. (The
normal distribution
appears as a straight
line.)
Excel (quantile-quantile) normal probability plot

Evaluating Normality Via Minitab
(continued)
Plot is not a straight
line, rises quickly in
the beginning, rises
slowly at the end and
shows the distribution
is skewed to the
right.
Normal Probability Plot From Minitab
70
60
50
40
30
20
10
0
99.9
99
95
90
80
70
60
50
40
30
20
10
5
1
0.1
3YrReturn%
Percent
Probability Plot of 3YrReturn%
Normal

 Conclusions
 The returns are right-skewed
 The returns have more values concentrated around
the mean than expected
 The range is larger than expected
 Normal probability plot is not a straight line
 Overall, this data set greatly differs from the
theoretical properties of the normal distribution
(continued)

The Uniform Distribution
 The uniform distribution is a probability
distribution that has equal probabilities
for all possible outcomes of the random
variable
 Also called a rectangular distribution

The Continuous Uniform Distribution:
otherwise
0
b
X
a
if
a
b
1



where
f(X) = value of the density function at any X value
a = minimum value of X
b = maximum value of X
The Uniform Distribution
(continued)
f(X) =

Properties of the
Uniform Distribution
 The mean of a uniform distribution is
 The standard deviation is
2
b
a
μ


12
a)
-
(b
σ
2


Uniform Distribution Example
Example: Uniform probability distribution
over the range 2 ≤ X ≤ 6:
2 6
0.25
f(X) = = 0.25 for 2 ≤ X ≤ 6
6 - 2
1
X
f(X)
4
2
6
2
2
b
a
μ 




1547
.
1
12
2)
-
(6
12
a)
-
(b
σ
2
2




Uniform Distribution Example
Example: Using the uniform probability
distribution to find P(3 ≤ X ≤ 5):
2 6
0.25
P(3 ≤ X ≤ 5) = (Base)(Height) = (2)(0.25) = 0.5
X
f(X)
(continued)
3 5
4

The Exponential Distribution
 Often used to model the length of time
between two occurrences of an event (the
time between arrivals)
 Examples:
 Time between trucks arriving at an unloading dock
 Time between transactions at an ATM Machine
 Time between phone calls to the main operator

X
λ
e
1
X)
time
P(arrival 



 Defined by a single parameter, its mean λ
(lambda)
 Exponential Probability Density Function:
 The probability that an arrival time is less than
some specified time X is
where e = mathematical constant approximated by 2.71828
λ = the population mean number of arrivals per unit
X = any value of the continuous variable where 0 < X < 
0
X
for
e
f(X) X

  λ


The Mean & Standard Deviation Of
 The mean (μ) of the exponential distribution is
given by:
 The standard deviation (σ) of the exponential
distribution is given by:


 /
1


 /
1


Exponential Distribution
Example
Example: Customers arrive at the service counter at
the rate of 15 per hour. What is the probability that the
arrival time between consecutive customers is less
than three minutes?
 The mean number of arrivals per hour is 15, so λ = 15
 Three minutes is 0.05 hours
 P(arrival time < .05) = 1 – e-λX = 1 – e-(15)(0.05) = 0.5276
 So there is a 52.76% chance that the arrival time
between successive customers is less than three
minutes

In Excel
Calculating the probability that an exponential
distribution with an mean of 20 is less than 0.1

The Exponential Distribution In Minitab
Cumulative Distribution Function
Exponential with mean = 0.05
x P( X <= x )
0.1 0.864665
Calculating the probability that an exponential distribution with a
mean of 1/20 is less than 0.1

Chapter Summary
In this chapter we discussed
 Key continuous distributions
 normal, uniform, exponential
 Finding probabilities using formulas and tables
 Recognizing when to apply different distributions
 Applying distributions to decision problems

Normal Approximation
To the Binomial
On Line Topic

Learning Objectives
In this topic, you learn:
 When it is appropriate to use the normal distribution to
approximate binomial probabilities
 How to use the normal distribution to approximate
binomial probabilities

Normal Approximation to the
Binomial Distribution
 The binomial distribution is a discrete
distribution, but the normal is continuous
 To use the normal to approximate the binomial,
accuracy is improved if you use a correction for
continuity adjustment
 Example:
 X is discrete in a binomial distribution, so P(X = 4)
can be approximated with a continuous normal
distribution by finding
P(3.5 < X < 4.5)

 The closer π is to 0.5, the better the normal
approximation to the binomial
 The larger the sample size n, the better the
normal approximation to the binomial
 General rule:
 The normal distribution can be used to approximate
the binomial distribution if
nπ ≥ 5
and
n(1 – π) ≥ 5
(continued)

 The mean and standard deviation of the
binomial distribution are
μ = nπ
 Transform binomial to normal using the formula:
(continued)
)
(1
n
n
X
σ
μ
X
Z
π
π
π





)
(1
n
σ π
π 


Using the Normal Approximation
to the Binomial Distribution
 If n = 1000 and π = 0.2, what is P(X ≤ 180)?
 Approximate P(X ≤ 180) using a continuity correction
adjustment:
P(X ≤ 180.5)
 Transform to standardized normal:
 So P(Z ≤ -1.54) = 0.0618
1.54
0.2)
)(1
(1000)(0.2
)
(1000)(0.2
180.5
)
(1
n
n
X
Z 







π
π
π
X
180.5 200
-1.54 0 Z

Topic Summary
In this topic we discussed:
 When it is appropriate to use the normal distribution to
approximate binomial probabilities
 How to use the normal distribution to approximate
binomial probabilities

Chap006.ppt

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