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Basic Business Statistics
(8th Edition)
Chapter 6
The Normal Distribution and
Other Continuous Distributions
© 2002 Prentice-Hall, Inc.
Chap 6-1
Chapter Topics

The normal distribution

The standardized normal distribution

Evaluating the normality assumption

The exponential distribution
© 2002 Prentice-Hall, Inc.
Chap 6-2
Continuous
Probability Distributions

Continuous random variable



Continuous probability distribution


Values from interval of numbers
Absence of gaps
Distribution of continuous random variable
Most important continuous probability
distribution

The normal distribution
© 2002 Prentice-Hall, Inc.
Chap 6-3
The Normal Distribution





“Bell shaped”
Symmetrical
Mean, median and
mode are equal
Interquartile range
equals 1.33 s
Random variable
has infinite range
© 2002 Prentice-Hall, Inc.
f(X)

X
Mean
Median
Mode
Chap 6-4
The Mathematical Model
f X  
1

e
1
2s
2
X





2s 2
f  X  : density of random variable X
  3.14159;
e  2.71828
 : population mean
s : population standard deviation
X : value of random variable    X   
© 2002 Prentice-Hall, Inc.
Chap 6-5
Many Normal Distributions
There are an infinite number of normal distributions
By varying the parameters s and , we
obtain different normal distributions
© 2002 Prentice-Hall, Inc.
Chap 6-6
Finding Probabilities
Probability is
the area under
the curve!
P c  X  d   ?
f(X)
c
© 2002 Prentice-Hall, Inc.
d
X
Chap 6-7
Which Table to Use?
An infinite number of normal distributions
means an infinite number of tables to look up!
© 2002 Prentice-Hall, Inc.
Chap 6-8
Solution: The Cumulative
Standardized Normal Distribution
Cumulative Standardized Normal
Distribution Table (Portion)
Z
.00
.01
Z  0
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
© 2002 Prentice-Hall, Inc.
0
Z = 0.12
Only One Table is Needed
Chap 6-9
Standardizing Example
Z
X 
s
6.2  5

 0.12
10
Standardized
Normal Distribution
Normal Distribution
s  10
 5
© 2002 Prentice-Hall, Inc.
sZ 1
6.2
X
Shaded Area Exaggerated
Z  0
0.12
Z
Chap 6-10
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
 5
© 2002 Prentice-Hall, Inc.
7.1
X
0.21
Shaded Area Exaggerated
Z  0
0.21
Z
Chap 6-11
Example:
P  2.9  X  7.1  .1664(continued)
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
© 2002 Prentice-Hall, Inc.
0
Z = 0.21
Chap 6-12
Example:
P  2.9  X  7.1  .1664(continued)
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
© 2002 Prentice-Hall, Inc.
0
Z = -0.21
Chap 6-13
Normal Distribution in PHStat


PHStat | probability & prob. distributions |
normal …
Example in excel spreadsheet
© 2002 Prentice-Hall, Inc.
Chap 6-14
Example:
P  X  8  .3821
Z
X 
s
85

 .30
10
Standardized
Normal Distribution
Normal Distribution
s  10
sZ 1
.3821
 5
© 2002 Prentice-Hall, Inc.
8
X
Shaded Area Exaggerated
Z  0
0.30
Z
Chap 6-15
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
© 2002 Prentice-Hall, Inc.
0
Z = 0.30
Chap 6-16
Finding Z Values for Known
Probabilities
What is Z Given
Probability = 0.1217 ?
Z  0
sZ 1
Cumulative Standardized
Normal Distribution Table
(Portion)
Z
.00
.01
0.2
0.0 .5000 .5040 .5080
.6217
0.1 .5398 .5438 .5478
0.2 .5793 .5832 .5871
Shaded Area
Exaggerated
© 2002 Prentice-Hall, Inc.
0
Z  .31
0.3 .6179 .6217 .6255
Chap 6-17
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
© 2002 Prentice-Hall, Inc.
Chap 6-18
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
© 2002 Prentice-Hall, Inc.
Chap 6-19
Assessing Normality

Construct charts



(continued)
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?
© 2002 Prentice-Hall, Inc.
Chap 6-20
Assessing Normality

Observe the distribution of the data set




(continued)
Do approximately
between mean 
Do approximately
between mean 
Do approximately
between mean 
2/3 of the observations lie
1 standard deviation?
4/5 of the observations lie
1.28 standard deviations?
19/20 of the observations lie
2 standard deviations?
Evaluate normal probability plot

Do the points lie on or close to a straight line with
positive slope?
© 2002 Prentice-Hall, Inc.
Chap 6-21
Assessing Normality

(continued)
Normal probability plot




Arrange data into ordered array
Find corresponding standardized normal quantile
values
Plot the pairs of points with observed data values
on the vertical axis and the standardized normal
quantile values on the horizontal axis
Evaluate the plot for evidence of linearity
© 2002 Prentice-Hall, Inc.
Chap 6-22
Assessing Normality
(continued)
Normal Probability Plot
for Normal Distribution
90
X 60
Z
30
-2 -1 0 1 2
© 2002 Prentice-Hall, Inc.
Look for Straight Line!
Chap 6-23
Normal Probability Plot
Left-Skewed
Right-Skewed
90
90
X 60
X 60
Z
30
-2 -1 0 1 2
-2 -1 0 1 2
Rectangular
U-Shaped
90
90
X 60
X 60
Z
30
-2 -1 0 1 2
© 2002 Prentice-Hall, Inc.
Z
30
Z
30
-2 -1 0 1 2
Chap 6-24
Exponential Distributions
P  arrival time  X   1  e
 X
X : any value of continuous random variable
 : the population average number of
arrivals per unit of time
1/: average time between arrivals
e  2.71828
e.g.: Drivers arriving at a toll bridge;
customers arriving at an ATM machine
© 2002 Prentice-Hall, Inc.
Chap 6-25
Exponential Distributions
(continued)

Describes time or distance between events


f(X)
Density function


Used for queues
f  x 
Parameters

 
© 2002 Prentice-Hall, Inc.
1

e

x

 = 0.5
 = 2.0
X
s 
Chap 6-26
Example
e.g.: Customers arrive at the check out line
of a supermarket at the rate of 30 per hour.
What is the probability that the arrival time
between consecutive customers to be
greater than 5 minutes?
  30
X  5 / 60 hours
P  arrival time >X   1  P  arrival time  X 

 1 1 e
30 5/ 60 

 .0821
© 2002 Prentice-Hall, Inc.
Chap 6-27
Exponential Distribution in
PHStat


PHStat | probability & prob. distributions |
exponential
Example in excel spreadsheet
© 2002 Prentice-Hall, Inc.
Chap 6-28
Chapter Summary

Discussed the normal distribution

Described the standard normal distribution

Evaluated the normality assumption

Defined the exponential distribution
© 2002 Prentice-Hall, Inc.
Chap 6-29