Download The Normal Distribution And Other continuous Cistributions

Statistics for Managers
Using Microsoft Excel
Chapter 6
The Normal Distribution And Other
Continuous Distributions
© 1999 Prentice-Hall, Inc.
Chap. 6 - 1
Chapter Topics
•The Normal Distribution
•The Standard Normal Distribution
•Assessing the Normality Assumption
•The Exponential Distribution
•Sampling Distribution of the Mean
•Sampling Distribution of the Proportion
•Sampling From Finite Populations
© 1999 Prentice-Hall, Inc.
Chap. 6 - 2
Continuous Probability
Distributions
•Continuous Random Variable:
Values from Interval of Numbers
Absence of Gaps
•Continuous Probability Distribution:
Distribution of a Continuous Variable
•Most Important Continuous Probability
Distribution: the Normal Distribution
© 1999 Prentice-Hall, Inc.
Chap. 6 - 3
The Normal Distribution
• ‘Bell Shaped’
• Symmetrical
f(X)
• Mean, Median and
Mode are Equal
• ‘Middle Spread’
Equals 1.33 s
• Random Variable has
Infinite Range
© 1999 Prentice-Hall, Inc.
m
X
Mean
Median
Mode
Chap. 6 - 4
The Mathematical Model
2
f(X) =
1
2p s
e
(-1/2)((X- m)/s)
f(X) =
frequency of random variable X
p
=
3.14159;
s
=
population standard deviation
X
=
value of random variable (- < X < )
m
=
population mean
© 1999 Prentice-Hall, Inc.
e = 2.71828
Chap. 6 - 5
Many Normal Distributions
There are
an Infinite
Number
Varying the Parameters s and m, we obtain
Different Normal Distributions.
© 1999 Prentice-Hall, Inc.
Chap. 6 - 6
Normal Distribution:
Finding Probabilities
Probability is the
area under the
curve!
P (c  X  d )=
?
f(X)
c
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d
X
Chap. 6 - 7
Which Table?
Each distribution
has its own table?
Infinitely Many Normal Distributions Means
Infinitely Many Tables to Look Up!
© 1999 Prentice-Hall, Inc.
Chap. 6 - 8
The Standardized Normal
Distribution
Standardized Normal Probability
Table (Portion)
m Z = 0 and s Z = 1
Z
.00
.01
.0478
.02
0.0 .0000 .0040 .0080
0.1 .0398 .0438 .0478
0.2 .0793 .0832 .0871
Z = 0.12
0.3 .0179 .0217 .0255
Probabilities
© 1999 Prentice-Hall, Inc.
Shaded Area
Exaggerated
Chap. 6 - 9
Standardizing Example
6 .2  5
X

m
Z=
=
= 0 . 12
s
10
Normal
Distribution
Standardized
Normal Distribution
s = 10
sZ = 1
m = 5 6.2 X
© 1999 Prentice-Hall, Inc.
m = 0 .12
Shaded Area Exaggerated
Z
Chap. 6 - 10
Example:
P(2.9 < X < 7.1) = .1664
z =
Normal
Distribution
z =
xm
s
x  m
s
=
2 .9  5
=  . 21
10
=
7 .1  5
= . 21
10
Standardized
Normal Distribution
s = 10
s =1
.1664
.0832 .0832
2.9 5 7.1 X
© 1999 Prentice-Hall, Inc.
-.21 0 .21
Shaded Area Exaggerated
Z
Chap. 6 - 11
Example: P(X  8) = .3821
z=
xm
s
Normal
Distribution
85
=
= .30
10
Standardized
Normal Distribution
s = 10
s =1
.5000
.1179
m =5
© 1999 Prentice-Hall, Inc.
8
X
.3821
m = 0 .30 Z
Shaded Area Exaggerated
Chap. 6 - 12
Finding Z Values
for Known Probabilities
What Is Z Given
P(Z) = 0.1217?
.1217
s =1
Standardized Normal
Probability Table (Portion)
Z
.00
.01
0.2
0.0 .0000 .0040 .0080
0.1 .0398 .0438 .0478
m = 0 .31 Z
Shaded Area
Exaggerated
© 1999 Prentice-Hall, Inc.
0.2 .0793 .0832 .0871
0.3 .1179 .1217 .1255
Chap. 6 - 13
Finding X Values
for Known Probabilities
Normal Distribution
Standardized Normal Distribution
s = 10
s =1
.1217
m =5
?
X
.1217
m = 0 .31
Z
X = m + Zs = 5 + (0.31)(10) = 8.1
© 1999 Prentice-Hall, Inc.
Shaded Area Exaggerated
Chap. 6 - 14
Assessing Normality
Compare Data Characteristics
to Properties of Normal
Distribution
• Put Data into Ordered Array
• Find Corresponding Standard
Normal Quantile Values
• Plot Pairs of Points
• Assess by Line Shape
© 1999 Prentice-Hall, Inc.
Normal Probability Plot
for Normal Distribution
90
X 60
Z
30
-2 -1 0 1 2
Look for Straight Line!
Chap. 6 - 15
Normal Probability Plots
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
© 1999 Prentice-Hall, Inc.
Z
30
Z
30
-2 -1 0 1 2
Chap. 6 - 16
Exponential Distributions
P (arrival time < X ) = 1 - e
e
-l x
= the mathematical constant
2.71828
l = the population mean of arrivals
X = any value of the continuous random
variable
e.g.
Drivers Arriving at a Toll Bridge
Customers Arriving at an ATM Machine
© 1999 Prentice-Hall, Inc.
Chap. 6 - 17
Exponential Distributions
Describes time or
distance between
events
 Used for queues
l = 0.5
f(X)
l = 2.0
Density function
1
f(x) =
l
e -x/l
X
Parameters
m = l, s = l
© 1999 Prentice-Hall, Inc.
Chap. 6 - 18
Estimation
•Sample Statistic Estimates Population Parameter
_
e.g. X = 50 estimates Population Mean, m
•Problems: Many samples provide many estimates of the
Population Parameter.
Determining adequate sample size: large sample give better
estimates. Large samples more costly.
How good is the estimate?
•Approach to Solution: Theoretical Basis is Sampling
Distribution.
© 1999 Prentice-Hall, Inc.
Chap. 6 - 19
Sampling Distributions
•Theoretical Probability Distribution
• Random Variable is Sample Statistic:
Sample Mean, Sample Proportion
• Results from taking All Possible Samples of
the Same Size
•Comparing Size of Population and Size of Sampling Distribution
Population Size = 100
Size of Samples = 10
13
Sampling Distribution Size = 1.7310
(Sampling Without Replacement)
© 1999 Prentice-Hall, Inc.
Chap. 6 - 20
Developing
Sampling Distributions
Suppose there’s a
population...
B
C
Population size, N = 4
Random variable, X,
is Age of individuals
Values of X: 18, 20, 22, 24
measured in years
D
A
© 1984-1994 T/Maker Co.
© 1999 Prentice-Hall, Inc.
Chap. 6 - 21
Population Characteristics
Summary Measure
N
m =
Population Distribution
 Xi
P(X)
N
.3
i =1
18 + 20 + 22 + 24
=
= 21
4
.2
.1
0
 X i  m )
N
s =
i =1
© 1999 Prentice-Hall, Inc.
N
2
= 2 . 236
A
B
C
D
(18)
(20)
(22)
(24)
X
Uniform Distribution
Chap. 6 - 22
All Possible Samples of Size n = 2
1st
Obs
2nd Observation
18
20
22
24
18 18,18 18,20 18,22 18,24
20 20,18 20,20 20,22 20,24
16 Sample Means
22 22,18 22,20 22,22 22,24
1st 2nd Observation
Obs 18 20 22 24
24 24,18 24,20 24,22 24,24
18 18 19 20 21
16 Samples
Samples Taken with
Replacement
© 1999 Prentice-Hall, Inc.
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
Chap. 6 - 23
Sampling Distribution
of All Sample Means
16 Sample Means
1st
Obs
Sample Means
Distribution
2nd Observation
18 20 22 24
P(X)
18 18 19 20 21
.3
20 19 20 21 22
.2
22 20 21 22 23
.1
24 21 22 23 24
# in sample = 2,
© 1999 Prentice-Hall, Inc.
0
_
18 19
20 21 22 23
24
X
# in Sampling Distribution = 16
Chap. 6 - 24
Summary Measures for the
Sampling Distribution
N
mx =
 Xi
i =1
N
18 + 19 + 19 +  + 24
=
= 21
16
Xi  mx )
N
sx =
=
© 1999 Prentice-Hall, Inc.
2
i =1
N
18 21) + 19 21)
2
2
16
+ + 24 21)
2
= 1.58
Chap. 6 - 25
Comparing the Population with its
Sampling Distribution
Population
N=4
Sample Means Distribution
n=2
m = 21, s = 2.236
m x = 21
P(X)
.3
P(X)
.3
.2
.2
.1
.1
0
0
A
B
C
(18)
(20)
(22)
© 1999 Prentice-Hall, Inc.
D X
s
x
= 1 . 58
_
18 19
20 21 22 23
24
(24)
Chap. 6 - 26
X
Properties of Summary
Measures
• Population Mean Equal to
Sampling Mean m x = m
• The Standard Error (standard deviation)
of the Sampling distribution is Less than
Population Standard Deviation
• Formula (sampling with replacement):
s x_ =
© 1999 Prentice-Hall, Inc.
s
n
As n increase,
s x_
decrease.
Chap. 6 - 27
Properties of the Mean
Unbiasedness
 Mean of sampling distribution equals
population mean
Efficiency
 Sample mean comes closer to population mean
than any other unbiased estimator
Consistency
 As sample size increases, variation of sample
mean from population mean decreases
© 1999 Prentice-Hall, Inc.
Chap. 6 - 28
Unbiasedness
P(X)
Unbiased
Biased
m
© 1999 Prentice-Hall, Inc.
X
Chap. 6 - 29
Efficiency
P(X) Sampling
Distribution
of Median
Sampling
Distribution of
Mean
X
m
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Chap. 6 - 30
Consistency
Larger
sample size
P(X)
B
Smaller
sample size
A
m
© 1999 Prentice-Hall, Inc.
X
Chap. 6 - 31
When the Population is Normal
Population Distribution
s = 10
Central Tendency
m_ = m
x
Variation
s
_
sx =
n
Sampling with
Replacement
m = 50
Sampling Distributions
n=4
s `X = 5
n =16
s`X = 2.5
m X-X = 50
© 1999 Prentice-Hall, Inc.
X
X
Chap. 6 - 32
Central Limit Theorem
As Sample
Size Gets
Large
Enough
Sampling
Distribution
Becomes
Almost Normal
regardless of
shape of
population
X
X
© 1999 Prentice-Hall, Inc.
Chap. 6 - 33
When The Population is
Not Normal
Central Tendency
Population Distribution
s = 10
mx = m
Variation
s
x
=
s
n
Sampling with
Replacement
m = 50
X
Sampling Distributions
n=4
s`X = 5
n =30
s`X = 1.8
X
m X = 50
© 1999 Prentice-Hall, Inc.
Chap. 6 - 34
Example: Sampling Distribution
X  m 7.8  8
Z=
=
= .50
s / n 2 / 25
X m
8 .2  8
Standardized
=
= . 50
Sampling Z =
s / n 2 / 25
Normal Distribution
Distribution
s X = .4
s=1
.3830
.1915 .1915
7.8 8 8.2
© 1999 Prentice-Hall, Inc.
Z
m =0
Chap. 6 - 35
Population Proportions
• Categorical variable (e.g., gender)
• % population having a characteristic
• If two outcomes, binomial distribution
 Possess or don’t possess characteristic
• Sample proportion (ps)
X
number of successes
Ps =
=
n
sample size
© 1999 Prentice-Hall, Inc.
Chap. 6 - 36
Sampling Distribution of
Proportion
Approximated by
normal distribution
 n·p  5
  n·(1 - p)  5
Mean
mP = p
Standard error
sP =
© 1999 Prentice-Hall, Inc.
p  1  p )
n
Sampling Distribution
P(ps)
.3
.2
.1
0
0
.2
.4
.6
8
1
ps
p = population proportion
Chap. 6 - 37
Standardizing Sampling
Distribution of Proportion
Z @
ps - m p
sp
Sampling
Distribution
p( 1  p )
n
Standardized
Normal Distribution
sp
s=1
mp
© 1999 Prentice-Hall, Inc.
=
ps - p
ps
Z
m =0
Chap. 6 - 38
Example: Sampling
Distribution of Proportion
 np  5
n( 1  p )  5
p
p
s
Z@
=
p( 1  p )
n
Sampling
Distribution
-
sp = .0346
.43 - .40
= .87
.40  ( 1  .40 )
200
Standardized
Normal Distribution
s=1
..3078
mp = .40
© 1999 Prentice-Hall, Inc.
.43
ps
m = 0 .87
Chap. 6 - 39
Z
Sampling from Finite Populations
• Modify Standard Error if Sample Size (n) is
Large Relative to Population Size (N)
n > .05·N (or n/N > .05)
• Use Finite Population Correction Factor
(fpc)
• Standard errors if n/N > .05:
sx =
© 1999 Prentice-Hall, Inc.
s
n

N n
N 1
sP =
p  1  p )

n
N  n)
 N  1)
Chap. 6 - 40
Chapter Summary
•Discussed The Normal Distribution
•Described The Standard Normal Distribution
•Assessed the Normality Assumption
•Defined The Exponential Distribution
•Discussed Sampling Distribution of the Mean
•Described Sampling Distribution of the Proportion
•Defined Sampling From Finite Populations
© 1999 Prentice-Hall, Inc.
Chap. 6 - 41