CONSTRUCTING RELATIVE & CUMULATIVE FREQUENCY DISTRIBUTIONS using EXCEL & WORD.
Statistics for Managers Using Microsoft Excel · PDF fileStatistics for Managers Using...
Transcript of Statistics for Managers Using Microsoft Excel · PDF fileStatistics for Managers Using...
© 1999 Prentice-Hall, Inc. Chap. 6 - 1
Statistics for Managers
Using Microsoft Excel
Chapter 6
The Normal Distribution And Other
Continuous Distributions
© 1999 Prentice-Hall, Inc. Chap. 6 - 2
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 - 3
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 - 4
The Normal Distribution
• ‘Bell Shaped’
• Symmetrical
• Mean, Median and
Mode are Equal
• ‘Middle Spread’
Equals 1.33 s
• Random Variable has
Infinite Range
Mean
Median
Mode
X
f(X)
m
© 1999 Prentice-Hall, Inc. Chap. 6 - 5
The Mathematical Model
f(X) = frequency of random variable X
p = 3.14159; e = 2.71828
s = population standard deviation
X = value of random variable (- < X < )
m = population mean
f(X) = 1 e
(-1/2) ((X- m)/s) 2
sp2
© 1999 Prentice-Hall, Inc. Chap. 6 - 6
Many Normal Distributions
Varying the Parameters s and m, we obtain
Different Normal Distributions.
There are
an Infinite
Number
© 1999 Prentice-Hall, Inc. Chap. 6 - 7
Normal Distribution:
Finding Probabilities
Probability is the
area under the
curve!
c d X
f(X)
P c X d ( ) ? =
© 1999 Prentice-Hall, Inc. Chap. 6 - 8
Which Table?
Infinitely Many Normal Distributions Means
Infinitely Many Tables to Look Up!
Each distribution
has its own table?
© 1999 Prentice-Hall, Inc. Chap. 6 - 9
Z Z
Z = 0.12
Z .00 .01
0.0 .0000 .0040 .0080
.0398 .0438
0.2 .0793 .0832 .0871
0.3 .0179 .0217 .0255
The Standardized Normal
Distribution
.0478 .02
0.1 .0478
Standardized Normal Probability
Table (Portion) m = 0 and s = 1
Probabilities
Shaded Area
Exaggerated
© 1999 Prentice-Hall, Inc. Chap. 6 - 10
Z m = 0
s Z = 1
.12
Standardizing Example
Normal
Distribution
Standardized
Normal Distribution
X m = 5
s = 10
6.2
12010
526.
.XZ =
=
=
s
m
Shaded Area Exaggerated
© 1999 Prentice-Hall, Inc. Chap. 6 - 11
0
s = 1
-.21 Z .21
Example:
P(2.9 < X < 7.1) = .1664
Normal
Distribution
.1664
.0832 .0832
Standardized
Normal Distribution
Shaded Area Exaggerated
5
s = 10
2.9 7.1 X
2110
592.
.xz =
=
=
s
m
2110
517.
.xz =
=
=
s
m
© 1999 Prentice-Hall, Inc. Chap. 6 - 12
Z m = 0
s = 1
.30
Example: P(X 8) = .3821
Normal
Distribution
Standardized
Normal Distribution
.1179
.5000
.3821
Shaded Area Exaggerated
.
X m = 5
s = 10
8
3010
58.
xz =
=
=
s
m
© 1999 Prentice-Hall, Inc. Chap. 6 - 13
Z .00 0.2
0.0 .0000 .0040 .0080
0.1 .0398 .0438 .0478
0.2 .0793 .0832 .0871
.1179 .1255
Z m = 0
s = 1
.31
Finding Z Values
for Known Probabilities
.1217 .01
0.3
Standardized Normal
Probability Table (Portion)
What Is Z Given
P(Z) = 0.1217?
Shaded Area
Exaggerated .1217
© 1999 Prentice-Hall, Inc. Chap. 6 - 14
Z m = 0
s = 1
.31 X m = 5
s = 10
?
Finding X Values
for Known Probabilities
Normal Distribution Standardized Normal Distribution
.1217 .1217
Shaded Area Exaggerated
X 8.1 = m + Zs = 5 + (0.31)(10) =
© 1999 Prentice-Hall, Inc. Chap. 6 - 15
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
Normal Probability Plot
for Normal Distribution
Look for Straight Line!
30
60
90
-2 -1 0 1 2
Z
X
© 1999 Prentice-Hall, Inc. Chap. 6 - 16
Normal Probability Plots
Left-Skewed Right-Skewed
Rectangular U-Shaped
30
60
90
-2 -1 0 1 2
Z
X
30
60
90
-2 -1 0 1 2
Z
X
30
60
90
-2 -1 0 1 2
Z
X
30
60
90
-2 -1 0 1 2
Z
X
© 1999 Prentice-Hall, Inc. Chap. 6 - 17
Exponential Distributions
P arrival time < X ( ) = 1 - e
e = the mathematical constant
2.71828
-l x
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 - 18
Describes time or
distance between
events
Used for queues
Density function
Parameters
Exponential Distributions
f(x) = 1
l e -x/l
m = l, s = l
f(X)
X
l = 0.5
l = 2.0
© 1999 Prentice-Hall, Inc. Chap. 6 - 19
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 - 20
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
Sampling Distribution Size =
(Sampling Without Replacement)
1.7310 13
© 1999 Prentice-Hall, Inc. Chap. 6 - 21
Population size, N = 4
Random variable, X,
is Age of individuals
Values of X: 18, 20, 22, 24
measured in years
© 1984-1994 T/Maker Co.
Developing Sampling Distributions
A
B C
D
Suppose there’s a
population...
© 1999 Prentice-Hall, Inc. Chap. 6 - 22
)2362
214
24222018
1
2
1
.N
X
N
X
N
ii
N
ii
=
=
=+++
=
=
=
=
ms
m
Population Characteristics
Summary Measure Population Distribution
.3
.2
.1
0 A B C D
(18) (20) (22) (24)
Uniform Distribution
P(X)
X
© 1999 Prentice-Hall, Inc. Chap. 6 - 23
1st
2nd
Observation
Obs 18 20 22 24
18 18,18 18,20 18,22 18,24
20 20,18 20,20 20,22 20,24
22 22,18 22,20 22,22 22,24
24 24,18 24,20 24,22 24,24
16 Samples
Samples Taken with Replacement
16 Sample Means
All Possible Samples of Size n = 2
1st 2nd Observation
Obs 18 20 22 24
18 18 19 20 21
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
1st 2nd Observation
Obs 18 20 22 24
18 18 19 20 21
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
© 1999 Prentice-Hall, Inc. Chap. 6 - 24
1st 2nd Observation
Obs 18 20 22 24
18 18 19 20 21
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
1st 2nd Observation
Obs 18 20 22 24
18 18 19 20 21
20 19 20 21 22
22 20 21 22 23
24 21 22 23 2418 19 20 21 22 23 24
0
.1
.2
.3
P(X)
X
Sample Means
Distribution
16 Sample Means
Sampling Distribution
of All Sample Means
# in sample = 2, # in Sampling Distribution = 16
_
© 1999 Prentice-Hall, Inc. Chap. 6 - 25
2116
241919181 =++++
=
= =
N
XN
ii
xm
)
) ) )581
16
212421192118222
1
2
.
N
XN
ixi
x
=+++
=
= =
ms
Summary Measures for the
Sampling Distribution
© 1999 Prentice-Hall, Inc. Chap. 6 - 26
18 19 20 21 22 23 24 0
.1
.2
.3 P(X)
X
Sample Means Distribution n = 2
Comparing the Population with its
Sampling Distribution
A B C D
(18) (20) (22) (24)
0
.1
.2
.3
Population
N = 4 m = 21, s = 2.236
P(X)
X
21=xm 581 .x =s
_
© 1999 Prentice-Hall, Inc. Chap. 6 - 27
• Population Mean Equal to
Sampling Mean
• The Standard Error (standard deviation)
of the Sampling distribution is Less than
Population Standard Deviation
• Formula (sampling with replacement):
Properties of Summary
Measures
mm =x
As n increase, decrease. s x =
s s
x n
_ _
© 1999 Prentice-Hall, Inc. Chap. 6 - 28
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
Properties of the Mean
© 1999 Prentice-Hall, Inc. Chap. 6 - 29
m
Unbiasedness
Biased Unbiased
P(X)
X
© 1999 Prentice-Hall, Inc. Chap. 6 - 30
m
Efficiency
Sampling
Distribution
of Median Sampling
Distribution of
Mean
X
P(X)
© 1999 Prentice-Hall, Inc. Chap. 6 - 31
m
Larger
sample size
Smaller
sample size
Consistency
X
P(X)
A
B
© 1999 Prentice-Hall, Inc. Chap. 6 - 32
m = 50
s = 10
Xm = 50
s = 10
X
mX
= 50- XmX
= 50- X
n =16 s`X = 2.5
n = 4 s`X = 5
When the Population is Normal
Central Tendency
Variation
Sampling with
Replacement
Population Distribution
Sampling Distributions
m x
m =
s s
x = n
_
_
© 1999 Prentice-Hall, Inc. Chap. 6 - 33
XX
Central Limit Theorem
As Sample
Size Gets
Large
Enough
Sampling
Distribution
Becomes
Almost Normal
regardless of
shape of
population
© 1999 Prentice-Hall, Inc. Chap. 6 - 34
nx
ss =
mm =x
n =30 s`X = 1.8 n = 4
s`X = 5
When The Population is
Not Normal
Central Tendency
Variation
Sampling with
Replacement
Population Distribution
Sampling Distributions
m = 50
s = 10
X
X50=mX
© 1999 Prentice-Hall, Inc. Chap. 6 - 35
Example: Sampling Distribution
Sampling
Distribution
Standardized
Normal Distribution
.1915
50252
887.
/
.
n/
XZ =
=
=
s
m
4.X =s
7.8 8 8.2 m = 0 Z
s = 1
.3830
.1915
50252
828.
/
.
n/
XZ =
=
=
s
m
© 1999 Prentice-Hall, Inc. Chap. 6 - 36
• Categorical variable (e.g., gender)
• % population having a characteristic
• If two outcomes, binomial distribution
Possess or don’t possess characteristic
• Sample proportion (ps)
sizesample
successes of number
n
XPs ==
Population Proportions
© 1999 Prentice-Hall, Inc. Chap. 6 - 37
Approximated by
normal distribution
n·p 5
n·(1 - p) 5
Mean
Standard error
pP =m
)n
ppP
=
1s
Sampling Distribution of
Proportion
p = population proportion
Sampling Distribution
P(ps)
.3
.2
.1
0 0 . 2 .4 .6 8 1
ps
© 1999 Prentice-Hall, Inc. Chap. 6 - 38
Standardizing Sampling
Distribution of Proportion
Sampling
Distribution
Standardized Normal Distribution
Z p p s @ s - m p
s p =
p -
n
)p(p 1
ps Z m = 0
sp
mp
s = 1
© 1999 Prentice-Hall, Inc. Chap. 6 - 39
Example: Sampling
Distribution of Proportion
51
5
)p(n
np
Sampling
Distribution Normal Distribution
Standardized
Z @ p s -
-
p =
.43 - .40 = .87
n
)p(p 1200
40140 ).(.
sp = .0346
ps
s = 1
m = 0 .87 Z
..3078
mp = .40 .43
© 1999 Prentice-Hall, Inc. Chap. 6 - 40
• 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:
1
=
N
nN
nx
ss
) ) )1
1
=
N
nN
n
ppPs
Sampling from Finite Populations
© 1999 Prentice-Hall, Inc. Chap. 6 - 41
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