Download Chap. 5: The Normal Distribution & Sampling Distributions

© 1998 Prentice-Hall, Inc.
Statistics for Managers Using
Microsoft Excel
Normal Distribution &
Sampling Distributions

5-1
Chapter 5
Learning Objectives
 Describe the normal probability distribution
 Solve probability problems involving the
normal distribution
 Describe the exponential distribution
 Describe the properties of estimators
 Explain sampling distribution
 State the central limit theorem
Statistics for Managers Using Microsoft Excel, 1/e
Data Types
© 1998 Prentice-Hall, Inc.
Data
Numerical
(Quantitative)
Discrete
5-2
Categorical
(Qualitative)
Continuous
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Continuous Random Variables
© 1998 Prentice-Hall, Inc.

Numerical outcome of an experiment

Weight of a student


Continuous random variable



Whole or fractional number
Obtained by measuring
Infinite number of values in interval

5-3
Observe 115, 156.8, 190.1, 225
Too many to list like discrete variable
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Continuous Random Variable
Examples
Experiment
Random
Variable
Possible
Values
Weigh 100 people
Weight
45.1, 78, ...
Measure part life
Hours
900, 875.9, ...
Ask food spending
Spending
54.12, 42, ...
Measure time between Inter-arrival 0, 1.3, 2.78, ...
arrivals
time
5-4
Statistics for Managers Using Microsoft Excel, 1/e
Continuous Probability Density
Function
© 1998 Prentice-Hall, Inc.

Mathematical formula
Frequency

Shows all values, X,
& frequencies, f(X)


f(X)
f(X) is not probability
Properties

f ( X ) dx  1
All X
(Area under curve)
f ( X )  0, a  X  b
5-5
(Value, Frequency)
a
X
b
Value
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Continuous Random Variable
Probability
d
P(c  X  d )   f ( X ) dx
c
f(X)
Probability is
area under curve!
a
5-6
c
d
b
X
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Continuous Probability
Distribution Models
Continuous
Probability
Distribution
Uniform
5-7
Normal
Exponential
Other
Statistics for Managers Using Microsoft Excel, 1/e
Normal Distribution - Importance
© 1998 Prentice-Hall, Inc.

Describes many random processes or
continuous phenomena

Can be used to approximate discrete
probability distributions



5-8
Binomial
Poisson
Basis for classical statistical inference
Statistics for Managers Using Microsoft Excel, 1/e
Normal Distribution
© 1998 Prentice-Hall, Inc.

‘Bell-shaped’ &
symmetrical

Mean, median, mode
are equal

‘Middle spread’ is
1.33 

5-9
f(X )
Random variable has
infinite range
X
Mean
Median
Mode
Statistics for Managers Using Microsoft Excel, 1/e
Probability Density Function
© 1998 Prentice-Hall, Inc.
f (X) 
f(X)


X

5 - 10
=
=
=
=
=
1
2
2
e
1 X    2
 

2  
frequency of random variable X
3.14159; e = 2.71828
population standard deviation
value of random variable (- < X < )
population mean
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Effect of Varying Parameters
( & )
(a = b) < c
b < (a = c)
Distributions A & B have same
mean which is less than C’s
Distributions A & C have same standard
deviation which is greater than B’s
f(X)
B
A
C
X
5 - 11
Statistics for Managers Using Microsoft Excel, 1/e
Normal Distribution Probability
© 1998 Prentice-Hall, Inc.
d
P(c  X  d)   f (X) dx?
Probability is
area under
curve!
c
f(X)
c
5 - 12
d
X
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Infinite Number of Tables
Normal distributions differ by
mean & standard deviation.
Each distribution would
require its own table.
f(X)
X
That’s an infinite number!
5 - 13
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Standardize the Normal Distribution
© 1998 Prentice-Hall, Inc.
X 
Z

Normal
Distribution
Standardized
Normal Distribution

= 1

X
=0
Z
One table!
5 - 14
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Standardizing Example
X   6.2  5
Z

 .12

10
Normal
Distribution
 = 10
= 5 6.2 X
5 - 15
Standardized
Normal Distribution
=1
= 0 .12
Z
Statistics for Managers Using Microsoft Excel, 1/e
Obtaining
the Probability
© 1998 Prentice-Hall, Inc.
Standardized Normal
Probability Table (Portion)
Z
.00
.01
=1
.02
0.0 .0000 .0040 .0080
.0478
0.1 .0398 .0438 .0478
0.2 .0793 .0832 .0871
= 0 .12
0.3 .1179 .1217 .1255
Probabilities
5 - 16
Z
Note: Shaded area
exaggerated in size
Statistics for Managers Using Microsoft Excel, 1/e
Example
P(3.8  X  5)
© 1998 Prentice-Hall, Inc.
X   3.8  5
Z

  .12

10
Standardized
Normal Distribution
Normal
Distribution
 = 10
=1
.0478
3.8 = 5
5 - 17
X
-.12  = 0
Shaded area exaggerated
Z
Statistics for Managers Using Microsoft Excel, 1/e
Example
P(2.9  X  7.1)
© 1998 Prentice-Hall, Inc.
Normal
Distribution
X 
2 .9  5
Z 

  . 21

10
X 
7 .1  5
Z 

 . 21

10
Standardized
Normal Distribution
 = 10
 = 1
.1664
.0832 .0832
2.9 5 7.1 X
5 - 18
-.2 1 0 .2 1
Shaded area exaggerated
Z
Statistics for Managers Using Microsoft Excel, 1/e
Example P(X  8)
© 1998 Prentice-Hall, Inc.
X  85
Z

 .30

10
Standardized
Normal Distribution
Normal
Distribution
 = 10
=1
.5000
.3821
.1179
=5
5 - 19
8
X
Shaded area exaggerated
=0
.30 Z
Statistics for Managers Using Microsoft Excel, 1/e
Example
P(7.1  X  8)
© 1998 Prentice-Hall, Inc.
X   7 .1  5
Z 

 . 21

10
X  85
Z 

 . 30

10
Normal
Distribution
Standardized
Normal Distribution
 = 10
 = 1
.0347
.1179
.0832
=5
5 - 20
7.1 8
X
 = 0
Shaded area exaggerated
.21 .30
Z
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Normal Distribution Thinking
Challenge
You work in Quality Control for
GE. Light bulb life has a normal
distribution with
= 2000 hours & = 200 hours.
What’s the probability that a
bulb will last

between 2000 & 2400
hours?

less than 1470 hours?
5 - 21
Statistics for Managers Using Microsoft Excel, 1/e
Solution*
P(2000  X  2400)
© 1998 Prentice-Hall, Inc.
X   2400 2000
Z

 2.0

200
Standardized
Normal Distribution
Normal
Distribution
 = 200
=1
.4772
 = 2000 2400
5 - 22
X
Shaded area exaggerated
=0
2.0
Z
Statistics for Managers Using Microsoft Excel, 1/e
Solution*
P(X  1470)
© 1998 Prentice-Hall, Inc.
X   1470  2000
Z

  2.65

200
Standardized
Normal Distribution
Normal
Distribution
=1
 = 200
.5000
.0040
1470  = 2000
5 - 23
X
.4960
-2.65  = 0
Shaded area exaggerated
Z
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Finding Z Values
for Known Probabilities
Standardized Normal
Probability Table (Portion)
What is Z given
P(Z) = .1217?
.1217
=1
Z
.00
.01
0.2
0.0 .0000 .0040 .0080
0.1 .0398 .0438 .0478
0.2 .0793 .0832 .0871
=0 ?
5 - 24
Z
0.3 .1179 .1217
Shaded area exaggerated
.1255
Statistics for Managers Using Microsoft Excel, 1/e
Finding Z Values
for Known Probabilities
© 1998 Prentice-Hall, Inc.
Standardized Normal
Probability Table (Portion)
What is Z given
P(Z) = .1217?
.1217
=1
Z
.00
.01
0.2
0.0 .0000 .0040 .0080
0.1 .0398 .0438 .0478
 = 0 .31
Z
0.2 .0793 .0832 .0871
0.3 .1179 .1217
5 - 25
Shaded area exaggerated
.1255
Statistics for Managers Using Microsoft Excel, 1/e
Finding X Values
for Known Probabilities
© 1998 Prentice-Hall, Inc.
Standardized Normal Distribution
Normal Distribution
 = 10
=1
.1217
= 5
?
X
.1217
 = 0 .31
Z
X    Z    5  .31  10   8.1
5 - 26
Shaded area exaggerated
Statistics for Managers Using Microsoft Excel, 1/e
Assessing Normality
© 1998 Prentice-Hall, Inc.


Compare data
characteristics to
properties of normal
distribution
Evaluate normal
probability plot


Create on computer
Plot of data values &
standardized quantile
values
Normal Probability Plot
for Normal Distribution
90
X 60
Z
30
-2 -1
0
1
2
Look for straight line!
5 - 27
Statistics for Managers Using Microsoft Excel, 1/e
Normal Probability Plots
© 1998 Prentice-Hall, Inc.
Left-Skewed
Right-Skewed
90
90
X 60
X 60
Z
30
-2 -1 0
1
2
-2 -1 0
Rectangular
90
X 60
X 60
Z
-2 -1 0
5 - 28
1
2
1
2
U-Shaped
90
30
Z
30
Z
30
-2 -1 0
1
2
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.

Describes time or
distance between
events



Exponential Distribution
Used for queues
Density function
1 x 
f (x)  e

f(x)
 = 2.0
 =0.5
Parameters
  ,   
5 - 29
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Exponential Distribution Probability
© 1998 Prentice-Hall, Inc.
P ( x  a)  1  e
f(x)
a 
P ( x  a)  e  a 
a
5 - 30
x
Statistics for Managers Using Microsoft Excel, 1/e
Exponential Distribution Example
© 1998 Prentice-Hall, Inc.
A college has a single
counter at the Registrar’s
Office. A student arrives
on average every 10
minutes following an
exponential distribution.
What is the probability
that more than 30
minutes pass without a
student’s arriving?
5 - 31
© 1995 Corel Corp.
Statistics for Managers Using Microsoft Excel, 1/e
Exponential Distribution Solution
© 1998 Prentice-Hall, Inc.
P ( x  a)  e
a 
P ( x  30)  e  30 10
 0.049787
 5%
© 1995 Corel Corp.
5 - 32
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Statistical Methods
© 1998 Prentice-Hall, Inc.
Statistical
Statistical
Methods
Methods
Descriptive
Descriptive
Statistics
Statistics
5 - 33
Inferential
Inferential
Statistics
Statistics
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Inferential Statistics
© 1998 Prentice-Hall, Inc.

Involves



Estimation
Hypothesis testing
Population?
Purpose

5 - 34
Make decisions about
population
characteristics
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Inference Process
© 1998 Prentice-Hall, Inc.
Estimates
& tests
Sample
statistic
(X, Ps )
5 - 35
Population
Sample
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Estimators
© 1998 Prentice-Hall, Inc.

Random variables used to estimate a
population parameter


Example: Sample meanX is an
estimator of population mean 


Sample mean, sample proportion, sample
median
IfX = 3 then 3 is the estimate of 
Theoretical basis is sampling
distribution
5 - 36
Statistics for Managers Using Microsoft Excel, 1/e
Sampling Distribution
© 1998 Prentice-Hall, Inc.

Theoretical probability distribution

Random variable is sample statistic

Sample mean, sample proportion etc.

Results from drawing all possible
samples of a fixed size

List of all possible [X, P(X) ] pairs

5 - 37
Sampling distribution of mean
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Developing
Sampling Distributions
Suppose there’s a
population ...

Population size, N = 4

Random variable, X,
is # errors in work

Values of X: 1, 2, 3, 4

Uniform distribution
5 - 38
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Population Characteristics
© 1998 Prentice-Hall, Inc.
Summary Measures
Population Distribution
N
 

X
i 1
N
i
.3
.2
.1
.0
 2 .5
1
N
 
5 - 39
 X
i 1
i
N
 
2
3
4
2
 1 . 12
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
All Possible Samples
of Size n = 2
16 Samples
1st 2nd Observation
Obs 1
2
3
4
1 1,1 1,2 1,3 1,4
2 2,1 2,2 2,3 2,4
3 3,1 3,2 3,3 3,4
4 4,1 4,2 4,3 4,4
Sample with replacement
5 - 40
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
All Possible Samples
of Size n = 2
16 Samples
16 Sample Means
1st 2nd Observation
Obs 1
2
3
4
1st 2nd Observation
Obs 1
2
3
4
1 1,1 1,2 1,3 1,4
1 1.0 1.5 2.0 2.5
2 2,1 2,2 2,3 2,4
2 1.5 2.0 2.5 3.0
3 3,1 3,2 3,3 3,4
3 2.0 2.5 3.0 3.5
4 4,1 4,2 4,3 4,4
4 2.5 3.0 3.5 4.0
Sample with replacement
5 - 41
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Sampling Distribution
of All Sample Means
16 Sample Means
Sampling
Distribution
1st 2nd Observation
Obs 1
2
3
4
1 1.0 1.5 2.0 2.5
2 1.5 2.0 2.5 3.0
3 2.0 2.5 3.0 3.5
4 2.5 3.0 3.5 4.0
5 - 42
P(X)
.3
.2
.1
.0
X
1.0 1.5 2.0 2.5 3.0 3.5 4.0
Statistics for Managers Using Microsoft Excel, 1/e
Summary Measures of
All Sample Means
© 1998 Prentice-Hall, Inc.
N
x 
 Xi
1.0  1.5    4.0

 2.5
16
i 1
N
 X
N
x 
i 1
 x 
2
i
N
1.0  2.5  1.5  2.5
2

5 - 43
2
16
  4.0  2.5
2
 .79
Statistics for Managers Using Microsoft Excel, 1/e
Comparison
© 1998 Prentice-Hall, Inc.
Population
.3
.2
.1
.0
Sampling Distribution
P(  X)
.3
.2
.1
.0
P(X)
1
5 - 44
2
3
4
X
1 1.5 2 2.5 3 3.5 4
  2.5
 x  2.5
  112
.
 x  .79
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.

Standard deviation of all possible
sample means,X


5 - 45
Standard Error
of Mean
Measures scatter in all sample means,X
Less than pop. standard deviation
Statistics for Managers Using Microsoft Excel, 1/e
Standard Error
of Mean
© 1998 Prentice-Hall, Inc.

Standard deviation of all possible
sample means,X

Measures scatter in all sample means,X

Less than pop. standard deviation

Formula (sampling with replacement):

5 - 46
x


n
Statistics for Managers Using Microsoft Excel, 1/e
Properties of Mean
© 1998 Prentice-Hall, Inc.

Unbiasedness


Efficiency


Sample mean comes closer to population
mean than any other unbiased estimator
Consistency

5 - 47
Mean of sampling distribution equals
population mean
As sample size increases, variation of sample
mean from population mean decreases
Statistics for Managers Using Microsoft Excel, 1/e
Unbiasedness
© 1998 Prentice-Hall, Inc.
P(X)
Unbiased
A
Biased
B

5 - 48
X
Statistics for Managers Using Microsoft Excel, 1/e
Efficiency
© 1998 Prentice-Hall, Inc.
P(X)
Sampling
Distribution
of Mean
Sampling
Distribution
of Median
B
A

5 - 49
X
Statistics for Managers Using Microsoft Excel, 1/e
Consistency
© 1998 Prentice-Hall, Inc.
P(X)
Larger
sample
size
Smaller
sample
size
B
A

5 - 50
X
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Sampling from
Normal Populations
Central Tendency
x  
Population Distribution
= 10
Dispersion

x 
n
Sampling with
replacement
 = 50
Sampling Distribution
n=4
X = 5
n =16
X = 2.5
 X- = 50
5 - 51
X
X
Statistics for Managers Using Microsoft Excel, 1/e
Standardizing Sampling
Distribution of Mean
© 1998 Prentice-Hall, Inc.
Sampling
Distribution
X  x X  
Z


x
n
Standardized
Normal Distribution
X
= 1
X
5 - 52
X
 =0
Z
Statistics for Managers Using Microsoft Excel, 1/e
Thinking Challenge
© 1998 Prentice-Hall, Inc.
You’re an operations
analyst for AT&T. Longdistance telephone calls
are normally distributed
with  = 8 min. &  = 2
min. If you select
random samples of 25
calls, what percentage of
the sample means would
be between 7.8 & 8.2
minutes?
5 - 53
© 1984-1994 T/Maker Co.
Statistics for Managers Using Microsoft Excel, 1/e
Sampling Distribution Solution*
© 1998 Prentice-Hall, Inc.
X   7 .8  8
Z

  .50
 n 2 25
Sampling
Distribution
X   8 .2  8
Z

 .50
 n 2 25
X = .4
Standardized
Normal Distribution
=1
.3830
.1915 .1915
7.8 8 8.2 X
5 - 54
-.50 0 .50
Z
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.
Sampling from
Non-Normal Populations
Central Tendency
Population Distribution
x  
= 10
Dispersion

x 
n

Sampling with
replacement
 = 50
Sampling Distribution
n=4
 X = 5
n =30
X = 1.8
 X- = 50
5 - 55
X
X
Statistics for Managers Using Microsoft Excel, 1/e
Central Limit Theorem
© 1998 Prentice-Hall, Inc.
As
sample
size gets
large
enough
( 30) ...
sampling
distribution
becomes
almost
normal.
X
5 - 56
Statistics for Managers Using Microsoft Excel, 1/e
Proportions
© 1998 Prentice-Hall, Inc.

Categorical variable (e.g., gender)

% population having a characteristic

If two outcomes, binomial distribution

5 - 57
Possess or don’t possess characteristic
Statistics for Managers Using Microsoft Excel, 1/e
Proportions
© 1998 Prentice-Hall, Inc.

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
5 - 58
Statistics for Managers Using Microsoft Excel, 1/e
Sampling Distribution
of Proportion
© 1998 Prentice-Hall, Inc.

Approximated by
normal distribution




n·p  5
n·(1 - p)  5
Mean
P  p
Standard error
P 
5 - 59
p  1  p 
n
Sampling Distribution
P(Ps)
.3
.2
.1
.0
.0
.2
.4
.6
.8
Ps
1.0
where p = Population
proportion
Statistics for Managers Using Microsoft Excel, 1/e
Standardizing Sampling Distribution:
Proportion
© 1998 Prentice-Hall, Inc.
Ps   P
Z

P
Ps  p
p  (1  p )
n
Standardized
Normal Distribution
Sampling
Distribution
P
= 1
P
5 - 60
Ps
 =0
Z
Statistics for Managers Using Microsoft Excel, 1/e
Thinking Challenge
© 1998 Prentice-Hall, Inc.
You’re manager of a
bank. 40% of depositors
have multiple accounts.
You select a random
sample of 200
customers. What is the
probability that the
sample proportion of
depositors with multiple
accounts would be
between 40% & 43% ?
5 - 61
© 1984-1994 T/Maker Co.
Statistics for Managers Using Microsoft Excel, 1/e
Solution*
P(.40  Ps  .43)
© 1998 Prentice-Hall, Inc.

n·p  5
n·(1 - p)  5
Z
Ps  p

p  (1  p )
n
Sampling
Distribution
.43  .40
 .87
.40  (1  .40 )
200
Standardized
Normal Distribution
=1
P = .0346
.3078
P = .40 .43
5 - 62
Ps
=0
.87
Z
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.

Modify standard error if sample size (n)
is large relative to population size (N)


5 - 63
Sampling from
Finite Populations
n > .05·N (or n/N > .05)
Use finite population correction (fpc)
factor
Statistics for Managers Using Microsoft Excel, 1/e
© 1998 Prentice-Hall, Inc.

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 (fpc)
factor

Standard errors if n/N > .05:

N n
x 

N 1
n
5 - 64
P 
p  1  p 

n
N  n
 N  1
Statistics for Managers Using Microsoft Excel, 1/e
Thinking Challenge
© 1998 Prentice-Hall, Inc.
You’re manager of a
bank. 40% of all 1000
depositors have multiple
accounts. You select a
random sample of 200
customers. What is the
probability that the
sample proportion of
depositors with multiple
accounts would be
between 40% & 43% ?
5 - 65
© 1984-1994 T/Maker Co.
Statistics for Managers Using Microsoft Excel, 1/e
Solution*
P(.40  Ps  .43)
© 1998 Prentice-Hall, Inc.
Z
Ps  p

p  (1  p )
N n

n
N 1
Sampling Distribution
.43  .40
 .97
.40  (1  .40)
1000  200

200
1000  1
Standardized Distribution
 =1
P = .0310
.3340
P = .40 .43
5 - 66
Ps
=0
.97
Z
Statistics for Managers Using Microsoft Excel, 1/e
Conclusion
© 1998 Prentice-Hall, Inc.






5 - 67
Described the normal probability
distribution
Solved probability problems involving
the normal distribution
Described the exponential distribution
Described the properties of estimators
Explained sampling distribution
Stated the central limit theorem
Statistics for Managers Using Microsoft Excel, 1/e