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


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


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


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


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


Many Normal Distributions

Varying the Parameters s and m, we obtain

Different Normal Distributions.

There are

an Infinite

Number


Normal Distribution:

Finding Probabilities

Probability is the

area under the

curve!

c d X

f(X)

P c X d ( ) ? =


Which Table?

Infinitely Many Normal Distributions Means

Infinitely Many Tables to Look Up!

Each distribution

has its own table?


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


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


0

s = 1

-.21 Z .21

Example:

P(2.9 < X < 7.1) = .1664

Normal

Distribution

.1664

.0832 .0832

Standardized

Normal Distribution


5

s = 10

2.9 7.1 X

2110

592.

.xz =

=

=

s

m

2110

517.

.xz =

=

=

s

m


Z m = 0

s = 1

.30

Example: P(X 8) = .3821

Normal

Distribution

Standardized

Normal Distribution

.1179

.5000

.3821


.

X m = 5

s = 10

8

3010

58.

xz =

=

=

s

m


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


Z m = 0

s = 1

.31 X m = 5

s = 10

?

Finding X Values

for Known Probabilities

Normal Distribution Standardized Normal Distribution

.1217 .1217


X 8.1 = m + Zs = 5 + (0.31)(10) =


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


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


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


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


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.

_


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


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...


)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


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


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

_


2116

241919181 =++++

=

= =

N

XN

ii

xm

)

) ) )581

16

212421192118222

1

2

.

N

XN

ixi

x

=+++

=

= =

ms

Summary Measures for the



18 19 20 21 22 23 24 0

.1

.2

.3 P(X)

X

Sample Means Distribution n = 2

Comparing the Population with its


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

_


• 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

_ _


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


m

Unbiasedness

Biased Unbiased

P(X)

X


m

Efficiency

Sampling

Distribution

of Median Sampling

Distribution of

Mean

X

P(X)


m

Larger

sample size

Smaller

sample size

Consistency

X

P(X)

A

B


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


m x

m =

s s

x = n

_

_


XX

Central Limit Theorem

As Sample

Size Gets

Large

Enough

Sampling

Distribution

Becomes

Almost Normal

regardless of

shape of

population


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


m = 50

s = 10

X

X50=mX


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


• 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


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


P(ps)

.3

.2

.1

0 0 . 2 .4 .6 8 1

ps


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


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


• 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


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

Date post:	09-Mar-2018
Category:	Documents
Upload:	dangtram
View:	225 times
Download:	1 times

Statistics for Managers Using Microsoft Excel · PDF fileStatistics for Managers Using...

Documents