Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd...

72
Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Transcript of Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd...

Page 1: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Chapter 4

Basic Probability And Probability Distributions

Business StatisticsA First Course

(3rd Edition)

Page 2: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Chapter Topics

• Basic Probability Concepts:Sample Spaces and Events, Simple Probability, and Joint Probability,

• Conditional Probability• Bayes’ Theorem

• The Probability Distribution for a Discrete Random Variable

Page 3: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Chapter Topics

• Binomial and Poisson Distributions

• Covariance and its Applications in Finance

• The Normal Distribution

• Assessing the Normality Assumption

Page 4: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Sample Spaces

Collection of all Possible Outcomes

e.g. All 6 faces of a die:

e.g. All 52 cards of a bridge deck:

Page 5: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Events• Simple Event: Outcome from a Sample Space

with 1 Characteristic

e.g. A Red CardRed Card from a deck of cards.

• Joint Event: Involves 2 Outcomes Simultaneously

e.g. An AceAce which is also a Red CardRed Card from a

deck of cards.

Page 6: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Visualizing Events

• Contingency Tables

• Tree Diagrams

Ace Not Ace Total

Red 2 24 26 Black 2 24 26

Total 4 48 52

Page 7: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Simple Events

The Event of a Happy Face

There are 55 happy faces in this collection of 18 objects

Page 8: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Joint Events

The Event of a Happy Face ANDAND Light Colored

3 Happy Faces which are light in color

Page 9: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Special Events

Null event

Club & diamond on 1 card draw

Complement of event

For event A,

All events not In A:

Null Event

'A

Page 10: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

3 Items: 3 Happy Faces Given they are Light Colored

Dependent or Independent Events

The Event of a Happy Face GIVEN it is Light Colored

E = Happy FaceLight Color

Page 11: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Contingency Table

A Deck of 52 Cards

Ace Not anAce

Total

Red

Black

Total

2 24

2 24

26

26

4 48 52

Sample Space

Red Ace

Page 12: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

2500

Contingency Table

2500 Employees of Company ABC

Agree Neutral Opposed | Total

MALE

FEMALE

Total

900 200

300 100

400 | 1500

600 | 1000

1200 300 1000 |

Sample Space

Page 13: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Tree Diagram

Event Possibilities

Red Cards

Black Cards

Ace

Not an Ace

Ace

Not an Ace

Full Deck of Cards

Page 14: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Probability

•Probability is the numerical

measure of the likelihood

that the event will occur.

•Value is between 0 and 1.•Sum of the probabilities of

all mutually exclusive and collective exhaustive events is 1.

Certain

Impossible

.5

1

0

Page 15: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Computing Probability• The Probability of an Event, E:

• Each of the Outcome in the Sample Space equally likely to occur.

e.g. P( ) = 2/36

(There are 2 ways to get one 6 and the other 4)

P(E) =Number of Event Outcomes

Total Number of Possible Outcomes in the Sample Space

=X

T

Page 16: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Computing Joint Probability

The Probability of a Joint Event, A and B:

e.g. P(Red Card and Ace)

P(A and B)

Number of Event Outcomes from both A and B

Total Number of Possible Outcomes in Sample Space

=

=

2 Red Aces 1

52 Total Number of Cards 26

Page 17: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

P(A2 and B1)

P(A1 and B1)

EventEvent Total

Total 1

Joint Probability Using Contingency Table

Joint Probability Marginal (Simple) Probability

P(A1)A1

A2

B1 B2

P(B1) P(B2)

P(A1 and B2)

P(A2 and B2) P(A2)

Page 18: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Computing Compound Probability

The Probability of a Compound Event, A or B:

e.g.

P(Red Card or Ace)

4 Aces + 26 Red Cards 2 Red Aces 28 7

52 Total Number of Cards 52 13

Numer of Event Outcomes from Either A or BP A or B

Total Outcomes in the Sample Space

Page 19: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

2500

Contingency Table

2500 Employees of Company ABC

Agree Neutral Opposed | Total

MALE

FEMALE

Total

900 200

300 100

400 | 1500

600 | 1000

1200 300 1000 |

Sample Space

Page 20: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion

Calculate the probability that an employee selected (at random) from this group will be:

1. A female opposed to the proposal

Page 21: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion

Calculate the probability that an employee selected (at random) from this group will be:

1. A female opposed to the proposal 600/2500 = 0.24

Page 22: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion

Calculate the probability that an employee selected (at random) from this group will be:

1. A female opposed to the proposal 600/2500 = 0.24

2. Neutral

Page 23: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion

Calculate the probability that an employee selected (at random) from this group will be:

1. A female opposed to the proposal 600/2500 = 0.24

2. Neutral 300/2500 = 0.12

Page 24: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion

Calculate the probability that an employee selected (at random) from this group will be:

1. A female opposed to the proposal 600/2500 = 0.24

2. Neutral 300/2500 = 0.12

3. Opposed to the proposal, GIVEN that

the employee selected is a female

Page 25: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion

Calculate the probability that an employee selected (at random) from this group will be:

1. A female opposed to the proposal 600/2500 = 0.24

2. Neutral 300/2500 = 0.12

3. Opposed to the proposal, GIVEN that

the employee selected is a female 600/1000 = 0.60

Page 26: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion

Calculate the probability that an employee selected (at random) from this group will be:

1. A female opposed to the proposal 600/2500 = 0.24

2. Neutral 300/2500 = 0.12

3. Opposed to the proposal, GIVEN that

the employee selected is a female 600/1000 = 0.60

4. Either a female or opposed to the

proposal

Page 27: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion

Calculate the probability that an employee selected (at random) from this group will be:

1. A female opposed to the proposal 600/2500 = 0.24

2. Neutral 300/2500 = 0.12

3. Opposed to the proposal, GIVEN that

the employee selected is a female 600/1000 = 0.60

4. Either a female or opposed to the

proposal ……….. 1000/2500 + 1000/2500 - 600/2500 =

1400/2500 = 0.56

Page 28: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion

Calculate the probability that an employee selected (at random) from this group will be:

1. A female opposed to the proposal 600/2500 = 0.24

2. Neutral 300/2500 = 0.12

3. Opposed to the proposal, GIVEN that

the employee selected is a female 600/1000 = 0.60

4. Either a female or opposed to the

proposal ……….. 1000/2500 + 1000/2500 - 600/2500 =

1400/2500 = 0.56

5. Are Gender and Opinion (statistically) independent?

Page 29: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

The pervious table refers to 2500 employees of ABC Company, classified by gender and by opinion on a company proposal to emphasize fringe benefits rather than wage increases in an impending contract discussion

Calculate the probability that an employee selected (at random) from this group will be:

1. A female opposed to the proposal 600/2500 = 0.24

2. Neutral 300/2500 = 0.12

3. Opposed to the proposal, GIVEN that

the employee selected is a female 600/1000 = 0.60

4. Either a female or opposed to the

proposal ……….. 1000/2500 + 1000/2500 - 600/2500 =

1400/2500 = 0.56

5. Are Gender and Opinion (statistically) independent?

For Opinion and Gender to be independent, the joint probability of each pair of A events (GENDER) and B events (OPINION) should equal the product of the respective unconditional probabilities….clearly this does not hold…..check, e.g., the prob. Of MALE and IN FAVOR against the prob. of MALE times the prob. of IN FAVOR …they are not equal….900/2500 does not equal 1500/2500 * 1200/2500

Page 30: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

P(A1 and B1)

P(B2)P(B1)

P(A2 and B2)P(A2 and B1)

EventEvent Total

Total 1

Compound ProbabilityAddition Rule

P(A1 and B2) P(A1)A1

A2

B1 B2

P(A2)

P(A1 or B1 ) = P(A1) +P(B1) - P(A1 and B1)

For Mutually Exclusive Events: P(A or B) = P(A) + P(B)

Page 31: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Computing Conditional Probability

The Probability of Event A given that Event B has occurred:

P(A B) =

e.g.

P(Red Card given that it is an Ace) =

P A B

P B

and

2 Red Aces 1

4 Aces 2

Page 32: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

BlackColor

Type Red Total

Ace 2 2 4

Non-Ace 24 24 48

Total 26 26 52

Conditional Probability Using Contingency Table

Conditional Event: Draw 1 Card. Note Kind & Color

26

2

5226

522

/

/

P(Red)

Red)AND P(Ace = Red) |P(Ace

Revised Sample Space

Page 33: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Conditional Probability and Statistical Independence

)B(P

)BandA(P

Conditional Probability:

P(AB) =

P(A and B) = P(A B) • P(B)

Multiplication Rule:

= P(B ) • P(A)

Page 34: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Conditional Probability and Statistical Independence (continued)

Events are Independent:P(A B) = P(A)

Or, P(A and B) = P(A) • P(B)

Events A and B are Independent when the probability of one event, A is not affected by another event, B.

Or, P(B A) = P(B)

Page 35: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Bayes’ Theorem

)B(P)BA(P)B(P)BA(P

)B(P)BA(P

kk

ii

11

)A(P

)AandB(P i

P(Bi A) =

Adding up the parts of A in all the B’s

Same Event

Page 36: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

A manufacturer of VCRs purchases a particular microchip, called the LS-24, from three suppliers: Hall Electronics, Spec Sales, and Crown Components. Thirty percent of the LS-24 chips are purchased from Hall, 20% from Spec, and the remaining 50% from Crown. The manufacturer has extensive past records for the three suppliers and knows that there is a 3% chance that the chips from Hall are defective, a 5% chance that chips from Spec are defective and a 4% chance that chips from Crown are defective. When LS-24 chips arrive at the manufacturer, they are placed directly into a bin and not inspected or otherwise identified as to supplier. A worker selects a chip at random.

    

What is the probability that the chip is defective?       A worker selects a chip at random for installation into a VCR and finds it is defective. What is the

probability that the chip was supplied by Spec Sales?   

Page 37: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

What are the chances of repaying a loan, given a college education?

Bayes’ Theorem: Contingency Table

Loan StatusEducation Repay Default Prob.

College .2 .05 .25

No College

Prob. 1

P(Repay College) =

? ? ?

? ?

P(College and Repay)

P(College and Repay) + P(College and Default)

= .80= .20.25

Page 38: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Discrete Random Variable

• Random Variable: outcomes of an experiment expressed numerically

e.g.

Throw a die twice: Count the number of times 4 comes up (0, 1, or 2 times)

Page 39: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Discrete Random Variable

•Discrete Random Variable: • Obtained by Counting (0, 1, 2, 3, etc.)

• Usually finite by number of different values

e.g.

Toss a coin 5 times. Count the number of tails. (0, 1, 2, 3, 4, or 5 times)

Page 40: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Discrete Probability Distribution Example

Probability distribution

Values probability

0 1/4 = .25

1 2/4 = .50

2 1/4 = .25

Event: Toss 2 Coins. Count # Tails.

T

T

T T

Page 41: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Discrete Probability Distribution

• List of all possible [ xi, p(xi) ] pairs

Xi = value of random variable

P(xi) = probability associated with value

• Mutually exclusive (nothing in common)

• Collectively exhaustive (nothing left out)0 p(xi) 1

P(xi) = 1

Page 42: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Discrete Random Variable Summary Measures

Expected value (The mean) Weighted average of the probability distribution

= E(X) = xi p(xi) E.G. Toss 2 coins, count tails, compute expected value:

= 0 .25 + 1 .50 + 2 .25 = 1

Number of Tails

Page 43: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Discrete Random Variable Summary Measures

Variance

Weighted average squared deviation about mean

= E [ (xi - )2]= (xi - )2p(xi)

E.G. Toss 2 coins, count tails, compute variance:

= (0 - 1)2(.25) + (1 - 1)2(.50) + (2 - 1)2(.25)

= .50

Page 44: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Important Discrete Probability Distribution Models

Discrete Probability Distributions

Binomial Poisson

Page 45: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Binomial Distributions

• ‘N’ identical trials E.G. 15 tosses of a coin, 10 light bulbs

taken from a warehouse

• 2 mutually exclusive outcomes on each trial E.G. Heads or tails in each toss of a coin,

defective or not defective light bulbs

Page 46: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Binomial Distributions

• Constant Probability for each Trial• e.g. Probability of getting a tail is the same each time we toss the coin and each light bulb has the same probability of being defective

• 2 Sampling Methods:• Infinite Population Without Replacement• Finite Population With Replacement

• Trials are Independent:

• The Outcome of One Trial Does Not Affect the Outcome of Another

Page 47: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Binomial Probability Distribution Function

P(X) = probability that X successes given a knowledge of n and p

X = number of ‘successes’ in sample, (X = 0, 1, 2, ..., n)

p = probability of each ‘success’

n = sample size

P(X)n

X ! n Xp pX n X!

( )!( )

1

Tails in 2 Tosses of Coin

X P(X) 0 1/4 = .25

1 2/4 = .50

2 1/4 = .25

Page 48: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Binomial Distribution Characteristics

n = 5 p = 0.1

n = 5 p = 0.5

Mean

Standard Deviation

E X np

np p

( )

( )1

0.2.4.6

0 1 2 3 4 5

X

P(X)

.2

.4

.6

0 1 2 3 4 5

X

P(X)

e.g. = 5 (.1) = .5

e.g. = 5(.5)(1 - .5)

= 1.118

0

Page 49: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Poisson Distribution

Poisson process:• Discrete events in an ‘interval’

The probability of one success in an interval is stable

The probability of more than one success in this interval is

0

• Probability of success is

Independent from interval to

Interval

E.G. # Customers arriving in 15 min

# Defects per case of light bulbs

P X x

x

x

( |

!

e-

Page 50: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Poisson Distribution Function

P(X ) = probability of X successes given = expected (mean) number of ‘successes’

e = 2.71828 (base of natural logs)

X = number of ‘successes’ per unit

P XX

X

( )!

e

e.g. Find the probability of 4 customers arriving in 3 minutes when the mean is 3.6.

P(X) = e-3.6

3.64!

4

= .1912

Page 51: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Poisson Distribution Characteristics

= 0.5

= 6

Mean

Standard Deviation

ii

N

i

E X

X P X

( )

( )1

0.2.4.6

0 1 2 3 4 5

X

P(X)

0.2.4.6

0 2 4 6 8 10

X

P(X)

Page 52: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Covariance

X = discrete random variable X

Xi = value of the ith outcome of X

P(xiyi) = probability of occurrence of the ith outcome of X and ith outcome of Y

Y = discrete random variable Y

Yi = value of the ith outcome of Y

I = 1, 2, …, N

)YX(P)Y(EY)X(EX iii

N

iiXY

1

Page 53: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Computing the Mean for Investment Returns

Return per $1,000 for two types of investments

P(XiYi) Economic condition Dow Jones fund X Growth Stock Y

.2 Recession -$100 -$200

.5 Stable Economy + 100 + 50

.3 Expanding Economy + 250 + 350

Investment

E(X) = X = (-100)(.2) + (100)(.5) + (250)(.3) = $105

E(Y) = Y = (-200)(.2) + (50)(.5) + (350)(.3) = $90

Page 54: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Computing the Variance for Investment Returns

P(XiYi) Economic condition Dow Jones fund X Growth Stock Y

.2 Recession -$100 -$200

.5 Stable Economy + 100 + 50

.3 Expanding Economy + 250 + 350

Investment

Var(X) = = (.2)(-100 -105)2 + (.5)(100 - 105)2 + (.3)(250 - 105)2

= 14,725, X = 121.35

Var(Y) = = (.2)(-200 - 90)2 + (.5)(50 - 90)2 + (.3)(350 - 90)2

= 37,900, Y = 194.68

2X

2Y

Page 55: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Computing the Covariance for Investment Returns

P(XiYi) Economic condition Dow Jones fund X Growth Stock Y

.2 Recession -$100 -$200

.5 Stable Economy + 100 + 50

.3 Expanding Economy + 250 + 350

Investment

XY = (.2)(-100 - 105)(-200 - 90) + (.5)(100 - 105)(50 - 90)

+ (.3)(250 -105)(350 - 90) = 23,300

The Covariance of 23,000 indicates that the two investments are positively related and will vary together in the same direction.

Page 56: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

The Normal Distribution

• ‘Bell Shaped’

• Symmetrical

• Mean, Median and

Mode are Equal

• ‘Middle Spread’

Equals 1.33

• Random Variable has

Infinite Range

Mean Median Mode

X

f(X)

Page 57: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

The Mathematical Model

f(X) = frequency of random variable X

= 3.14159; e = 2.71828

= population standard deviation

X = value of random variable (- < X < )

= population mean

21

2

2

1

2

X

f X e

Page 58: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Varying the Parameters and , we obtain Different Normal Distributions.

There are an Infinite Number

Many Normal Distributions

Page 59: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Normal Distribution: Finding Probabilities

Probability is the area under thecurve!

c dX

f(X)

P c X d( ) ?

Page 60: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Infinitely Many Normal Distributions Means Infinitely Many Tables to Look Up!

Each distribution has its own table?

Which Table?

Page 61: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

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

Solution (I): The Standardized Normal Distribution

.0478.02

0.1 .0478

Standardized Normal Distribution Table (Portion) = 0 and = 1

Probabilities

Shaded Area Exaggerated

Only One Table is Needed

Page 62: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Z = 0.12

Z .00 .01

0.0 .5000 .5040 .5080

.5398 .5438

0.2 .5793 .5832 .5871

0.3 .5179 .5217 .5255

Solution (II): The Cumulative Standardized Normal Distribution

.5478.02

0.1 .5478

Cumulative Standardized Normal Distribution Table (Portion)

Probabilities

Shaded Area Exaggerated

Only One Table is Needed

0 and 1

Page 63: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Z = 0

Z = 1

.12

Standardizing Example

Normal Distribution

Standardized Normal Distribution

X = 5

= 10

6.2

12010

526 ..XZ

Shaded Area Exaggerated

Page 64: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

0

= 1

-.21 Z.21

Example:P(2.9 < X < 7.1) = .1664

Normal Distribution

.1664

.0832.0832

Standardized Normal Distribution

Shaded Area Exaggerated

5

= 10

2.9 7.1 X

2110

592.

.xz

2110

517.

.xz

Z

Page 65: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Z = 0

= 1

.30

Example: P(X 8) = .3821

Normal Distribution

Standardized Normal Distribution

.1179

.5000

.3821

Shaded Area Exaggerated

.

X = 5

= 10

8

3010

58.

xz

Page 66: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Z .00 0.2

0.0 .0000 .0040 .0080

0.1 .0398 .0438 .0478

0.2 .0793 .0832 .0871

.1179 .1255

Z = 0

= 1

.31

Finding Z Values for Known Probabilities

.1217.01

0.3

Standardized Normal Probability Table (Portion)

What Is Z Given Probability = 0.1217?

Shaded Area Exaggerated

.1217

Page 67: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Z = 0

= 1

.31X = 5

= 10

?

Recovering X Values for Known Probabilities

Normal Distribution Standardized Normal Distribution

.1217 .1217

Shaded Area Exaggerated

X 8.1 Z= 5 + (0.31)(10) =

Page 68: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

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

Page 69: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Assessing Normality

Normal Probability Plot for Normal Distribution

Look for Straight Line!

30

60

90

-2 -1 0 1 2

Z

X

Page 70: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

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

Page 71: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Chapter Summary• Discussed Basic Probability Concepts:

Sample Spaces and Events, Simple Probability, and Joint Probability

• Defined Conditional Probability

• Discussed Bayes’ Theorem

• Addressed the Probability of a Discrete Random Variable

Page 72: Chapter 4 Basic Probability And Probability Distributions Business Statistics A First Course (3rd Edition)

Chapter Summary

• Discussed Binomial and Poisson Distributions

• Addressed Covariance and its Applications in Finance

• Covered Normal Distribution

• Discussed Assessing the Normality Assumption