BCOR 1020 Business Statistics Lecture 6 – February 5, 2007.

BCOR 1020Business Statistics

Lecture 6 – February 5, 2007

Overview

• Chapter 4 Example

• Chapter 5 – Probability– Random Experiments– Probability

Chapter 4 - Example

• Problem 4.22 list the rents paid by a random sample of 30 students who live off campus. The sorted data is below.

• Using Excel, we can quickly calculate the sample average and standard deviation…

• Using these, we can find standardized values (zi)…

• Find the Quartiles and Construct a Boxplot…

500 560 570 600 620 620 650 660 670 690 690 700 700 710 720

720 730 730 730 730 740 740 760 800 820 840 850 930 930 1030

7.724X 3.114S

-1.97 -1.44 -1.35 -1.09 -0.92 -0.92 -0.65 -0.57 -0.48 -0.3 -0.3 -0.22 -0.22 -0.13 -0.04

-0.04 0.047 0.047 0.047 0.047 0.134 0.134 0.309 0.659 0.834 1.009 1.097 1.797 1.797 2.672

Chapter 4 - Example

• Ordered Data…

• Median = 720• Q1 = 660• Q3 = 760• IQR = 100

500 560 570 600 620 620 650 660 670 690 690 700 700 710 720

720 730 730 730 730 740 740 760 800 820 840 850 930 930 1030

Chapter 5 - Probability

Chapter 5 – Random Experiments

Sample Space:• A random experiment is an observational

process whose results cannot be known in advance and whose outcomes will differ based on random chance.

• The sample space (S) for the experiment is the set of all possible outcomes in the experiment.– A discrete sample space is one with a countable (but

perhaps infinite) number of outcomes.– A continuous sample space is one where the

outcomes fall on a continuous interval (often the result of a measurement).

Chapter 5 – Random Experiments

Sample Space:• Discrete Sample Space Examples:

The sample space describing a Wal-Mart customer’s payment method is…

S = {cash, debit card, credit card, check}

• Continuous Sample Space Examples:The sample space for the length of a randomly chosen

cell phone call would be…

S = {all X such that X > 0} or written as S = {X | X > 0}.

The sample space to describe a randomly chosen student’s GPA would be S = {X | 0.00 < X < 4.00}.

Chapter 5 – Random Experiments

Some sample spaces can be enumerated:• For Example:

– For a single roll of a die, the sample space is:

S = {1, 2, 3, 4, 5, 6}.

– When two dice are rolled, the sample space is the following pairs: {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6),

(2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

S =

Chapter 5 – Random Experiments

Some sample spaces are not easily enumerated:• For Example:

– Consider the sample space to describe a randomly chosen United Airlines employee by

6 home bases (major hubs), 2 genders,21 job classifications, 4 education

levels– There are: 6 x 22 x 21 x 4 = 1008 possible outcomes.

Chapter 5 – Random Experiments

Events:• An event is any subset of outcomes in the

sample space.• A simple event or elementary event, is a single

outcome.– A discrete sample space S consists of all the simple

events (Ei):

S = {E1, E2, …, En}

• A compound event consists of two or more simple events.

Chapter 5 – Random Experiments

Example of a Simple Event:• Consider the random experiment of tossing

a balanced coin. What is the sample space?

S = {H, T}

• What are the chances of observing a H or T?

These two elementary events are equally likely.

Clickers

When you buy a lottery ticket, the sample space S = {win, lose} has only two events.

Are these two events equally likely to occur?

A = Yes

B = No

Chapter 5 – Random Experiments

Example of Compound Events:• Recall: a compound event consists of two or more

simple events.• For example, in a sample space of 6 simple events, we

could define the compound events…

• These are displayed in a Venn diagram:

A = {E1, E2}

B = {E3, E5, E6}

• Many different compound events could be defined.

• Compound events can be described by a rule.

ClickersRecall our earlier example involving the roll of two dice where the sample space is given by…

If we define the compound event A = “rolling a seven” on a roll of two dice, how many simple events does our compoundevent consist of?

A = 4 B = 6

C = 7 D = 36

{(1,1), (1,2), (1,3), (1,4), (1,5), (1,6),

(2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

S =

Chapter 5 – Probability

Definition:• The probability of an event is a number that

measures the relative likelihood that the event will occur.– The probability of event A [denoted P(A)], must lie

within the interval from 0 to 1:

0 < P(A) < 1

If P(A) = 0, then the event cannot occur.

If P(A) = 1, then the event is certain to occur.

Chapter 5 – ProbabilityDefinitions:• In a discrete sample space, the probabilities of

all simple events must sum to unity:

P(S) = P(E1) + P(E2) + … + P(En) = 1

• For example, if the following number of purchases were made by…

credit card: 32%

debit card: 20%

cash: 35%

check: 18%

Sum = 100%

Probability

P(credit card) = .32

P(debit card) = .20

P(cash) = .35

P(check) = .18

Sum = 1.0

Chapter 5 – Probability

• Businesses want to be able to quantify the uncertainty of future events.– For example, what are the chances that next month’s

revenue will exceed last year’s average?

• The study of probability helps us understand and quantify the uncertainty surrounding the future.– How can we increase the chance of positive future

events and decrease the chance of negative future events?

Chapter 5 – Probability

What is Probability?

• There are three approaches to probability:– Empirical – Classical – Subjective

Empirical Approach:• Use the empirical or relative frequency approach to assign

probabilities by counting the frequency (fi) of observed outcomes defined on the experimental sample space.

• For example, to estimate the default rate on student loans

P(a student defaults) = f /nnumber of defaults

number of loans=

Chapter 5 – Probability

Empirical Approach:• Necessary when there is no prior knowledge of events.• As the number of observations (n) increases or the

number of times the experiment is performed, the estimate will become more accurate.

Law of Large Numbers:• The law of large numbers is an important

probability theorem that states that a large sample is preferred to a small one.

Chapter 5 – Probability

Example: Law of Large Numbers:• Flip a coin 50 times. We would expect the

proportion of heads to be near .50.– However, in a small finite sample, any ratio can be

obtained (e.g., 1/3, 7/13, 10/22, 28/50, etc.).– A large n may be needed to get close to .50.

• Consider the results of simulating 10, 20, 50, and 500 coin flips…

Chapter 5 – Probability

Chapter 5 – Probability

Classical Approach:• In this approach, we envision the entire sample

space as a collection of equally likely outcomes.• Instead of performing the experiment, we can

use deduction to determine P(A).• a priori refers to the process of assigning

probabilities before the event is observed.• a priori probabilities are based on logic, not

experience.

Chapter 5 – Probability

Classical Approach:• For example, the two dice experiment has 36

equally likely simple events. The probability that the sum of the two dice is 7, P(7), is

number of outcomes with 7 dots 6( ) 0.1667

number of outcomes in sample space 36P A

• The probability is obtained a priori using the classical approach as shown in this Venn diagram for 2 dice:

ClickersConsider the Venn Diagram for the roll of two dice from the previous example:What is the probability that the two dice sum to 4, P(4)?

A = 0.083

B = 0.111

C = 0.139

D = 0.167

E = 0.194

Chapter 5 – ProbabilitySubjective Approach:• A subjective probability reflects someone’s

personal belief about the likelihood of an event. – Used when there is no repeatable random experiment.– For example,

• What is the probability that a new truck product program will show a return on investment of at least 10 percent?

• What is the probability that the price of GM stock will rise within the next 30 days?

– These probabilities rely on personal judgment or expert opinion.

• Judgment is based on experience with similar events and knowledge of the underlying causal processes.