review of the quantitative business analysis section of the ets exam * (probability and statistics...

Review of the Quantitative BusinessAnalysis Section of the ETS Exam*

(Probability and Statistics and Management Science)

• Probability and Saistics– Measure of set operations– Conditional/joint probabilities– Counting rules

• Multiplication • Addition • Permutations • Combinations • Permutations when not all objects are different

– Measures of central tendency and dispersion – Distributions (including normal and binomial) – Sampling and estimation – Hypothesis testing – Correlation and regression – Time-series forecasting – Statistical concepts in quality control

*Slides available at H:\ditri\Teaching\Review for the ETS exam

• Management science– Linear programming – Project scheduling (including PERT and CPM) – Inventory and production planning – Special topics

• queuing theory• simulation, and • decision analysis

Review of the Quantitative BusinessAnalysis Section of the ETS Exam

Probability and Statistics

• Measure of set operations– “Set” is a collection of objects– Sets are defined by the elements in the set (usually

numbers)– Sets are usually labeled with letters and there will

be a universal set (U) containing all elements in question

– In statistics we are frequently interested in the set of real numbers from 0 to 1

– Usually interested in “subsets” that meet our requirements out of the universal set of 4 stems on an exam question we want the subset that contains the right answer

– There can be a null set, that is one that does not meet all the requirements and is empty

– Venn diagrams can be used to describe set operations

Venn Diagrams

BA

A union B

A intersection B

BA

A

A complement

Conditional/Joint Probabilities

• Probability– Real number from 0 to 1 mapping the likelihood of

an “event” to the set of real numbers• An event is a set of possible outcomes• Events come in two flavors

– Independent• Knowledge of one event does not provide

insight into another event– Non independent events

• Correlated events where knowledge of the outcome of one event changes the likelihood of another event

• Conditional probabilities

• Joint probabilities– Multidimensional situation where we are interested

in more than one numeric characteristic

FlipsofNumberHeadsofNumberHeadsP

____)(

26/12/1

52/1)(Re

)_&(Re)(

)()\( dP

HeartsAcedPAP

BAPABP

Counting Rules

• Multiplication– (Decision tree) where subsequent outcomes

depend on prior outcomes– N (total number of events) = n1 * n2 … nk where

• ni is the number of events at each stage

• Addition– Parallel activities– N = n1 + n2 + ….nk

• Permutations (number of distinct orders)– Ordering n distinct items– An orange, apple, and a grapefruit

• OAG, OGA, AOG, AGO, GAO, and GOA

• With 0! = 1

n1

n2

n1 n2

))...(3)(2)(1(! nnnnnnnN

Counting Rules(continued)

• Permutations (continued)– Subset of r items from n distinct items

– Number of distinct poker hands (including order dealt with each card regarded as unique)

• Combinations– Distinct hands without regard to order

– Or “n choose r”– Permutations when not all objects are different

)!(!rn

nN

200,875,311)!552(

!52)!(

!

rnnN

960,598,2)!552(!

!52)!(!

!

rrnrnN

rn

N

!!...2!1!

nknnnN

Measures of Central Tendency and Dispersion

• Random variables described by distributions– Captures the fact that more than one value per variable and

some values are more common than others

• Measures of central tendency– Mean– Mode (most frequent)– Median (“middle” value)

• Measures of dispersion– Range (largest minus the smallest)– Standard deviation

00.050.1

0.150.2

0.250.3

0.350.4

0.45

-4 -2 0 2 4

N

XN

ii

1

N

XN

ii

1

2)(

Distributions(including normal and binomial)

• As the name implies, we are describing how the frequency of various outcomes is distributed

• Comes in two flavors– Probability distribution function

• Measures the rate at which we accumulate probability (analogous to velocity)

• Usually labeled f(x)– Cumulative distribution function

• Measures the total accumulated probability (analogous to distance)

• For discrete random variables

• For continuous random variables

)()( xXPxF

j

jxpxF )()(

dssfxFs

)()(

Normal Probability and Cumulative Distribution Functions

00.050.1

0.150.2

0.250.3

0.350.4

0.45

-4 -2 0 2 4

0

0.2

0.4

0.6

0.8

1

1.2

-4 -2 0 2 4

Binomial Distribution

• Discrete, considering “yes” or “no” phenomena• Common in quality where products are “good” or

“bad”• Usual case to have a sample of size n inspected

and we are interested in the probability of seeing k (k = 0, 1, 2, …n) defects

• Formally

• Probability and Cumulative Distribution functions

00.050.1

0.150.2

0.250.3

0.350.4

0.450.5

0 2 4 6

knk ppkn

kXP

)1()(

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8

Sampling and Estimation

• Defined measures of central tendencies and dispersion earlier

• Often, the parameters describing the population are not assessable– Might be too large to measure– Measuring process can be destructive– Measurement process itself might be biased (as is the case

with how the census is compiled in your country)• Addressed by drawing a representative subset (a sample)

and use inference to estimate the analogous parameters in the population

• We can estimate mu and sigma in the population , using the X-bar and the standard deviation of the sample

• X-bar and S are unbiased estimators of mu and sigma—that is, on average they accurately predict the estimated parameters

,

,

n

XX

n

ii

1

1

)(1

2

2

n

XXs

n

ii

Hypothesis Testing

• A testable idea• Based on the premise that science never knows,

only rejects the null hypothesis at higher levels of statistical significance

• My favorite example—our legal system– ‘People are presumed innocent until it is too

unlikely (“beyond a reasonable doubt”) that the person is actually innocent

– Upon rejecting the null, the person is sent to jail (there is always a chance the perp was indeed innocent)

• Commonly used to test for differences in means (ANOVA)– Question: is the difference in sales levels for

different cereal box designs the result of random chance or does one design sell better than another

– H0: Sales levels are the same– HA: There is a difference in sales levels

• Rejection of the null is based on the “p value” the probability of seeing a difference of the observed magnitude or more by chance

Correlation and Regression

• Correlation– If random variables are not independent, we are

interested in describing the relationship– Pearson Correlation Coefficient (r)

Y

XPositive r: Y increasesas X increases.

Y

XNegative r: Y decreasesas X increases.

Y

Xr = -1: a perfect negative correlation between X and Y.

Y

Xr = 1: a perfectpositive correlation between X and Y.


• Regression– Captures the relationship between an independent

and dependent variable– Goal is to achieve a “best fit” of a math model

XY 10

Y

X

Y = 0 + 1X1

Distance

SSE y xi

ni i

0 1

2

1

^ ^

Hours of Service

Hol

e D

evia

tion


• Regression– Objective is to explain variation in the data set– Coefficient of determination (R2) describes the

proportion of variation explained by the model– In addition to R2, quality of model is assessed by

• Significance of the model (measured with an F statistic)

• Significance of estimated beta values (measured with t statistics)

• Assurance the residuals exhibit “white noise”– Normally distributed– Mean zero– Constant variance– Independent

Time-series Forecasting

• Two issues– Creating forecasts– Evaluating forecasts

• Time-series methods – Average sets of data to make predictions of future

events– Do not address causal issues—data sets “speak”

for themselves– Issue is seeking a balance between “responsive”

forecasting methods (respond to changes rapidly but are noisy) and filtering (averaging out of random fluctuations might be over damped)

• Nomenclature:At = actual demand in period tFt = forecast for period tn = number of observations in

forecastt = number of the current period


• Simple moving average

• Weighted moving average

– With

• Exponential smoothing– Simple formula

– with

– Forecast is adjusted for the error– Masks a very sophisticated weighting scheme that considers

all data points– Can be modified (Holt’s and Winter’s methods) to consider

trends and seasonal effects

nAAAAF ntttt

t

321

ntnttt AwAwAwF 2211

11

n

iiw

)( 111 tttt FAFF

10


• Wide assortment of forecasting methods (and possible parameters)

• Wide assortment of evaluation methods• All look at averaging the errors

• Bias is a simple average of the errors• Mean Absolute Deviation is an average of the absolute

values of the errors• Standard Deviation and Mean Squared Error measures are

averages of the squared errors• Mean Average Percentage Error is an average of the error

scaled period by period• Two major groups

– Absolute (average of errors)– Relative (average errors are scaled)

tttt FAeError

Major Evaluation Procedures

EvaluationMethod

What isDone

AbsoluteVersion

RelativeCounter Part

Bias

Mean Absolute Deviation

Standard Deviation

et = Ft-Dt

et = /Ft-Dt/

et = (Ft-Dt)2

MAD = / e /

Ni=1

N

i

SD = ( F - D )

N - 1i=1

N

i i2

BiasRelative Forecast Error =

AD*

Mean Deviation = MAD AD*

Coefficient of Variation = SD

AD*

Mean AveragePercentage Error

)100(1

ND

DF

= MAPE

N

t t

tt

*AD =D

N

tt

N

1

N

e = Bias

i

N

=1i

Mean SquaredError et = (Ft-Dt)2

N

DF = MSE

N

iii

1

2

Statistical Concepts in Quality Control

• Been there, done that– Process control charts are nothing but the sampling

distributions turned on their sides– Mean as the center and then upper and lower

control limits drawn at three standard deviations (need to decide on the size of the sample)

– Acceptance sampling is based on the binomial distribution discussed earlier• Again, decide on the sample size• Set “cutoff” values for the number of defects

found in the sample

Process Control Charts

UCL

LCL

Mean

Control Chart. Used to continuously monitor a process.

Control Charts

TypeUsage

X and R bar charts Continuous data

“P” charts Data in binary form

“C” charts Data In integer form

Illustrates when there is variation due to an assignable cause

Statistical Quality Control Models

Statistical Q.C. Models

AcceptanceSampling

O.C.Curves

SingleSampling

DoubleSampling

SequentialSampling

ProcessControl

MeanChart

ControlCharts

RangeChart

ProportionDefective

DefectsPer Unit

Linear Programming

Definition:

Allocation of scarce resources among competingactivities.

Activities are variables - Xi

where Xi is "how many" of activity i

Resources are in the form of constraints

Maximize:c1X1 + c2X2 + . . . + cnXn

Subject to:a11X1 + a12X2 + . . . + a1nXn < b1a21X1 + a22X2 + . . . + a2nXn < b2

.

.

.am1X1 + am2X2 + . . . + amnXn < bm

X1 , X2 , . . . Xn > 0

Linear Progamming Topic Summary

Issue is the optimal allocation of scare resources among competing activities.

Variables are the activities

Constraints are the resources

Linear programming requires three assumptions:

AdditivityDivisibilityLinearity in the variables

Feasible region is the set of numbers that will satisfy allconstraints simultaneously

Optimization involves finding the best point(s) in the feasibleregion. Best is defined by an objective function that is madeas large or as small as possible.

Solution procedures include graphical methods (suitable for2 decision variables) and computer algorithms.

Dual or shadow prices measure the marginal value of oneadditional unit of a resource.

The dual price remains constant over a range of right hand side values. This range is defined by other constraints. Within this range

Price Shadow*RHSZ We can evaluate the effect of changing a cost coefficient on the objective function for “small” changes in the coefficients. A small change is one where the solution (value of the Xs) stays the same. For “small” changes

X* iiCZ

A reduced cost is the amount that a coefficient will have to be changed to make it worthwhile to engage that activity. i.e., if X = 0, how much do we have to change X’s

coefficient in the objective function in order to make it worthwhile to engage in that activity (X > 0).

We then considered a smattering of classical applications: Mixing problems Transportation problems Assignment problems

Project Scheduling(including PERT and CPM)

• Some history– Both date from the 1950’s

• CPM => DuPont• PERT => Polaris submarine project

• Fundamentals– activities are unique branches (or nodes)– capturing

• time and• precedence

– additional precedence relationships can be modeled using dummy activities

– CPM (critical path method) is deterministic– Project Evaluation and Review Technique is

stochastic

Example Problem

• Issue is determining the critical path• Activities where the slack is zero• Activities that determine the expected

completion time for the network

A(2)

0

0

2

2

B(5)

2

2

7

7

C(4)

2

3

6

7

D(3)

7

7

10

10

Converting Intuition into a Mean and Standard Deviation

6

4 bmate

Optimistictime (a)

Most likely(modal) time (m)

Expectedtime (te)

Pessimistictime (b)

6ab

te

Mean and Variance of theEntire Project

• Based on summing the expected times and expected variability along the critical path

• Assumptions:– Activity times are independent– There is one critical path– Project completion time is normally

distributed• Te is the sum of the tes on the critical path

– Where tei is the expected activity time for each of the N activities along the critical path.

N

iee i

tT1

Measure of the Variability in the Expected Completion Time for the

Entire Project

• Nature adds variability through the variance not the standard deviation

BABABA 22

Te Te

Normal Distribution

N

itT iee

1

2

Inventory and Production PlanningForms of Inventory

• Consider the transformation process:– Purchasing =>

• Operations =>– Distribution

– Raw materials (RM)• Work-in-Progress (WIP)

– Finished Goods (FG)• Maintenance, Repair and Operating

Expenses (MRO)

• Manufacturing Inventory– RM + WIP + MRO

Classical Production Planning Hierarchy

Business Plan

Sales Plan

Production Plan (aggregate production plan)

Master Production Scheduling

Material Requirements Planning

Purchasing(external factory)

Shop Floor Control

Rough Cut Capacity Planning

Capacity Requirements Planning

Forecasts

Cycle Inventory Levels

Time BetweenOrders(f)

Q

Time

Q/2 = Average Cycle inventory

Demand Rate

On

Hand

Inventory

in

Units

Types of Costs

• Inventory holding rate– Time value of money– Insurance– Storage space– Risk

• Obsolescence• Loss• Damage

• Ordering Cost/Setup Cost• Stockout Costs• Backorder cost

Measuring Inventory Management

Turnover = Annual Sales (at cost) Average Inventory

Generally assumed that larger numbers are better, if . . .customer service stays high.

Coverage = Average Inventory per period sales

Weeks of Supply = Average Inventory (at cost) weekly sales

Generally assumed that smaller numbers are better if . . .customer service stays high.

MRP - The System

MPS

MRPBOMFile

Inventory Records

PlannedOrders

A

C B

C

(2 req’d)

Item A

Gross Req’ts

Sched. Rec’ts

On Hand

Planned Orders

1 2 3 4 5 6

Order Q. = 20; Lead T. = 1; Safety S. = 0

5 15 18 8 12 42

21

Item B

Gross Req’ts

Sched. Rec’ts

On Hand

Planned Orders

1 2 3 4 5 6

Order Q. = 40; Lead T. = 2; Safety S. = 0

3220

Item C

Gross Req’ts

Sched. Rec’ts

On Hand

Planned Orders

1 2 3 4 5 6

Order Q. = LFL; Lead T. = 1; Safety S. = 10

50

Special TopicsQueuing Theory

• Part of field of Stochastic Processes (those based on uncertainty)

• Among others, Markov processes– Works from “states” where issue is

probability of jumping to another brand– Seeks long term equilibrium state

• “Birth Death” processes– Describe maintenance problems where– Machines “die” when they fail and are

“reborn” when repaired• Interested in cost effective trade-off—

customer’s time versus “servers”

The Queuing System

• Arrivals—entities entering the system from a population (follows a statistical distribution—usually exponential)

• Population– Finite, limited size or group of customers

• Statistical properties change when someone enters the queuing system

– Infinite size (usual case)• Queue discipline

– Service rules (FIFO, LIFO, etc)– “Balking”– Number of servers– Line switching

Common Waiting Line Models

Common Waiting Line Models

Model LayoutSourcePopulation Service Pattern

1 Single channel Infinite Exponential

2 Single channel Infinite Constant

3 Multichannel Infinite Exponential

4 Single or Multi Finite Exponential

These four models share the following characteristics: Single phase Poisson arrival FCFS Unlimited queue length

Special TopicsSimulation

SimulationDefined

• A simulation is a computer-based model used to run experiments on a real system– Typically done on a computer– Determines reactions to different operating

rules or change in structure– Can be used in conjunction with traditional

statistical and management science techniques

Major Phases in a Simulation Study

Start

Define Problem

Construct Simulation Model

Specify values of variables and parameters

Run the simulation

Evaluate results

Validation

Propose new experiment

Stop

Data Collection & Random No. Interval Example

Suppose you timed 20 athletes running the 100-yard dash and tallied the information into the four time intervals below

Seconds 0-5.996-6.997-7.998 or more

Freq.41042

You then count the tallies and make a frequency distribution

%20502010

Then convert the frequencies into percentages

Finally, use the percentages to develop the random number intervals

RN Intervals00-1920-6970-8990-99

Accum. %207090100

You then can add the frequencies into a cumulative distribution

Types of Simulation Models

• Continuous– Based on mathematical equations– Used for simulating continuous values for

all points in time– Example: The amount of time a person

spends in a queue• Discrete

– Used for simulating specific values or specific points

– Example: Number of people in a queue

Desirable Features of Simulation Software

• Be capable of being used interactively as well as allowing complete runs

• Be user-friendly and easy to understand• Allow modules to be built and then connected • Allow users to write and incorporate their own

routines• Have building blocks that contain built-in

commands• Have macro capability, such as the ability to develop

machining cells• Have material-flow capability • Output standard statistics such as cycle times,

utilization, and wait times• Allow a variety of data analysis alternatives for

both input and output data• Have animation capabilities to display graphically

the product flow through the system• Permit interactive debugging

Advantages of Simulation

• Often leads to a better understanding of the real system

• Years of experience in the real system can be compressed into seconds or minutes

• Simulation does not disrupt ongoing activities of the real system

• Simulation is far more general than mathematical models

• Simulation can be used as a game for training experience

• Simulation provides a more realistic replication of a system than mathematical analysis

• Simulation can be used to analyze transient conditions, whereas mathematical techniques usually cannot

• Many standard packaged models, covering a wide range of topics, are available commercially

• Simulation answers what-if questions

Disadvantages of Simulation

• There is no guarantee that the model will, in fact, provide good answers

• There is no way to prove reliability• Building a simulation model can take a great deal

of time• Simulation may be less accurate than

mathematical analysis because it is randomly based

• A significant amount of computer time may be needed to run complex models

• The technique of simulation still lacks a standardized approach

Overview of Decision Analysis

• Definitions– i indexes decision alternatives– j indexes states of nature

– Decision Alternative (Ai)– States of Nature (Sj)– Payoff (Vij) (intersection of Ai and Sj)– Regret (Rij)

• Without information (game theory)– MaxiMax– MaxiMin– MiniMax regret

• Decisions with uncertainty– Expected value of perfect information– Payoff tables (newsperson problem)

• Decision trees

Decision Analysis-Decision Trees

• A decision tree is a graphical representation of every possible sequence of decision and random outcomes (states of nature) that can occur within a given decision making problem.

• A decision tree is composed of a collection of nodes (represented by circles and squares) interconnected by branches (represented by lines).


General Form of a Decision Tree

Two flavors of nodes: decision and event


• The value assigned to an event node is the expectation of the values that correspond to adjacent nodes.

Evaluation of event nodes

V1

V2

V3

V4

p1

p2

p3

V4 = V1 x p1 + V2 x p2 + V3 x p3

Example Decision Tree

Do not retain E.I.

3.5

3.0

2.71.0

12.5

12.4-0.5

-0.25

25

Prob. Low = 0.1

Prob. Modest = 0.4

Prob. High = 0.5

Prob. Low = 0.1

Prob. Modest = 0.4

Prob. High = 0.5

Prob. Low = 0.1

Prob. Modest = 0.4

Prob. High = 0.5

Small Development

Medium Development

Large Development

2.9

11.3

12.35

12.35

3.5

3.0

2.71.0

12.5

12.4-0.5

-0.25

25

Prob. Low = 0.261

Prob. Modest = 0.522

Prob. High = 0.217

Prob. Low = 0.261


Prob. High = 0.217

Prob. Low = 0.261


Prob. High = 0.217

Small Development

Medium Development

Large Development

3.065

9.478

5.174

9.478

3.5

3.0

2.71.0

12.5

12.4-0.5

-0.25

25

Prob. Low = 0.070


Prob. High = 0.465

Prob. Low = 0.070


Prob. High = 0.465

Prob. Low = 0.070


Prob. High = 0.465

Small Development

Medium Development

Large Development

2.895

11.651

11.477

11.651

3.5

3.0

2.71.0

12.5

12.4-0.5

-0.25

25

Prob. Low = 0.029


Prob. High = 0.735

Prob. Low = 0.029


Prob. High = 0.735

Prob. Low = 0.029


Prob. High = 0.735

Small Development

Medium Development

Large Development

2.794

12.088

18.309

18.309

13.415

Retain E.I. (-.5)

12.915

Proportion of the time E.I. predicts “low” = 0.230

Proportion of the time E.I. predicts “modest” = 0.430

Proportion of the time E.I. predicts “high” = 0.340

Final Word

Good Luck!!

review of the quantitative business analysis section of the ets exam * (probability and statistics...

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