BCOR 1020 Business Statistics Lecture 15 – March 6, 2008.

BCOR 1020Business Statistics

Lecture 15 – March 6, 2008

Overview

• Review for Midterm Exam…– Key Definitions– Visual Descriptions– Numerical Descriptions– Probability– Discrete Distributions– Continuous Distributions

Midterm Exam – Review• Statistics vs. statistics

– Descriptive statistics refers to the collection, organization, presentation and summary of data

– Inferential statistics refers to generalizing from a sample to a population, estimating unknown parameters, drawing conclusions, and making decisions.

• Sample vs. Population:– statistics vs. parameters

• Data Types:– Attribute– Discrete Numerical– Continuous Numerical

Midterm Exam – Review

• Levels of Measurement:– Nominal – Ordinal– Interval – Ratio

Level of Level of MeasurementMeasurement CharacteristicsCharacteristics ExampleExample

NominalNominal Categories onlyCategories only Eye color (Eye color (blueblue, , brownbrown, , greengreen, , hazelhazel))

OrdinalOrdinal Rank has meaningRank has meaning Bond ratings (Aaa, Aab, C, D, F, Bond ratings (Aaa, Aab, C, D, F, etc.)etc.)

IntervalInterval Distance has meaningDistance has meaning Temperature (57Temperature (57oo Celsius) Celsius)

RatioRatio Meaningful zero existsMeaningful zero exists Accounts payable ($21.7 million)Accounts payable ($21.7 million)

Midterm Exam – Review• Time series data

– Each observation in the sample represents a different equally spaced point in time (e.g., years, months, days).

• Cross-sectional data– Each observation represents a different individual unit

(e.g., person) at the same point in time.

• Sampling Concepts & Methods:– When to sample vs. when to census– Probability samples (simple random, systematic,

stratified, etc.)– Nonprobability samples (judgment, convenience)

Midterm Exam – Review• Visual Descriptions

– Dot Plots– Frequency Distributions and Histograms – Including

Modal Class, Symmetry & Skewness– Simple Line Charts – Time Series– Bar charts – Including Pareto Charts– Scatter Plots – Cross-sectional Data– Tables– Deceptive Graphs

Midterm Exam – Review• Numerical Description

– Central Tendency – Mean, Median, Mode, Midrange, etc.

– Skewness:• If Median/Mode > Mean, skewed left• If Median/Mode = Mean, symmetric• If Median/Mode < Mean, skewed right

– Dispersion – Variance & Standard Deviation, Coefficient of Variation, etc.

• The Empirical Rule (Symmetric Distributions):– Approximately 68.26% within – Approximately 95.44% within– Approximately 99.73% within

1 2 3

Midterm Exam – Review• Standardized Variables:

– Unusual observations & outliers

• Percentiles and Quartiles:– Find Q1, Q2, Q3

– Midhinge:– Midspread (IQR):– Coefficient of Quartile Variation (CQV):– Boxplots – Including Quartiles, Median, IQR, Unusual

Observations, etc.

ix

iz

Midterm Exam – Review• Random Experiments:

– Sample Space – discrete or continuous– Events – simple and compound

• Probability:– Definition & Characteristics– Empirical, Classical, Subjective Approaches

• Venn Diagrams

Midterm Exam – Review• Rules of Probability (illustrated w/ Venn

Diagrams):– Compliments– Unions – Law of Addition– Mutually Exclusive Events– Conditional Probabilities, Independence, Intersection –

Multiplication Rule– Independent vs. Mutually Exclusive

• Contingency Tables and Trees as Tools (Example):– Marginal probabilities, conditional probabilities, etc.

Midterm Exam – Review

)()()()( BAPBPAPBAP

)()()|( BPBAPBAP

)()|()|()()( BPBAPABPAPBAP

)()()( BPAPBAP

)'(1)( APAP )(1)'( APAP and

)()()( BPAPBAP

In general.

For mutually exclusive events.

)()()|( APBAPABP and

)()|( APBAP )()|( BPABP and

If A and B are independent:

Conditional Probabilities:

Midterm Exam – Review• Example: this table gives expense ratios by fund

type for 21 bond funds and 23 stock funds.

Find P(B), P(H), P(H|S), )( HSP

4411)( HSP23

11)|( SHP

4414)( HP44

21)( BP

Midterm Exam – Review• Random variable –a function or rule that assigns

a numerical value to each outcome in the sample space of a random experiment.– Capital letters are used to represent random variables

(e.g., X, Y).– Lower case letters are used to represent values of the

random variable (e.g., x, y).

• A discrete random variable has a countable number of distinct values. Values are integers.

• A continuous random variable has an uncountable (infinite) number of distinct values. Values fall on an interval.

Midterm Exam – Review• Probability Distributions:• A discrete probability distribution is a rule (function) that

assigns a probability to each value of a discrete random variable X.

• To be a valid probability, each probability must be between

0 P(xi) 1 • and the sum of all the probabilities for the values of X must be

equal to unity.

1

( ) 1n

ii

P x

• For a continuous random variable, the probability density

function (PDF) is an equation that shows the height of the curve f(x) at each possible value of X over the range of X.

1)( dxxf

Midterm Exam – Review

dxxfxXE

)()(

• The expected value, E(X), is a measure of central tendency.

1

( ) ( )n

i ii

E X x P x

• For a discrete random variable,

• For a continuous random variable,

• The variance, V(X), is a measure of dispersion.

2 2

1

( ) [ ] ( )n

i ii

V X x P x

• For a discrete random variable,

Midterm Exam – Review• Example: On any given day, the number of

prescriptions submitted by a random customer at a pharmacy (X) is described by the probabilities in the following table:

x 0 1 2 3 4 5

P(x) 0.05 0.40 0.25 0.15 0.10 0.05

a) Find E(X)

b) Find the probability that a randomly-selected customer will submit at least 4 prescriptions.

c) Find the probability that a randomly-selected customer will have at least one prescription.

Midterm Exam – Review

)05(.5)1(.4)15(.3)25(.2)4(.1)05(.0)( xxP

25.4.45.5.4.0

95.05.1

x 0 1 2 3 4 5

P(x) 0.05 0.40 0.25 0.15 0.10 0.05

a) E(X) =

b) P(at least 4 prescriptions) =

c) P(> 1 prescription) =

15.05.10.

)0(1)0(1)0( PXPXP

0.2

)5()4()4( PPXP

Midterm Exam – Review

• Common Distributions– Binomial– Poisson– Normal– Exponential

• Be able to recognize the experimental conditions leading to these distributions.

• For each of these distributions, be able to use the – PDF and/or CDF– formulas for and .

Midterm Exam – Review• If X denotes the number of “successes” observed

in n Bernoulli trials, then we say that X has the Binomial distribution with parameters n and .– This is often denoted X~b(n,).

nx

CxfxXP xnxxnx

nxnxxn

,,2,1,0

,)1()1()()( )!(!!

nXE )(

)1( n

Midterm Exam – Review

• If X denotes the number of “occurrences” of interest observed on a given interval of length 1 unit of a Poisson Process with parameter > 0, then we say that X has the Poisson distribution with parameter .

,3,2,1,0,)()( ! xxfxXP x

ex

)(XE

Midterm Exam – Review

• If events per unit of time follow a Poisson distribution, the waiting time until the next event follows the Exponential distribution with parameter .

0,)( xexXP x

0,)( xexf x

1)( XE

1

0,1)( xexXP x

PDF:

Midterm Exam – Review• Generally, we will use the Normal random

variable when we make the assumption that our population is normally distributed.– Denoted N(, )– “Bell-shaped” Distribution– Domain is – < X < + – Defined by two parameters, (the mean) and (the

standard deviation)

• To find probabilities or percentiles with a normal distribution…(i) Standard normal transformation:

(ii) Cumulative Normal Tables (Handouts – Included on Exam)

XZ