QM 2113 - Spring 2002QM 2113 - Spring 2002

Business StatisticsBusiness Statistics

Exercises with Exercises with Normally Normally

ProbabilitiesProbabilities

Student ObjectivesStudent Objectives

Calculate probabilities Calculate probabilities associated with normally associated with normally distributed random variablesdistributed random variables

Apply normal distribution Apply normal distribution calculations to various calculations to various decision making situationsdecision making situations

First, an First, an AnnouncementAnnouncement

Visits this week by major firmsVisits this week by major firms– Cardinal Health– Acxiom

Cardinal Health (Kathy White)Cardinal Health (Kathy White)– Who are they?– Why do we care?– When/where

AcxiomAcxiom– Who are they and what do they do?– When/where

Working with Normal Working with Normal DistributionsDistributions

First, sketchFirst, sketch– Number line with

• Average (i.e., )• Also x value of concern

– Curve approximating histogram Identify areas of importanceIdentify areas of importance Then determine how many Then determine how many

standard deviations standard deviations x valuex value is from is from

Now use the tableNow use the table Finally, put it all togetherFinally, put it all together

Mechanics: Some Mechanics: Some Calculation ExercisesCalculation Exercises Let x ~ N(34,3) as with the mpg Let x ~ N(34,3) as with the mpg

problemproblem DetermineDetermine

– Tail probabilities• F(30) which is the same as P(x ≤ 30)• P(x > 40)

– Tail complements• P(x > 30)• P(x < 40)

– Other• P(32 < x < 33)• P(30 < x < 35)• P(20 < x < 30)

Recall About the Recall About the Normal TableNormal Table

The The outsideoutside values are z-scores values are z-scores– That is, how many standard deviations a given

x value is from the average– Use these values to look up probabilities

The The bodybody of the table indicates of the table indicates probabilitiesprobabilities

Note: This is not a “z table”!Note: This is not a “z table”! We can (and do) also work in reverseWe can (and do) also work in reverse

– Given a probability, determine z– Once we have z we can determine what x value

corresponds to that probability

Keep In MindKeep In Mind Probability = proportion of area under Probability = proportion of area under

the normal curvethe normal curve What we get when we use tables is What we get when we use tables is

always the area between the mean and z always the area between the mean and z standard deviations from the meanstandard deviations from the mean

Because of symmetryBecause of symmetry P(x > ) = P(x < ) = 0.5000

Tables show probabilities rounded to 4 Tables show probabilities rounded to 4 decimal placesdecimal places– If z < -3.09 then probability ≈ 0.5000– If z > 3.09 then probability ≈ 0.5000

Theoretically, P(x = a) = 0Theoretically, P(x = a) = 0 P(30 ≤ x ≤ 35) = P(30 < x < 35)

Why Is This Why Is This Important?Important?

Some practical applicationsSome practical applications– Process capability analysis– Decision analysis– Optimization (e.g., ROP)– Reliability studies– Others

Most importantly, the normal Most importantly, the normal distribution is the basis for distribution is the basis for understanding statistical inferenceunderstanding statistical inference

Hence, bear with this; it should be Hence, bear with this; it should be apparent soonapparent soon

HomeworkHomework

Rework (as necessary) Rework (as necessary) exercises assigned from exercises assigned from Chapter 5Chapter 5

Work problems on Exam #3 Work problems on Exam #3 from Spring 2000from Spring 2000

Review for midterm examReview for midterm exam