Applications of the Normal Applications of the Normal Distribution ModelDistribution Model(The Confidence Interval)(The Confidence Interval)

©Dr. B. C. Paul 2003 revision 2009©Dr. B. C. Paul 2003 revision 2009

Note – The concepts found in these slides are considered to be Note – The concepts found in these slides are considered to be common knowledge to those trained in the field of statistics. Most common knowledge to those trained in the field of statistics. Most basic statistics texts have coverage of these concepts. The specific basic statistics texts have coverage of these concepts. The specific table depicted in these slides is from Fundamental Concepts in the table depicted in these slides is from Fundamental Concepts in the Design of Experiments by Charles R. Hicks, 1982 published by Holt Design of Experiments by Charles R. Hicks, 1982 published by Holt Reinhardt and Winston, although most basic statistics texts include Reinhardt and Winston, although most basic statistics texts include

similar tables for a standard normal distribution.similar tables for a standard normal distribution.

Preparing for RainfallPreparing for Rainfall

Wendy Wetone has just designed a storm Wendy Wetone has just designed a storm sewer system for a new housing projectsewer system for a new housing project Culverts and intakes will handle a 2.5 inch Culverts and intakes will handle a 2.5 inch

rainfall in 24 hoursrainfall in 24 hours The average big rainfall even in the area is The average big rainfall even in the area is

only 1.25 inchesonly 1.25 inches Wendy is ok Right?Wendy is ok Right? If the roads and homes in an area are If the roads and homes in an area are

going to wash out maybe being ready for going to wash out maybe being ready for an average rain isn’t good enoughan average rain isn’t good enough

Reality for Major Rainfall EventsReality for Major Rainfall Events

Average is 1.25 inches, but suppose Average is 1.25 inches, but suppose there is a 1 inch standard deviationthere is a 1 inch standard deviation

μ = 1.25

σ = 1

How would we knowSomething like this?

We built a modelFrom weatherRecords.

Is there enough of a chance up hear that I should be getting heart-burn over this design?

We Know How to Solve This OneWe Know How to Solve This One

Normal Distribution Normal Distribution is fully defined by a is fully defined by a formulaformula

We only need to We only need to know the average know the average (in this case 1.25) (in this case 1.25) and the variance and the variance (standard deviation (standard deviation squared – easy squared – easy when standard when standard deviation is one)deviation is one)

ex

xfY

2

2

2

2

1)(

What That Formula DoesWhat That Formula Does

Y is a probability value (chance of Y is a probability value (chance of occurrence)occurrence)

X in this case is a rainfall eventX in this case is a rainfall event Rather obviously we are interested in Rather obviously we are interested in

rainfall events greater than 2.5 inchesrainfall events greater than 2.5 inches Guess that means x is 2.5Guess that means x is 2.5

Problem – Formula gives probability for Problem – Formula gives probability for only a discrete value – ie it will give us the only a discrete value – ie it will give us the probability of a 2.5 inch rain eventprobability of a 2.5 inch rain event We are in fact worried about any event that We are in fact worried about any event that

exceeds our design capacityexceeds our design capacity

That’s not a Problem for Us Smart That’s not a Problem for Us Smart EngineersEngineers

Just Integrate the Function from 2.5 Just Integrate the Function from 2.5 inches on upinches on up In fact most statistical modeling is done In fact most statistical modeling is done

on cumulative probability distributions on cumulative probability distributions (ie integrated areas on the probability (ie integrated areas on the probability density function)density function)

Just one little problemJust one little problem Normal probability density function is Normal probability density function is

one of those beasts that the math one of those beasts that the math teachers don’t like to talk about – can’t teachers don’t like to talk about – can’t get an analytical integrated solutionget an analytical integrated solution

That’s Only a Problem for That’s Only a Problem for MathematiciansMathematicians

We have numeric integrationWe have numeric integration Ok maybe that is a problem if we Ok maybe that is a problem if we

have to integrate that thinghave to integrate that thing Remember – desk top computers are Remember – desk top computers are

recent vintagerecent vintage Do you have a numeric integration package Do you have a numeric integration package

on your computer even now?on your computer even now? Normal Distribution dates from 1733 Normal Distribution dates from 1733

so know someone created tables of so know someone created tables of numeric integrationnumeric integration

Which Normal Distribution?Which Normal Distribution? Probability Density function is a function of Probability Density function is a function of

mean and standard deviationmean and standard deviation Infinite possibilitiesInfinite possibilities

Tabulated Standard Normal Probability Tabulated Standard Normal Probability DistributionDistribution Has a mean of 0Has a mean of 0 A standard Deviation of 1 (by George we got one)A standard Deviation of 1 (by George we got one)

Turns out there are simple formulas for Turns out there are simple formulas for converting parameters from any normal converting parameters from any normal distribution to standard normaldistribution to standard normal Then its just a table look upThen its just a table look up If you have the software – it may even do the If you have the software – it may even do the

look-uplook-up

Normal Distribution TableNormal Distribution Table

Converting to a Value on Standard Converting to a Value on Standard Normal DistributionNormal Distribution

What we want to know is what are the What we want to know is what are the chances of a rainfall event exceeding chances of a rainfall event exceeding our drainage system designour drainage system design Ie what percentage of big rainstorms will Ie what percentage of big rainstorms will

exceed 2.5 inches (on a distribution with exceed 2.5 inches (on a distribution with mean of 1.25 and standard deviation of 1)mean of 1.25 and standard deviation of 1)

Convert 2.5 inches to an equivalent Convert 2.5 inches to an equivalent value on standard normal distributionvalue on standard normal distribution The area above that value in the curve will The area above that value in the curve will

be the same as our actual distribution.be the same as our actual distribution.

Magic Conversion FormulaMagic Conversion Formula

x

Z

1

25.15.225.1

Now Its Look Up TimeNow Its Look Up Time

Prob = 0.8944

ResultsResults

Table shows that from minus infinity Table shows that from minus infinity to 1.25 there is 0.8944to 1.25 there is 0.8944 Ie 0.1056 is above 1.25Ie 0.1056 is above 1.25

English TranslationEnglish Translation There is a 10.56% chance that a large There is a 10.56% chance that a large

rainfall event will exceed the design rainfall event will exceed the design capacity of our drainage systemcapacity of our drainage system

Sounds like Wendy might be doing some Sounds like Wendy might be doing some design work overdesign work over

Basis for Rainfall EventsBasis for Rainfall Events

10% chance called a 10 year storm 10% chance called a 10 year storm (distribution of years largest storms)(distribution of years largest storms)

0.05% chance called a 20 year storm0.05% chance called a 20 year storm 0.01% chance called a 100 year storm0.01% chance called a 100 year storm When say it is designed for a 100 year flood it When say it is designed for a 100 year flood it

doesn’t mean it only happens every 100 yearsdoesn’t mean it only happens every 100 years It means 1% chance in any given yearIt means 1% chance in any given year Problem with other thinking is if you had a big flood Problem with other thinking is if you had a big flood

5 years ago that must mean there is no chance it 5 years ago that must mean there is no chance it will ever happen again in your lifetime (Wrong!)will ever happen again in your lifetime (Wrong!)

Ore Grade ControlOre Grade Control

Orville Orman is planning a truck fleet to Orville Orman is planning a truck fleet to haul his copper ore out of his minehaul his copper ore out of his mine Some rock will have so little copper in it that Some rock will have so little copper in it that

it would cost more to process than its worthit would cost more to process than its worth This stuff is going to get put asideThis stuff is going to get put aside

Other pay rock will be carried to the Other pay rock will be carried to the processing plantprocessing plant

Commonly have ore and waste truck Commonly have ore and waste truck fleets but need to know how much of fleets but need to know how much of each type of rock you will have to design.each type of rock you will have to design.

Orville’s OreOrville’s Ore

Orville knows average grade is Orville knows average grade is 0.95% Cu0.95% Cu

Standard Deviation is say 0.5% CuStandard Deviation is say 0.5% Cu Cut-Off Grade (point at which ore Cut-Off Grade (point at which ore

costs more to process than Cu will costs more to process than Cu will sell for) is 0.25%sell for) is 0.25%

What percentage of Orville’s ore is What percentage of Orville’s ore is below cut-off grade?below cut-off grade?

The SituationThe Situation

μ = 0.95

σ = .5

0.25

How much ofMy rock isDown here?

Oh We Are HotOh We Are Hot

Our critical x value is 0.25Our critical x value is 0.25 We will convert this to a “Z score” We will convert this to a “Z score”

from the standard normal distributionfrom the standard normal distribution We will then look up in the table how We will then look up in the table how

much of our distribution is from much of our distribution is from minus infinity to our Zminus infinity to our Z

We will then tell our truck planners We will then tell our truck planners how much rock to prepare forhow much rock to prepare for

Crunch AwayCrunch Away

5.0

95.025.04.1

Go to the Table

Table Says! 0.0808

About 8.1% of our Rock is Below Cut-Off

Previous ExamplesPrevious Examples

Called One Tailed TestsCalled One Tailed Tests Our Civil Engineers were concerned Our Civil Engineers were concerned

about events larger than some amountabout events larger than some amount An upper tail testAn upper tail test

Our Mining Engineers were concerned Our Mining Engineers were concerned about tonnage below cut-offabout tonnage below cut-off

A lower tail testA lower tail test What if interest in either too much or What if interest in either too much or

too littletoo little Typical of a machine tolerance problemTypical of a machine tolerance problem

ToleranceTolerance

Benjamin Bidwell would like to bid on a Benjamin Bidwell would like to bid on a DOD order for machined shaftsDOD order for machined shafts The spec says 1 inch +/- 0.005 inchesThe spec says 1 inch +/- 0.005 inches Benjamin knows his men and equipment can Benjamin knows his men and equipment can

put any chosen part size within a standard put any chosen part size within a standard deviation of 0.0025 inchesdeviation of 0.0025 inches

He figures he can put in a winning bid provided He figures he can put in a winning bid provided no more than 3% of the pieces he makes have no more than 3% of the pieces he makes have to be rejectedto be rejected

Can Benjamin put in a winner bid on this Can Benjamin put in a winner bid on this order?order?

The SituationThe Situation

σ = 0.0025

μ = 1 1.0050.995

How manyProducts areOut hereIn theTails?

We Know What to DoWe Know What to Do

Convert those Convert those limits to Z scoreslimits to Z scores

Start with the top Start with the top limitlimit

0025.0

1005.12

Table Look Up Says 0.9773 or 2.27% will be too largeNow we use our knowledge – this distribution and

tolerance isSymmetric - ie 2.27% on the bottom endThat Sucks - about 4.54% of products will be out

of Spec

SummarySummary We represent our distribution of outcomes with a We represent our distribution of outcomes with a

mathematical modelmathematical model Then we can easily check the limits on the model Then we can easily check the limits on the model

to see if we can expect our outcomes to be to see if we can expect our outcomes to be acceptableacceptable

We can be doing one tailed tests to see how We can be doing one tailed tests to see how much will be too much or too littlemuch will be too much or too little

We can do two tailed tests to check tolerancesWe can do two tailed tests to check tolerances Obviously these types of models could be fit to Obviously these types of models could be fit to

anythinganything Voting pollsVoting polls Drug reactionsDrug reactions Percentage of Criminal RehabilitatedPercentage of Criminal Rehabilitated

Your ApplicationYour Application

You would feel deprived if you didn’t You would feel deprived if you didn’t get a chance to try thisget a chance to try this Do Homework Unit 2 #1Do Homework Unit 2 #1

The problem involves whether you The problem involves whether you need a coal bunker to handle extra need a coal bunker to handle extra production on those rare times when production on those rare times when you produce more coal than your you produce more coal than your hoist can handle.hoist can handle.