Applications of the Normal Distribution Model (The Confidence Interval) ©Dr. B. C. Paul 2003 revision 2009 Note – The concepts found in these slides are considered to be common knowledge to those trained in the field of statistics. Most basic statistics texts have coverage of these concepts. The specific table depicted in these slides is from Fundamental Concepts in the Design of Experiments by Charles R. Hicks, 1982 published by Holt Reinhardt and Winston, although most basic statistics texts include similar tables for a standard normal distribution.

Preparing for Rainfall Wendy Wetone has just designed a storm sewer system for a new housing project Culverts and intakes will handle a 2.5 inch rainfall in 24 hours The average big rainfall even in the area is only 1.25 inches Wendy is ok Right? If the roads and homes in an area are going to wash out maybe being ready for an average rain isn’t good enough

Reality for Major Rainfall Events Average is 1.25 inches, but suppose there is a 1 inch standard deviation Average is 1.25 inches, but suppose there is a 1 inch standard deviation μ = 1.25 σ = 1 How would we know Something like this? We built a model From weather Records. Is there enough of a chance up hear that I should be getting heart-burn over this design?

We Know How to Solve This One Normal Distribution is fully defined by a formula Normal Distribution is fully defined by a formula We only need to know the average (in this case 1.25) and the variance (standard deviation squared – easy when standard deviation is one) We only need to know the average (in this case 1.25) and the variance (standard deviation squared – easy when standard deviation is one)

What That Formula Does Y is a probability value (chance of occurrence) X in this case is a rainfall event Rather obviously we are interested in rainfall events greater than 2.5 inches Guess that means x is 2.5 Problem – Formula gives probability for only a discrete value – ie it will give us the probability of a 2.5 inch rain event We are in fact worried about any event that exceeds our design capacity

That’s not a Problem for Us Smart Engineers Just Integrate the Function from 2.5 inches on up In fact most statistical modeling is done on cumulative probability distributions (ie integrated areas on the probability density function) Just one little problem Normal probability density function is one of those beasts that the math teachers don’t like to talk about – can’t get an analytical integrated solution

That’s Only a Problem for Mathematicians We have numeric integration Ok maybe that is a problem if we have to integrate that thing Remember – desk top computers are recent vintage Do you have a numeric integration package on your computer even now? Normal Distribution dates from 1733 so know someone created tables of numeric integration

Which Normal Distribution? Probability Density function is a function of mean and standard deviation Infinite possibilities Tabulated Standard Normal Probability Distribution Has a mean of 0 A standard Deviation of 1 (by George we got one) Turns out there are simple formulas for converting parameters from any normal distribution to standard normal Then its just a table look up If you have the software – it may even do the look-up

Normal Distribution Table

Converting to a Value on Standard Normal Distribution What we want to know is what are the chances of a rainfall event exceeding our drainage system design Ie what percentage of big rainstorms will exceed 2.5 inches (on a distribution with mean of 1.25 and standard deviation of 1) Convert 2.5 inches to an equivalent value on standard normal distribution The area above that value in the curve will be the same as our actual distribution.

Magic Conversion Formula

Now Its Look Up Time Prob =

Results Table shows that from minus infinity to 1.25 there is Ie is above 1.25 English Translation There is a 10.56% chance that a large rainfall event will exceed the design capacity of our drainage system Sounds like Wendy might be doing some design work over

Basis for Rainfall Events 10% chance called a 10 year storm (distribution of years largest storms) 0.05% chance called a 20 year storm 0.01% chance called a 100 year storm When say it is designed for a 100 year flood it doesn’t mean it only happens every 100 years It means 1% chance in any given year Problem with other thinking is if you had a big flood 5 years ago that must mean there is no chance it will ever happen again in your lifetime (Wrong!)

Ore Grade Control Orville Orman is planning a truck fleet to haul his copper ore out of his mine Orville Orman is planning a truck fleet to haul his copper ore out of his mine Some rock will have so little copper in it that it would cost more to process than its worth Some rock will have so little copper in it that it would cost more to process than its worth This stuff is going to get put aside This stuff is going to get put aside Other pay rock will be carried to the processing plant Other pay rock will be carried to the processing plant Commonly have ore and waste truck fleets but need to know how much of each type of rock you will have to design. Commonly have ore and waste truck fleets but need to know how much of each type of rock you will have to design.

Orville’s Ore Orville knows average grade is 0.95% Cu Orville knows average grade is 0.95% Cu Standard Deviation is say 0.5% Cu Standard Deviation is say 0.5% Cu Cut-Off Grade (point at which ore costs more to process than Cu will sell for) is 0.25% Cut-Off Grade (point at which ore costs more to process than Cu will sell for) is 0.25% What percentage of Orville’s ore is below cut-off grade? What percentage of Orville’s ore is below cut-off grade?

The Situation μ = 0.95 σ = How much of My rock is Down here?

Oh We Are Hot Our critical x value is 0.25 Our critical x value is 0.25 We will convert this to a “Z score” from the standard normal distribution We will convert this to a “Z score” from the standard normal distribution We will then look up in the table how much of our distribution is from minus infinity to our Z We will then look up in the table how much of our distribution is from minus infinity to our Z We will then tell our truck planners how much rock to prepare for We will then tell our truck planners how much rock to prepare for

Crunch Away Go to the Table Table Says! About 8.1% of our Rock is Below Cut-Off

Previous Examples Called One Tailed Tests Called One Tailed Tests Our Civil Engineers were concerned about events larger than some amount Our Civil Engineers were concerned about events larger than some amount An upper tail test An upper tail test Our Mining Engineers were concerned about tonnage below cut-off Our Mining Engineers were concerned about tonnage below cut-off A lower tail test A lower tail test What if interest in either too much or too little What if interest in either too much or too little Typical of a machine tolerance problem Typical of a machine tolerance problem

Tolerance Benjamin Bidwell would like to bid on a DOD order for machined shafts Benjamin Bidwell would like to bid on a DOD order for machined shafts The spec says 1 inch +/ inches The spec says 1 inch +/ inches Benjamin knows his men and equipment can put any chosen part size within a standard deviation of inches Benjamin knows his men and equipment can put any chosen part size within a standard deviation of inches He figures he can put in a winning bid provided no more than 3% of the pieces he makes have to be rejected He figures he can put in a winning bid provided no more than 3% of the pieces he makes have to be rejected Can Benjamin put in a winner bid on this order? Can Benjamin put in a winner bid on this order?

The Situation σ = μ = How many Products are Out here In the Tails?

We Know What to Do Convert those limits to Z scores Convert those limits to Z scores Start with the top limit Start with the top limit Table Look Up Says or 2.27% will be too large Now we use our knowledge – this distribution and tolerance is Symmetric - ie 2.27% on the bottom end That Sucks - about 4.54% of products will be out of Spec

Summary We represent our distribution of outcomes with a mathematical model We represent our distribution of outcomes with a mathematical model Then we can easily check the limits on the model to see if we can expect our outcomes to be acceptable Then we can easily check the limits on the model to see if we can expect our outcomes to be acceptable We can be doing one tailed tests to see how much will be too much or too little We can be doing one tailed tests to see how much will be too much or too little We can do two tailed tests to check tolerances We can do two tailed tests to check tolerances Obviously these types of models could be fit to anything Obviously these types of models could be fit to anything Voting polls Voting polls Drug reactions Drug reactions Percentage of Criminal Rehabilitated Percentage of Criminal Rehabilitated

Your Application You would feel deprived if you didn’t get a chance to try this You would feel deprived if you didn’t get a chance to try this Do Homework Unit 2 #1 Do Homework Unit 2 #1 The problem involves whether you need a coal bunker to handle extra production on those rare times when you produce more coal than your hoist can handle. The problem involves whether you need a coal bunker to handle extra production on those rare times when you produce more coal than your hoist can handle.