Stat 541 Lecture 6: Normal Distribution & Checking Normality at VCU by James M. Davenport , Study notes of Statistics

Download Stat 541 Lecture 6: Normal Distribution & Checking Normality at VCU by James M. Davenport and more Study notes Statistics in PDF only on Docsity! 1 Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 1 Virginia Commonwealth University STAT 541 APPLIED STATISTICS FOR ENGINEERS & SCIENTISTS Instructor: Dr. James M. Davenport Lecture # 6 Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 2 Today’s Lecture Information in today’s lecture corresponds to the following sections in your textbook (in addition to my notes): • 5.1 & 6.1: Normal Distribution (continued), Checking Normality, The Nature of Statistical Inference, Populations & Samples, Random Sampling Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 3 Finding Probabilities for non standard normals You can prove that if X has N ( μ , σ2 ) , then the distribution of is N ( 0 , 1 ) . XZ μ σ − = Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 4 Example: percent production in spec. Certain shafts are manufactured to meet engineering specs. But there is variation in the produced items. X = diameter of shaft and X has N ( μ , σ2) where μ = 0.251 and σ = 0.001 Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 5 Example: percent production in spec. Engineering Spec’s call for 0.25 + 0.002 . Given that X has a normal distribution with mean and standard deviation as given above, what proportion of this population of manufactured rotor shafts is “in spec” ? Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 6 Example: percent production in spec. X has N ( μ = 0.251 , σ2 = (0.001)2 ) P[ 0.25 – 0.002 < X < 0.25 + 0.002 ] P[ 0.248 < X < 0.252 ] 0.248 0.252XP μ μ μ σ σ σ − − −⎡ ⎤< <⎢ ⎥⎣ ⎦ 2 Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 7 Example: percent production in spec. . [ ] ( ) ( ) 0.248 0.251 0.252 0.251 0.001 0.001 3.00 1.00 1 3 0.8413 {1 0.9987} 0.8413 0.0013 0.8400 P Z P Z − −⎡ ⎤< <⎢ ⎥⎣ ⎦ = − < < = Φ −Φ − = − − = − = Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 THE STANDARD NORMAL CURVE Plot of the Standard Normal p.d.f. z Variable D en si ty Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 9 Checking for Normality Much of what we will be doing for the remainder of the semester is based upon the normal distribution. How can be check data to see if it has arisen from an underlying normal dist.? Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 10 Checking for Normality There are several ways to check for an underlying normal distribution. The first is a Q-Q type plot of the data against the theoretical counterparts. This is called a normal probability plot and is available in most all statistical packages. Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 11 Checking for Normality The second are numerical analytical “tests” that most statistics packages perform, such as the Shapiro-Wilk’s test. We will not discuss these in this course, but will examine the output of NCSS on occasion. Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 12 Example of a Q-Q type plot -- Lilliefors Plots This is a graphical method that many statistical packages will provide (in various different versions) when you ask for a normal probability plot. These are best illustrated by an example. 5 Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 25 Checking for Normality The expected standard normal quantiles, are computed using the following definition for : Common choices are a = and a = . ip Z ip ( ) ( )2 1i i a p n a − = − + 1 2 1 3 Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 26 22.5 23.0 23.5 24.0 24.5 25.0 25.5 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 Normal Probability Plot for the EPA Mileage Data Expected Normal Percentile M ile s pe r G al lo n ( Using Expected Normal Quantiles for a = 1/2 ) Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 27 22.5 23.0 23.5 24.0 24.5 25.0 25.5 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 Normal Probability Plot for the EPA Mileage Data Expected Normal Percentile M ile s pe r G al lo n ( Using Expected Normal Quantiles for a = 1/3 ) Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 28 Normal Prob. Plots • The next slide provides what NCSS gives as a Normal Probability Plot, when you request this as output. • Let’s see how to get this using NCSS. Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 29 23.0 23.5 24.0 24.5 25.0 25.5 -2.0 -1.0 0.0 1.0 2.0 Normal Probability Plot of mileage Expected Normals M ile ag e Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 30 What is Statistics: Statistics: – Nature – Philosophy – Thinking – Inference (The Statistical Method) 6 Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 31 What is Statistics: Some have defined it as the science of drawing conclusions from data in a reliable manner when faced with uncertainly. Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 32 H.G. Wells (circa 1940) … “ statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write.” Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 33 Walter Shewhart - 1939 The long-range contribution of statistics depends not so much on getting a lot of highly trained statisticians into industry as it does in creating a statistically minded generation of physicists, chemists, engineers and others who will in any way have a hand in developing and directing the production processes of tomorrow. Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 34 Statistical Thinking Variability is omnipresent whenever measurement values are observed on a variable. Here are three sources of variability; there are others: • Process variability • Measurement variability • Sampling variability Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 35 0 5 10 15 20 25 30 -60 -40 -20 0 20 40 60 Histogram for Newcomb's Speed of Light Measurements; Data Recorded in 1882 Translated and Scaled Time Measurement Fr eq ue nc y C ou nt Numbers are actually the TIME in seconds, Multiplied by 10 to the 9, minus 24800 Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 36 Statistical Thinking What makes the discipline of statistics useful and unique is that it is concerned with the process of getting data and understanding these problems in the presence of variable measurements. 7 Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 37 Distribution So whenever we “observe” a measurement repeatedly, we almost never get the same value. Instead we obtain what is called a distribution of values Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 38 Distribution - definition The pattern of variation of a variable is called its distribution. The distribution records and provides the numerical values of the variable and how often each value occurs. Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 39 0 5 10 15 20 25 30 -60 -40 -20 0 20 40 60 Histogram for Newcomb's Speed of Light Measurements; Data Recorded in 1882 Translated and Scaled Time Measurement Fr eq ue nc y C ou nt Numbers are actually the TIME in seconds, Multiplied by 10 to the 9, minus 24800 Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 40 Distributions Our study of statistics will involve many aspects of dealing with distributions (both data distributions and theoretical distributions): • Location – mean, etc. • Variation – variance & standard deviation • Comparisons (between and among) Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 41 We need data! • How is this data generated? • Where does it come from, and does that matter? Yes. • What are the properties of the data, and does it matter? Yes. Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 42 The Statistical Method - Inferential Procedure Two very, very important elements are: 1. The POPULATION 2. The SAMPLE 10 Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 55 Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 56 • 90 - discard • 00 - discard • 13 - # 1 • 40 - # 2 • 51 - # 3 05 - # 8 • 50 - # 4 94 - discard • 12 - # 5 69 - # 9 • 68 - # 6 51 - duplicate • 55 - # 7 28 - # 10 Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 57 No. Selected (in Index No. of ascending order) Order of Selection 5 # 8 12 # 5 13 # 1 28 # 10 40 # 2 50 # 4 51 # 3 55 # 7 68 # 6 69 # 9 Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 58 Create your own table of random digits Using Excel or any other spreadsheet that has a function to generate uniform numbers in the interval (0,1) =RAND() Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 59 Create your own table of random digits • Create your own table of random digits --- Excel???? • Microsoft seems to have heard the complaints and is being a little more open on what it is using. See • http://support.microsoft.com/kb/828795 Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 60 Create your own table of random digits • I like John von Newmann’s take on random numbers– • "Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin." 11 Lecture 6 - VCU's Stat 541 James M. Davenport, Copyright 2008 61 Use random generators on-line • http://www.fourmilab.ch/hotbits • http://www.Random.org • http://www.lavarand.org 511 513“ J1ea] 103 [102] 101 =|_Coox St. FE 3 Siz Cooxf AVE, 105/106 | gs] 108} 109 S10 5] 508 Lawnence! 107 St. 113} t12 Wt ios 507 oe | 8 3 St. a Caneny a 506 sos | & Favette Ave —jar| 115 116) 197 | 8 = ScaRRITT WILLIAMS St. 302 Sor (yp | 204 [203] 202 | 201 5 ee ALLER |5r, ALLEN 7] wo 400 FT 4 a zt 7° Vos. 2 Vine | Sr. S | £/208 | 209 | 210 408 z 407 wi Vine LST. J (Srave|Hwy, ay [212]2"4] 213] 202 | 2! Se 302 305] 306 | 307 “| = Pine| ST. ZIPIne 404 1405 | 406 1301 Toa taag| “Gos 8 CEDAR sr. ry a) gf 401 [317 [ais [313 309 eB} OE, og} St Seebce | 342 | St aoa gao2 | gl z 310 ( “LF A Slsisiayal au | sv. FIGURE 3.3 Map of a census tract in Cedar Rapids, lowa