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# ENGINEERING STAT STAT 305

ISU

GPA 3.5

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This 21 page Class Notes was uploaded by Giovani Ullrich PhD on Saturday September 26, 2015. The Class Notes belongs to STAT 305 at Iowa State University taught by Staff in Fall. Since its upload, it has received 21 views. For similar materials see /class/214408/stat-305-iowa-state-university in Statistics at Iowa State University.

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Date Created: 09/26/15

ChapterZ Data Collection Section 22 7 Samnling in studies with an identifiable concrete 39 quot of items Simple Random Sample of size n 7 sample collected from the population in such a manner that every collection of 71 items in the population is equally likely to compose the sample 39 It is important that samples are taken randomly to ensure correct statistical results and the act of simple random sampling can protect from conscious and subconscious human bias 39 It is common practice to use 71 to represent sample size and N to represent population size Selecting a random sample 39 Methods using Random Digits 7 use of random number generators built into computers When computers were not widely available it was common to use printed random digit tables Random Digits Table 22 p 35 is part ofthe larger Random Digits Table B1 p785786 12159 66144 05091 13446 45653 13684 66024 91410 51351 22772 30156 90519 95785 47544 66735 35754 11088 67310 19720 08379 59069 01722 53338 41942 65118 71236 09132 70343 25812 62275 54107 58081 82470 59407 13475 95872 16268 78436 39521 64247 99681 81295 06315 28212 45029 57701 96327 85436 33614 29070 To use a table of random digits 1 Give every item in your population a label ie number your items 1 through N 2 Select labels by moving left to right top to bottom circlingM digits at a time M is the number of digits in N Note Ignore repeats and unused labels 0 Example Suppose that a population ofN 520 objects are numbered 1 through 520 nd the labels numbers of the items that make up a random sample of size n 8 Section 23 7 Principles for Effective E neu39 Response Variable 7 one that is monitored as characterizing system performance output of interest Supervised 0r Managed Variable 7 one over which an investigator exercises power choosing the settings for use 39 Controlled variable 7 a supervised variable that is held constant only has 1 setting 39 Experimental variable 7 a supervised variable with different settings in a study Extraneous Variable 7 one whether thought of or not that could affect the response but is not of interest to the experimenter Wavs of dealing with variables Treat them as controlled variables ie hold them constant Use blocking 7 handle the extraneous variables as experimental variables by including them in the study at several different levels A block of experimental units is a homogenous group in which to compare levels of the primary experimental variables 0 Example Use blocking on the brick molds Put bricks into blocks according to the mold they came from Study only within the blocks molds Use randomization not all extraneous variables can be supervised 7 often means that experimental objects are divided up between the experimental conditions at random or that the order of testing is randomly determined 0 Example Randomly assign bricks to the heat settings Do not apply the same heat to all the bricks from one mold Replication 7 having more than one observation for a given setting of experimental variables this is not simply remeasuring an experimental unit 39 Replication is fundamental in experimentation It helps establish the reproducibility of experimental results and gives an idea of the size of experimental error Section 24 7 Commom Experimental Plans Completely Randomized Experiment 7 all experimental variables are of primary interest none are used for just blocking and randomization is used at every possible point of choosing the experimental protocol Randomized Complete Block Experiment 7 one experimental variable is a blocking factor and within each block every setting of the primary experimental variable appears at least once Randomization is employed at all possible points 25 63 5 58 10 55 15 61 175 62 20 37 25 38 3O 45 35 46 40 19 Bivariate Fit ofyByx j 7a 39 601 50quot gt4 4o 30 20quot IG 1 1 I 0 20 30 4D 50 x MLinear Fit Linear Eit 4 J y 66417699 0900885 x Lsum39mgry bf Fit RSquam 0632518 RSquaIe Adj 0586583 Robt Mean Square Error 9124307 Mean of Response 484 Obserya ons 01 Sum Wgts 10 Analy s mm v A i x 1 J lt Source DF Sum of Squares Mean Square F Ratio 39 Model 1 1146376 114638 137698 39 Error I 8 6660239 8325 Prob gt F C Total 9 18124000 00059 Parameter Estimates j u Term Estimate Std Error t Ratio Probgtltl Intercept 66417699 5648129 1176 lt0001 x 0900885 0242776 3 71 00059 x y Predicted y Residuals y 25 63 641654867 41654867 5 58 61 9132743 39132743 10 55 574088496 24088496 15 61 529044248 809557522 t 175 62 506522124 113477876 20 37 484 114 25 38 438955752 589557 52 3O 45 39391 1504 560884956 35 46 348867257 11 1132743 40 19 303823009 11382301 11673 305 Wk gt Fi 20 4 we 1f leaf2 J a UMP 24122 329w 77W 731 5222 235L254 X WhJ Z g W G m 275 2424 fma me Mr Wax2122221 X xsq Y 25 625 63 5 25 58 uid7 g 9 Mg 1 0 100 55 a s 39 quot gal5039 15 225 61 gsMe 2 fr g 175 30625 62 20 400 37 4M 1 a 25 625 38 39 t 30 900 45 yquotb b 25 2 Z 35 1225 45 40 1600 19 I Response y 39 I IWhoie Model x xsq I ISummary of Fit I RSquare 0658473 RSquare Adj 0560894 Root Mean Square Error 9403518 Mean of Response 484 Observations or Sum Wgts 10 I I Analysis of Variance Source DF Sum of Squares Mean Square F Ratio Model 2 11934169 596708 67481 Error 7 6189831 88426 Prob gt F C Total 9 18124000 00233 I Parameter Estimates I Term Estimate Std Error t Ratio Pr0bgtltl Intercept 6152298 8883684 693 39 00002 x 0208514 0981695 021 08378 xsq 0016541 0022678 073 04895 IE ect Tests r I Source Nparm DF Sum of Squares F Ratio Prob gt F x 1 1 3989307 00451 08378 xsq 1 1 47040796 05320 04895 IResiduai by Predicted Plot I 1 5 A 3925 10quot n 39 E 5 39 39 gt 10 39 39 391 5 I i l 10 20 30 40 50 60 70 y Predicted 39 3 MamM Ww w I v 39 39 1194 792 9 939quot quotn39 W 9 ar7w7 W 3qu v 39 2 0 Ir XL 39 y 97 g mg 91 W 33 344 W 4 742 gag 00 U 3 200 m 3 may fc zfaz 4 03 Hon 5 55 x 43 w m 45 go I700 lff 7 Sh MZ yrs a q do We 7 9 WW a 744 0 300 m y a 39 2 a H a m g e39 13 7kg h I my 1017 7 BM 6 IrResponse hardness LWhole Model I pctcu I temp I I Summary of Fit I RSquare 0899073 V RSquare Ad39 0876645 39 39 Root Mean quare Error 3790931 Hayms m Mean of Response 6630833 Observations or Sum Wgts 12 I Analysis of Variance I re MAM Source DF Sum of Squares Mean Square F Ratio Model 1 2 11521887 576094 400868 PYMIW S Error 9 1293404 14371 Prob gt F s C Total 11 12815292 ltooo1 5 500 quot2 M7 Parameter Estimates I Term Estimate 39 Std Error t Ratio Probgtitl W Intercept 161 33646 1143285 1411 ltooo1 26 ny We 5 pctcu 3296875 1675371 1 97 00806 39 temp 00855 0009788 874 lt0001 I Effect Tests I Source Nparm DF Sum of Squares F Ratio Prob gt F pctou 1 1 556512 38724 00806 temp 1 1 10965375 763013 lt0001 pctcu temp hardness Predicted hardness 002 1000 789 764958333 002 1100 651 679458333 002 1200 552 593958333 002 1300 564 508458333 01 1000 809 791333333 01 1 100 697 705833333 01 1200 574 620333333 01 1300 554 534833333 018 1000 853 817708333 018 1100 718 V 732208333 018 1200 607 646708333 018 1300 589 561208333 W439HWMB Distributions I Residual hardness I I WWW l fiznlk ss ciQZ z yggJaME m H o 1 2 Normal Quantile Plot 39 P wa I Bivariate Fit of Residual hardness By pctcu E w 73 N N 01 O 01 U1 1 l I I Residual hardnes I n I W 65 3 i5 pctcu C O Bivariate Fit of Residual hardness By temp 75 5 l O 039 L l I 3 039 I Residual hardness I 7 11bo 1 bo 1500 temp 900 1660 1 400 lBivariate Fit of Residual hardness By Predicted hardness E m 7 5 O I lduel hardnes m 01 L I 25 Res Tl I C 80 U1 C 0 O 1 0 Predicted hardness 90 55997 7L39IZim25 fv42 6 pctcu temp hardness tmpsq 002 1000 789 1000000 Hamw 4252 002 1100 651 1210000 I 002 1200 552 1440000 rgglpmg WW 002 1300 564 1690000 01 1000 809 1000000 Fwdm 01 1100 697 1210000 c pgr age 01 1200 574 lt 1440000 H 01 1300 554 1690000 0905444 018 1000 853 1000000 Warm 018 1100 718 1210000 CW 018 1200 607 1440000 a 018 1300 589 1890000 W 190432 I Response hardness I Whole Model pctcu temp tmpsq Summary of Fit 1 39 RSquare 098288 RSquare Adj 097646 Root Mean Square Error 1656034 Mean of Response 6630833 Observations 0r Sum Wgts 12 Analysis of Variance 39 I Source DF Sum of Squares Mean Square F Ratio Model 3 12595896 419863 1530980 Error 8 219396 2742 Prob gt F C Total 11 12815292 39 lt0001 Parameter Estimates Term Estimate Std Error t Ratio Probgtlt Intercept 55324479 6282414 881 lt0001 pctcu 3296875 7318705 450 00020 39 temp 0773583 01 10036 703 00001 I tmpsq 00002992 0000048 626 00002 1 Effect Tests J Source Nparm DF Sum of Squares F Ratio Prob gt F pctcu 1 1 5565125 202925 00020 temp 1 1 13554500 494248 00001 tmpsq 1 1 10740083 391624 00002 1 Residual by Predicted Plot 39 l m 3 2 39 39 8 0 1 I O39m 39Wf m mwiwmm7 1 quot E 2 39 g 39 50 60 70 80 90 hardness Predicted 6 7 Ag 5 Example 2 Example 1 on pg 2 in Vardeman amp Jobe Gears were either laid flat or hung and placed in a continuous carburizing furnace The engineer then measured the amount of gear distortion by measuring quotthrust face runout 00001 in Runout values were obtained for 38 laid gears and 39 hung gears and are given in the following table Gears Laid Gears Hung 588999910 1010 7881010101011 1111 11111111111111121212 1213131315 1717171718 1213131313 1414141515 1919 2021212122222223 1515 161717181927 232323242727283136 We want to answer How should the gears be placed in the furnace in order to minimize distortion Note le 480 le 700 2x2 6612 2x2 14384 Laid Gears Hung Gears Median i np5 385 5 195 i np5 3955 20 Median 512 512 12 Median 18 Sample 1 1 X 480 1263 X 700 1795 Mean 38 89lt Range R27 522 R36 729 2 2 Sample 2 6612 7 4 14 834 2 14384 7 733 Variance s 7W7 39 s T4739892 Sample 5 s2 14384 385 s 692 Standard Deviation Quantiles Gears Laid Q25 i 3825 5 10 Q25 x10 10 Q5 i 385 5 195 Q5 20 195 x19 195 19 x20 5 12 512 12 Q75 i 3875 5 29 Q75 x29 15 IQR 15 10 5 UF 15 155 225 LF 10 155 25 Gears Hung Q25 i 3925 5 1025 Q25 11 1025 x10 1025 10 x11 7511 2512 1125 Q5 i 395 5 20 X20 Q75 i 3975 5 2975 Q75 30 2975 x29 2975 29 x30 2522 7523 2275 IQR 2275 1125 115 UF 2275 15115 40 LF 1125 15115 6 Boxglots GearsLaid GearsHung v v v v v v 10 15 20 25 30 35 SiderbyrSide StemandLeaf Plots Gears Laid The decimal point is at the 05889999 1000111111122223333444 1555567789 2 27 Gears Hung The decimal point is 1 digits to the right of the 0 788 1 00001112333 1 57777899 2 011122233334 2 778 3 1 3 6 Backto Back 9999885 0 788 444333322221111111000 1 00001112333 987765555 1 57777899 2 011122233334 7 2 778 ISI 1 ISI 6 Chapter 1 Introduction What is Statistics 39 Statistics is the scienti c application of mathematical principles to the collection analysis and presentation of data at the foundation of all of statistics is data 39 Engineers and scientists are constantly exposed to data that they are expected to make sense of and statistics is a tool that if used properly allows us to gain knowledge about how physical systems work 39 Engineering statistics is the study of how best to 1 Collect engineering data 2 Summarize or describe engineering data 3 Draw formal inferences and practical conclusions on the basis of engineering data all the while recognizing the reality of variation Section 12 7 Basic Terminology Types of Statistical Studies Observational study 7 A study in which the investigator s role is basically passive A process is watched and data is recorded without any intervention from the person conducting the study 0 Example A researcher keeps track of how many cars drive on a certain stretch of road over a onehour period to study why so many accidents occur there Experimental study 7 A study in which the investigator s role is active Process variables are manipulated and the study environment is regulated 0 Example A researcher tests the fracture strength of bricks by subjecting them to different temperatures and measuring the fracture point Experimental studies are more commonly seen when collecting engineering data Experimental studies are more efficient and reliable it s quicker to manipulate variables and watch the response than to passively observe and make it easier to infer causality to say one thing causes another in an experiment Additional Terms Population 7 The ENTIRE group of objects about which one wishes to gather information in a statistical study Sample 7 Group of objects of which one actually gathers data 0 Example A manufacturing line produces 100 microchips each day Of interest is the quality of these microchips It may be costly or impossible to test ALL 100 microchips the population so instead a group of 5 microchips the sample is randomly chosen for quality testing In a perfect world we would always have access to the entire population of data however that is almost never the case This is the reason that there is always uncertainty involved in statistics We take a sample and use it to make guesses about the entire population Note that the larger the sample the better the guess Types of Data Qualitative 0r Categorical data 7 any data where an object is assigned a category 0 Examples malefemale conformingnonconforming largemediumsmall etc Quantitative 0r Numerical data 7 can be counts or measurements strictly of a numerical nature 0 Examples number of times a die is rolled until a 5 is seen count weight or height of an object etc Additional Data Description Univariate data 7 observations are made on only a single characteristic of each sampled item Multivariate data 7 observations are made on more than one characteristic of each sampled item A special case of this involves two characteristics called bivariate data 0 Examples temperature and viscosity of a uid bivariate data resistance diameter and length of electrical wire multivariate data Repeated measures data 7 multivariate data that consist of several determinations of basically the same characteristic In the special case of bivariate responses the term paired data is used 0 Examples measuring someone s blood pressure before they take medication and again after they take medication paired data measuring the hardness of a part using three different instruments repeated measures data Types of Data Structures Complete factorial study 7 several variables are of interest and data are collected under each possible combination of settings of these variables The variables are called factors and the settings are called levels Fractional factorial study 7 data are collected for only some of the combinations that would make up a complete factorial study wn V F V rll md wn now temperature humlddty and pressure affect tlne fracture strengtln of bncks Two temperatures are oflnterest say T1 and T2 two numrdt levels are of rnterest say H1 and H2 and two pressures are oflnterest say P1 and P2 Y md measured under all posslble eombrnatrons of temperature humlddty and pressure 2 x 2 x 2 8 eomblnatlons llsted below Comblnauon Temperature Humldl Pressure 1 T1 H1 P1 2 T1 H1 P2 3 T1 H2 P1 4 T1 H2 P2 5 T2 H1 P1 6 T2 H1 P2 7 T2 H2 P1 8 T2 H2 P2 Seetlon 1 3 7 Measurement Its Importance and Dif cultx Valrdmeasurement e appropnately represents tlne feature ofan object that ls of ee rmportan Preclslon 7 small varratron m repeated measurement ofthe same object Aeeurate unblasedmeasurement system 7 produeestlne true or eorreetvalue on average ofthe quanuty belng measured Calrbratron ls an aeumty almed atrmpromng measurement aeeuraey Instruments usedln measurement should be ealrbrated agarnst a standard MeasurementTar etshootln Analo 0 o o o Not Aeeurate Not Preclse Accurate Not pmse Not Aeeurate Preclse Aeeurate Preerse Statistics 305 Formulas for Inference in Simple Linear Regression 39 MOdek 19 160 iili 5i Least squares estimates estimates of the slope and intercept are the least squares slope and intercept from Section 41 Estimate of 02 1 1 2 A 2 z 2 ELF Wn22y y 428 Residuals as in Section 41 81 9 gt Standardized residuals V 6 e quot 739 air5 9 F v1 i quot Inference for 81 con dence interval b1 it ELF i w ilz ttestofHaz 1 T b1 hzw 2395 3LF I Inference for My n 51931 con dence interval quot 392 Mt 5m 21 ttest ofHuyl ii sets A 1 2 12 yit SLF 1EZT39I i2 Simple Regression and ANOVA 39 SSE 2y 172 n 2siF SSR SSTot SSE 217 02 SSR SSTot we 0 Prediction intervals R2 For testing Ho A 0 t2 SSE1 MSR 39 F SSEn 2 MSE X 25 63 5 58 10 55 15 61 175 62 20 37 25 3s 30 45 35 46 4o 19 Bivariate Fit ofyByx j 75 6 504 gt5 4o 30 20 10 l 1 I I 0 1O 20 3O 4O 50 X quotLinear Fit Linear th y 66417699 0900885 x salinity of Fit RSquare 0632518 RSquare Adj 0586583 Root Mean Square Error 9124307 Mean of Response 484 Observations 0139 Sum Wgts 10 An lysis fva anwj J Source DF Sum of Squares Mean Square F Ratio Model 1 1 1463761 114638 13 7698 Error 8 6660239 8325 Prob gt F C Total 9 18124000 00059 Ll arameter Estimates Tenn Estimate Std Error tRatio Probgtt Intercept 66417699 5648129 1176 lt0001 x 0900885 0242776 371 00059 i y Predicted y Residuals y 25 63 641654867 11654867 5 58 61 9132743 39132743 10 55 574088496 24088496 15 61 529044248 809557522 175 62 506522124 113477876 20 37 484 1 14 25 38 438955752 68955752 30 45 393911504 5 60884956 3 5 46 343867257 1 11 132743 40 19 11382301 303823009 5 305 I Wk gt 4 we 5 W 5 39 UMP am J aUZW Z 5W 411554 1 W agom9 3 Ma 1451 faa me W MexM12981 30 Chapter 4 Describing Relationships Between Variables Example 5 Surface Fitting and Brownlee39s Stack Loss Data Table 48 contains part of a set of data on the operation of a plant for the oxidation of ammonia to nitric acid that appeared rst in Brownlee s Statistical Theory and Methodology in Science and Engineering In plant operation the nitric oxides produced are absorbed in a countercurrent absorption tower 39 39 The air ow variable x represents the rate of operation of the plant The acid concentration variable x3 is the percent circulating minus 50 times 10 The response variable y is ten times the percentage of ingoing ammonia that escapes from the absorption column unabsorbed ie an inverse measure of overall plant ef ciency For purposes of understandingtpredicting and possibly ultimately optimizing plant performance it would be useful to have an equation describing how y depends on x1 x2 and x3 Surface tting via least squares is a method of developing such an empirical equation Printout 4 shows results 39om a MINITAB run made to obtain a tted equation of the form 5quot 150 blxl bzxz bsxa Table 48 Brownlee39s Stack Loss Data ii x2i xai 39 Observation x Cooling Water Acid yi Number Air Flow Inlet Temperature Concentration Stack Loss 1 80 27 88 37 2 62 22 87 18 3 62 23 87 18 4 62 24 93 19 5 62 24 93 p 20 6 39 58 23 87 15 7 58 18 80 14 8 58 1398 89 14 9 58 17 88 13 10 58 18 82 39 39 1 1 11 58 19 93 12 12 50 18 89 8 13 50 18 86 7 14 50 19 72 8 15 50 19 79 8 16 50 20 80 9 17 15 56 20 32 Response y I Whole Model 1R1 11 x2 1sq Summary of Fit J RSquare 0979859 RSquare Adj 0975211 Root Mean Square Error 1124554 Mean of Response 1447059 Observations 0r Sum Wgts 17 I Analysis of Variance I Source DF Sum of Squares Mean Square F Ratio Model 3 79979521 266598 2108127 Error 13 1644008 1265 Prob gt F C Total 16 81623529 lt0001 I Parameter Estimates I Term Estimate Std Error t Ratio Probgtt Intercept 1540929 1260267 122 02431 x1 0069142 0398419 017 08649 x2 05278044 0150079 352 00038 x1 sq 00068183 0003178 215 00514 x1 y Predicted y StdErr Pred y Residual y 39Studentized Resid y 80 37 36947 1121 0053 0573 62 18 18125 0407 0125 0119 62 18 18653 0462 0653 0637 62 19 19181 0553 0181 0185 62 20 19181 0553 0819 0837 58 15 15657 0513 0657 0656 58 14 13018 0475 0982 0964 58 14 13018 0475 0982 0964 58 13 12490 0595 0510 0534 58 11 13018 0475 2018 1980 58 12 13546 0378 1 546 1459 5039 8 7680 0493 0320 0317 50 7 7680 0493 0680 0673 50 8 8208 0499 0208 0206 50 8 8208 0499 0208 0206 50 9 8735 0548 0265 0269 56 15 12657 0298 2343 2161 WV 9 Mow Ler 37er 7 M M 5 24254 s 58 10 55 9 15 61 W 395 175 62 39 20 37 25 38 3 4 30 45 gr Z we 35 46 40 19 39 f 1 BivariateFitofyByx r j 7G 4 0443 W 50quot gt40 3 Sham772276 hi EW 20 39 391 10 1 1 1 1 1 0 10 20 30 40 50 V x k x iinearFit I X Wag15244quot LinearEit I 39 geg gq M473 y 66417699 0900885 x i f J summgiyofmt RSquam 0632518 3M RSquare Adj 0586583 6 RootMean SquareError 9124307 quotquot5 3M Mean ofResponse 484 3 W Obsexva ons or Sum Wgts 10 7 LF Analysisoiva nceAMQVA f a f v Source DF Sum of Squares Mean Square FRatio F0 kiwi a 23 5 0 H41 I w Model 1 11463761 114638 1398 gt 4461 5148125176 P Value Error 8 6660239 rob gt F c Total 9 18124000 00059 7 Vim Paragl tereqimak quot 39 r Term Estimate Std Error Ratio Probgtltl Intemept 66417699 5648129 1176 lt000139 4er 584178456 TValwz x 0900885 0242776 00059 W5 FWMM For 425 H0 6 0 x y Predictedy Residualsy 3 25 63 641654867 11654867 939 l5 0 5 58 619132743 39132743 10 55 574088496 24088496 15 61 529044248 809557522 175 62 506522124 113477876 20 37 484 114 25 38 438955752 58955752 30 45 393911504 560884956 35 46 348867257 111132743 40 19 303823009 41382301 W 4 39 j 1

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