ENGR EXPERIMENTATION II
ENGR EXPERIMENTATION II ME 311
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ME 311 Experimentation Spring 2006 IVE 311 MchanicalEngI39neen39ng Universiw oIKemucky The Concept Of Uncertainty Af icts all measurements When scientists and engineers talk about the errors associated with a measurement they really mean uncertainty An Example What time is it now Look at your watch not the clock IVE 311 MchanicalEngI39neen39ng Universiw oIKemucky DOE Interlude Statistical Uncertainty IVE 311 MchanicalEngI39neen39ng Universiw oIKemucky Systematic vs Random Uncertainty Why isn t there a single answer to the time Not all timepieces set to an identical time Calibration both in initial setting and clock rate Different people can report different times from the same clock Measurement errors digital readouts makes this harderto mess up but still possible what happens if the display is broken What if someone forgot to adjust for daylight savings time Systematic error IVE 311 MchanicalEngI39neen39ng Universiw oIKemucky Systematic vs Random Uncertainty You can estimate the random uncertainty with multiple measurements and looking at the distribution of results Systematic errors biases are more difficult need to be estimated based on other information Exaggerate the source of the effect if possible Think there s a temperature effect Crank up the heat Figure out multiple ways to ask the same basic question redundancy independent measurement techniques IVE 311 MohanicalEngI39neen39ng V Unite my oIKemucky Political Polls We are now very familiar with political polls take Zogby in PA date unknown Bush 47 Kerry 47 i 4 This is a 95 confidence interval thus there is a 95 chance that the actual values fall within 43 to 51 for each candidate If one candidate has a 2 lead thus within the margin of error that candidate has a 5095 chance of having a 2 lead in the public at large IVE 311 MohanicalEngI39neen39ng Systematic vs Random Uncertainty Which of these are sources of systematic uncertainties and which produce random errors Clocks run slightly slower in the winter than in the summer due to temperature effects The second hand on different clocks clicks forward at slightly different instants within a second some at the beginning some in the middle and some at the end IVE 311 MohanicalEngI39neen39ng V Unite my oIKemucky Universiw oIKemucky Samples vs Entire Population Political polls try to gauge the nature of the entire population by extracting a subset sample and conducting an experiment on that If the subset is a representative sample then the characteristics of the sample should reflect the characteristics of the full population This method is applied all the time Engineering product quality control Medicine rate of adverse reactions to a medication Politics political polls Entertainment television ratings IVE 311 MohanicalEngI39neen39ng Universiw oIKemucky Random Error in Polling Assuming a representative sample is chosen Poll N people get uncertainty oixN Usingthet table 10 1000000 Uncertainty depends only on sample size and NOT the population size For suf ciently large population ME SitiMunlniul Engineering UNVEUW gummy systematic and random of random error Too close to call MK The Florida Election Fiasco Alarmingly relevant example of measurement error lfvoting machines have 01 counting error and they re olten not that good what might you expect error to be in a state with 6 million votes evidence 1 of 6 million is 60000 a 0 1 is 6000 Vote count was closer than this 100200 Margin was different with every re count Systematic errors butter y ballots K Harris Did anyone really win this election By engineering standards no but politics ain t engineeringl ME ai i Muhlniul Slimreuer UNVUUKV gummy Gaussian Normal Or Bellshaped Distribution Gaussian distribution is produced in general when measuremen s aggregate many random stochastic events in a process that is not changing over time stationary Examples are everywhere Distribution of heights of the people in the room Answers to our question about time lOs test grades not always production samples etc BUT not all things are simple Gaussians thus all ofthe other t ns we discussed Distribution of incomes in US people take to commute to school each day Grades on engineering exams often bimodal Standard Deviationquot characterizes the spread in the values 23 683 ofthe values lie within one stan ar deviation 1o 954 ofthe values lie within 2 standard deviations 2 U 997 ofthe values lie within 3 standard deviations 3U ME 3MMunlniul Engineering UnivurxilrulKAnlunky MK Statistics Summarize Distribution Properties M531 1 Muhlniul Slimreuer Univullly gummy Men 69 i 3 inches standard deviation is 3 inches 68 between 5 6quot and 6 0quot 95 between 53quot and 6 3quot 997 between 50quot and 6 6quot Women 655 i 25 inches 68 between 52quot and 57quot 95 between 4 115quot and 595quot 997 between 49quot and 6 0quot Example Typical American Heights Only1 in 700 men taller than 6 6quot Only1 in 700 women taller than 6 0 UK IVE 3 11 MechanicalEngineering Uniter Kentucky y a Some Criteria To Evaluate Claims Is the sample representative Are the uncertainties given as well as the data What possible bias could exist in the result Is the claim consistent with other data Extraordinary claims require extraordinary proof You don t understand your data if you don t understand your uncertainties IVE311MechanicaIEnginee ng Unive If my oIKemuc ky Limitations of Statistics 0 Nothing is certain Most statistical models have tails a finite but very tiny probabilitythat almost anything can happen Eg the Gaussian distribution has tails to infinity Ergo predicting events on the tails whose probability is small is often not accurate Chances that a shuttle will fail during a mission Chances that a Cat 5 hurricane will hit New Orleans 0 It s easy to misuse and misunderstand statistics The government issues 23 car seats to every family because the average family has 23 children It s not always obvious when statistics have been misused or deliberately abused IVE 3 11 MechanicalEngineering Unitmm oIKemucky Comparing data with modelspredictions Uncertainties in data points show up as error bars in plots Angular Size on the Sky 9 1I2 lI3 l l MAXIMAJ a Relative Signal Strength N l T r n l l l t l 200 400 500 800 1000 Angular Frequency on slw Angular scale of cosmic microvmve back round uctuations Is the theoretical model solid line consistent with the data IVE 3 11 MechanicalEngineering a Unitmm oIKemucky Error Bars Error bars on a plot denote the confidence level for that particular datum typically 95 but the confidence level or 0 factor should be explicitly stated Most plotting packages allow for error bar plots Excel MATLAB Class poll ME 311 Mechanical Engineering University of Kentucky Political Survey Results Respondents 25 total Fall 2004 I Bush I Kerry I Kerry Didn39t Vote I Undecided I Badnarik undfgde i e Badnarik 4 Kerry Didn39t Bull Kerry 42 ME 311 Mechanical Engineering I University of Kentucky Political Survey In Ill 39 A39F nrrnr Just the actual voters pie charts not a good n IIUVV IIIQIHIII UI UIIUI Badnarik 4 Kerry 48 quotlo Bush 48 ln ME 311 Mechanical Engineering University of Kentucky Statistical Error 23 voters gives a 95 confidence margin of error of 20 difficult to express in a pie chart use bar or line plot instead note that we ll use a typical error of 5 since our sample is unconventionall small Kerry 9a Badnarik Percent Respondents Note that the error on he smallest group goes negative Percentage Voters ME 311 Mechanical Engineering 0 University of Kentucky Comparison w Results When we compare with the actual election we find that the survey actually proved accurate well within the margin of error given a larger sample so Final results fall within the 95 confidence interval 39 of our prediction I Bush I Kerry Class Actual ME 311 Mechanical Engineering 9 University of Kentucky OK Why Doesn t This Matter The national presidential poll statistic is often almost always in fact used in the wrong way it doesn t matter what the national respondents say only what the outcome of the electoral college state elections is State by state polls would be more useful here This is an example of the statistics being correct but used or interpreted incorrectly Why isn t it done the correct way instead Cost Confusing to electorate Less impact to media ME 311 Mechanical Engineering University of Kentucky State by State Electoral College Results Bush 286 53 Kerry 252 47 ME 311 Mechanical Engineering 0 University of Kentucky County by County Results Looks skewed but remember it is population that matters ME 311 Mechanical Engineering University of Kentucky Weighted by Population Weighted by area Proportional shading The way you look at the data makes a big difference on how you Viewthe data ME 311 Mechanical Engineering D University of Kentucky Finale HW 3 Due next week Lab 2 starts this week lab plans due at end of section Some statistics review follows on following slides Return to ME 310 fort and ztables ME 311 Mechanical Engineering u University of Kentucky ME 311 Mechanical Engineering University of Kentucky Statistics Review Sample Mean Samplt Mean If the n observations in a sample are donated by x x2 the sample mean is l x 39 21 ME 311 Mechanical Engineering quot University of Kentucky From Montgomery Design and Analysis of Experiments 2001 Consider me Oring tensile strength Dxpcilmcm Liescribed in Clmprer i The data from me 1 1 i inrh Ftp i 1 psi for the eight obscrmiions on strength is 7iigr 7 ll 71037 i047 i1140 i 3440 T 10550 psi A physical interpretation of the sample mean as a measure afioeution is shown in Fig Nnictliatthc1impicmc39 L5 can be thought 3qu ii quotbalance poll Timis if c observation repre uiispoundofmass i iattlic mime tliCi zwls a Fulcrum ior Dated at Twmiitl exactly balance mi v cm 01 weights N me Montgomery Design and Analysis oEXpeIimems 2001 IVE 31 1 MechanicalEngineen39ng n Uniepsin olKemucky Variance and Standard Deviation Sample Variance and Sample Standard Deviation I39 he I obscivuiium in u gtLimple lI C dcnmctl by i r1 x then the sample variance is 271 l39iit 39 quot quot i 1013 me Montgomery Design and Analysis oEXpeIimems 2001 IVE 31 1 MechanicalEngineen39ng Uniepsin olKemucky Population Mean Population Mean For a nite population with N measurements the mean is 121 The sample mean is a reasonable estimate of the population mean me Montgomery 09519quot and Analysis oEXpeIimems 2001 IVE 311 MechanI39caIEngI39neering a Unitepsin olKemucky Example 1000 1050 i050 1070 1391 N4 7quot2 1 3 gtlt V4 ifs pq 1 6 I 3937 V me Montgomery 09519quot and Analysis oEXpeIimems 2001 IVE 311 MechanI39caIEngI39neering Unitepsin olKemucky Example cont Population Variance 4 3 quoti quot quot1 X Population Variance i 11x x 4 3 lili 4 in When the population is nite and consists ofN values 3 14m x 4 we may de ne the population variance as 4 lIii l 1 2 5 Illiii 15 225 l lil7l 15 125 7 in i39 324 1 N HIT is 314 N44 Lil INN The sample variance is a reasonable estimate of the iltl257psi3 2 y 1057 Hill si population variance From MUnlgUmery Design and Analysis UfExperimerils 2001 ME 311 Mechanical Engineering ME 311 Mechanical Engineering gt University ofKentucky A University of Kentucky Multivariate Data Multivariate Example Table 2 9 Wire Bond Dam The dot diagram stemandleaf diagram histogram and box plot are descriptive displays for univariate Pull Sfcngih Viictcngth Die Highi Pull Sjicngth WircLtng1h Die Zetgln data39 that is they convey descriptive information 17 f 39 39 E ii 120 iii l78 l 400 abOU t a Slngle Varlable39 4 ill vi l7 4mm 0 bill Many eng1neer1ng problems 1nvolve collectmg and 5 1 l 3 l X i 4 200 l0 3493 ii 140 analyzmg mult1var1ate data or data on several 7 1 7 20 4050 h w X l 31 ll 443 l JOE d1fferent var1ables v 0 ion 2 541 in an W will 2 5053 i7 390 In eng1neer1ng stud1es 1nvolv1ng mult1var1ate data n 4 v 4 2213 6 mi l1 ll Jill Z5 2ll Alli often the ob ect1ve 1s to determ1ne the relat10nsh1ps n n 500 l l among the variables or to build an empirical model From lllUnlgUmely Design and Analysis UfExperimerils 2001 ME 311 Mechanical Engineering ME 311 Mechanical Engineering F University ofKentucky University of Kentucky 70 u 707 7 go so 0 so 50 L I z n E 40 4 g 30 i 39 30 39 m m r I h n H l in in s 39 39 o n 0 5 10 15 an 0 100 200 300 1100 SUD r100 Length Die height I Pull simngn versus length m Pull slwiiglli L39elsllsdle might Figure l 17 Scatter diagrams uml lwx plots lor the ll39C bond pull strength lum ll39l Table l 1 la Pull strength Cl xllx39 length lil Pull strength icrsus klIL height From Montgomery Design and Analysis of Experiments 2001 ME 311 Mechanical Engineering r University of Kentucky Correlation Coefficient Sample Correlation Coefficient Given 1 pairs of 111111 yr XI ppm Clem l39 is de ned by 39l39 the sample correlation coef 26 with l S 15 1 ME 311 Mechanical Engineering A Univershy of Kentucky Correlation Examples v x x t u 39 o o u n n u n 39 o u n u 39 n 39 I 39 c I 0 u 39 u n I n 39 o 39 0 a u I 39 39 39 o n n I 1 i m 5 M 1 m 5 m 1 m l is near 0 i lt1an in r is war 0 y anrl J are unrelated x are ncnllnearly rnlaled Figure 218 SCtlllCl dlilgl39illlh for tlillbimil ulucs vflhe sumplc correlation costlicicm Ul i lt ncnr l 7 r ix ncnr tr 1 is I1cli ll391m are unrelated til I39 is near ll 139 And are nonlincru39ly related ME 311 Mechanical Engineering University of Kentucky Plotting Multivariate Data Example Table 210 Data on Shampoo Foam Scent Color Residue Region Quality 5x 84 Foam Scam Color RCSlLlHL Region Scull Hull color l 34 Regulus 01 H 5M 37 Region 71W u 175 n lbs so 0mm in n1ln l1l5 nlN7 n5l1 7 n lm m 5 83 From Montgomery Design and so Analysis of Experiments 2001 ME 311 Mechanical Engineering nivershy of Kentucky 1O Foam 5925 1575 5 90 a 6175 4125 g 4925 l 30 u 3 575 ResldUtJ V Fn m 1 75 n 39 Figure 220 A mm diagram 01 simmpo quainy versus 1 2i foam 929198 39 139 1 839066 on I Quality I u 39 1 0 lt0 lt0 lt9 3 lt3 o 2 3 9 7 quotL 39i 00 q j b 596 5 b 595 x Fig1 2 19 Matrix Discuttcr plots for the shampoo dam From Montgomery Design and in Table 2 10 Analysis orExperimems 2001 UK ME 311 Mechanical Engineering University of Kentucky ME 311 Experimentation ll Spnng2006 22 and 23 Factorial Experiments ME 311 Mechanical Engineering University of Kentucky Analysis of MultiFactored Experiments Twolevel 2 Factor Factorial Designs ME 311 Mechanical Engineering University of Kentucky The 2quot Factorial Design Special case of the general factorial design kfactors all at two levels The two levels are usually called low and high they could be either quantitative or qualitative Very widely used in industrial experimentation Form a basic building block for other very useful experimental designs DNA Special shortcut methods for analysis ME 311 Mechanical Engineering University of Kentucky Chemical Process Example Factor Treatment Replicate A B Combination I II III Total A low B low 28 25 27 80 A high B low 36 32 32 100 A low E high 18 19 23 60 A high H high 31 30 29 90 A reactant concentration B catalyst amount y recovery response ME 311 Mechanical Engineering University of Kentucky The Si plest Case The 22 b60 a 0 High 181923 l313029 2 pounds 3 m E 7 s E 5 Low 1 quotWmquot 1so a oo 28 25 27 36 32 32 i i 4 Low High 15 25 Reacta nt concentration A Figure 61 Treatment combinations in the 22 design and denote the low and high levels of a factor respectively Low and high are arbitrary terms Geometrically the four runs form the corners of a square Factors can be quantitative or qualitative although their treatment in the nal model will be different UK ME 311 Mechanical Engineering University of Kentucky Estimating Effects of Each Factor Estimate of the effect of A a1b1 aob1 estimate of effect of A at high B a1b0 aob0 estimate of effect of A at low B sum2 estimate of effect of A over all B Estimate of the effect of B a1b1 a1b0 estimate of effect of B at high A aob1 aob0 estimate of effect of B at high A sum2 estimate of effect of B over all A ME 311 Mechanical Engineering 0 University of Kentucky Estimating Interaction of Factors Estimate the interaction of A and B a1b1 aob1 estimate of effect of A at high B a1b0 aob0 estimate of effect of A at low B difference2 estimate of effect of B on the effect of A Called the interaction of A and B a1b1 a1b0 estimate of effect of B at high A aob1 aob0 estimate of effect of B at lowA difference2 estimate of the effect of A on the effect of B Called the interaction of B and A UK ME 311 Mechanical Engineering University of Kentucky Interactions Note that the two differences in the interaction estimate are identical by definition the interaction ofA and B is the same as the interaction of B and A In a given experiment one of the two literary statements of interaction may be preferred by the experimenter to the other but both have the same numerical value ME 311 Mechanical Engineering 0 University of Kentucky Remarks on effects and estimates The use of all four yields in the estimates of the effect of A the effect of B and the effect of the interaction of A and B all four yields are needed and are used in each estimates The effect of each of the factors and their interaction can be and are assessed separately this in an experiment in which both factors vary simultaneously Note further that with respect to the two factors studied the factors themselves together with their interaction are logically all that can be studied These are among the merits of these factorial designs 1v 311 Memamcal Engmewg Univerle omenmm Semantics Recall that simply written B is called a main effect Our estimate of B is often simply written B AB is called an interaction effect Our estimate of AB is often simply written AB So the same letter is used generally without confusion to describe the factor to describe its effect and to describe our estimate of its effect Keep in mind that it is only for economy in writing that we sometimes speak of an effect rather than an estimate of the effect We should always remember that all quantities formed from the results are merely estimates A is called a main effect Our estimate of A is often A 1v 311 Memanical Engineering Universin omenmm Table of signs The following table is useful Notice that in estimating A the two treatments with A at high level are compared to the two treatments with A at low level Similarly B This is of course logical Note that the signs of treatments in the estimate of AB are the products of the signs of A and B Note that in each estimate plus and minus signs are equal in num er 1v 311 Memamcal Enqmwmq unwasm omenmm Also recall from the Pareto chart exercise that a change of scale by multiplying each yield by a constant multiplies each estimate by the constant but does not affect the relationship of estimates to each other Addition of a constant to each yield does not affect the estimates The numerical magnitude of estimates is not important here it is their relationship to each other 1v 311 Memamcal Enqmwmq unwasm omenmm Modern notation and Yates order Modern notation a0b01 a0b1b a1b0a a1b1ab We also recall Yates standard order of treatments and yields each letter in turn followed by all combinations of that letter and letters already introduced This will be the preferred order for the purpose of analysis of the yields It is not necessarily the order in which the experiment is conducted that will be discussed later For a twofactor twolevel factorial design Yates order is 1 a b ab U ing rgodern notation and Yates order the estimates of e ec s c e e om A 1abab2 B 1 a b ab2 AB 1 a b ab2 ME 311 Mechanical Engineering 0 University of Kentucky Analysis of MultiFactored Experiments Twolevel 3 Factor Factorial Designs ME 311 Mechanical Engineering University of Kentucky Three factors each at two levels Example The variable is the yield of a nitration process The yield forms the base material for certain dye stuffs and medicines m him A time of addition of nitric acid 2 hours 7 hours B stirring time 12 hour 4 hours C heel absent present Treatments or yields in a old notation b new notation a aoboco aobo 1 aobico aob1 1 a1bo o a1bo 1 a1b1c0 a1b1 31 b 1 c b be a ac ab abc Yates order 1 a b ab c ac bc abc ME 311 Mechanical Engineering 0 University of Kentucky Figure 55 Effects in The 23 Factorial Design A7A 7A BJ7E B CC fc etcetc cl Thremracmr imaramiun Geometric prescnlalion of contrasts corresponding m the main effects and ixircractinns 1 Ike 23 design UK ME 311 Mechanical Engineering University of Kentucky Estimating effects in threefactor twolevel designs 23 Estimate ofA 1 a 1 estimate of A with B low and C low 2 ab b estimate of A with B high and C low 3 ac c estimate of A with B low and C high 4 abc bc estimate of A with B high and C high aabacabc 1b cbc4 1ababcacbcabc4 in Yates order Note that nothing new is added here the algebra just becomes a little more complex M 311 Manama sngmamg w Univerle oYKenmd Estimate of AB Effect of A With B high effect of A with B low all at C high plus effect of A With B high effect of A with B low all at C low Note that interactions are averages Just as our estimate of A is an average of response to A over all B and all C so our estimate of AB is an average response to AB over all C AB 43 2 14 1ababcacbclabcl4 in Yates order or abciabc1 abacbcl4 M 311 Memam39cal Engmamg quot Universin oYKenmd Estimate of ABC interaction of A and B at C high minus interaction of A and B at C ow ABC 4 3 2 14 lababcacbcabc4 in Yates order or a bcabc 1abacbc4 ME 311 Manama Engineering quot Univerle oYKenmd This is our first encounter with a threefactor interaction It measures the impact on the yield of the nitration process of interaction AB when C heel goes from C absent to C present Or it measures the impact on yield of interaction AC when B stirring time goes from 12 hour to 4 hours Or finally it measures the impact on yield of interaction BC when A time 0 addition of nitric acid goes from 2 hours to 7 hours As with twofactor twolevel factorial designs the formation of estimates in threefactor twolevel factorial designs can be summarized in a table M 311 Manama Engineering quot Univerle oYKenmd Sign Table for a 23 design IV 311 Memamcal Enqmwmq umsz omenmm Example Yield of nitration process discussed earlier 1 b b be abc Y 72 34 20 30 67 92 34 37 A main effect of nitric acid time 125 B main effect of stirring time 485 AB interaction of A and B 060 C main effect of heel 060 AC interaction of A and C 015 BC interaction of B and C 045 ABC interaction of A B and C 050 NOTE ac largest yield AC smallest effect 11 311 Memamcal 511111112an 1K WWW We describe several of these estimates though on later analysis of this example taking into account the unreliability of estimates based on a small number eight of yields some estimates may turn out to be so small in magnitude as not to contradict the conjecture that the corresponding true effect is zero The largest estimate is 485 the estimate of B an increase in stirring time from 12 to 4 hours is associated with a decline in yield The interaction AB 06 an increase in stirring time from 12 to 4 hours reduces the effect of A whatever it is A 125 on yield IV 311 Memamcal Enqmwmq umsz omenmm Or equivalently an increase in nitric acid time from 2 to hours reduces makes more negative the already negative effect B 485 of stirring time on yield Finally ABC 05 Going from no heel to heel the negative interaction effect AB on yield becomes even more negative Or going from low to high stirring time the positive interaction effect AC is reduced Or going from low to high nitric acid time the ositive interaction effect BC is reduced A three descriptions of ABC have the same numerical value but the chemist would select one of them then say it better IV 311 Memamcal Enqmwmq umsz omenmm Number and kinds of effects We have already introduced the notation 2quot This means a factor design with each factor at two levels The number of treatments in an unreplicated 2k design is 2quot The following table shows the number of each kind of effect for each of the six two level designs shown across the top IV 311 Memamcal Enqmwmq Umvers m1 omenmm 7 factor interaction In a 2k design the number of r factor effects is C r krk r IVE 311 Memamcal Engmwmg quot Umversln ofKenmd Notice that the total number of effects estimated in any design is always one less than the number of treatments In a 22 design there are 224 treatments we estimate 22 1 3 effects In a 23 design there are 238 treatments we estimate 231 7 effects IV 311 Memamcal Enqmwmq Umversln omenmm en en 33 53 P ms oEPMoHp j ME 311 Experimentation ll Blocking amp Confounding Blocking occurs when you perform blocks of runs in different sets eg days Confounding is a result of blocking IV 311 mmanrcal Engineering Unrversin oYKerde Spring 2006 Lecture 8 1K 3 Blocking Blocking is a technique for dealing with controllable nuisance variables Two cases are considered Replicated designs Unreplicated designs The examples here will use 2k experiments but the principles apply to any DoE IV 311 Mammal Enqmwmq Univerle oYKerde Blocking Example In an unreplicated 2k there are 2k treatment combinations Consider 3 factors at 2 levels each ie 8 treatment combinations If each requires 2 hours to run 16 hours will be required Over such a long time period there could be say a change in personnel let s say we run 8 hours Monday and 8 hours Tuesday Hence 4 observations on each of two days or4 observations in each of 2 plants or4 observations in each of 2 potentially different plots of land or4 observations by 2 different technicians Replace one large block by 2 smaller blocks IV 311 Mammal Enqmwmq Univerle oYKerde Consider the TCs l a b ab c ac bc abc Now separate them into the following sample blocks How many different blocks of 2 are possible Recall that the block with the 1 observation everything at low level is called the Principal BlocK it has equal stature with other blocks but is useful to identify M T C T C T Assume all Monday yields are higher than 1 c 1 a 1 a Tuesday yields by a near constant but unknown amount X X is in units ofthe dependent or a ac ab b ab b b be c ac ac c response variable under study ab abc abc bc bc abc What is the consequences of having 2 smaller Which is preferable Why Does it matter mocks UK IV 311 IV 311 mica Enqmeermq m1 omenmm Estimate of Main Effects Consider the 3rd option 1 1xa babx Zl cacx bcxabc usual estimate X s cancel out l UsualABC lab abc ac bcabc Usual estimate 4 1 lgtlt ab abgtlt A 14 1a bab clac bcabc Z C ac X be X abo NOWBECOMES Usual estimate x quotK IVE 3 11 UK IVE 311l emamcaIEngmeerm Univerle omenmm Confounding We would find that we estimate A B AB C BC ABC X Switch Monday amp Tuesday and ABC X becomes ABC X 0 Replacement of one complete block by 2 smaller blocks requires the sacrifice confounding of at least one effect IV 311 Mammal Enqmeemrq w Univers m1 omenmm Some combinations are worse than others Con founded E eets a a c gt B C AB AC C 4 out of 7 effects confounded instead of 1 out of 7 Consider again the following 3 blockings Which effects are confounded for each arrangement M T M T M T 1 c 1 a 1 a a ac ab b ab b b bc c ac ac c ab abc abc bc bc abc Con founded E eets Only C Only AB Only ABC IV 311 mmanrcal Engineering quot Umvers m1 omenmm Recall X is nearly constant If X varies significantly with tc s it interacts with AIBIC etc and should be included as an additional factor UK IV 311 Mammal Enqmwmq Univerle omenmm IV 311 Mammal Enqmwmq quot Um versm omenmm Basic idea can be viewed as follows STUDY IMPORTANT FACTORS UNDER MORE HOMOGENEOUS CONDITIONS V th the influence of some ofthe heterogeneity in yields caused by unstudied factors confined to one effect generally the one were least interested in estimating and often one were willing to assume equals zero usually the highest order interaction We reduce the Experimental Error by creating 2 smaller blocks at the expense of confounding one effect UK 1v 311 Memamcal Engmwmg Univerle omenmmi All estimates not los can bejudged against less variability and hence we get narrower confidence intervals smaller B error for given cc error etc For large k in 2 confounding is popular Why I it is difficult to create large homogeneous blocks 2 loss of one effect is not thought to be important eg in 27 we give up I out of 127 effects perhaps ABCDEFG 1v 311 Memamcal Engmwmg Univerle omenmmi Partial Confounding 23 with 4 replications Confound Confound Confound Confound ABC B AC BC ab b ab b b ab 3 ab 1v 311 Memamcal Enqmwmq Univerle omenmmi Can estimate A B C from all 4 replications 32 units of reliability AB from Repl I 3 4 24 units of AC from 1 2 4 reliability BC from 1 2 3 ABC from 2 3 4 1v 311 Memamcal Enqmwmq Univerle omenmmi Example from Johnson and Leone Statistics and Experimental Design in Engineering and Physical Sciences 1976 Wiley Dependent Variable Variables Weight loss of ceramic ware A Firing Time B Firing Temperature C Formula ofingredients IV 311 Mmamcai Enqmwmq Univerle oYKerde Multiple Confounding Further blocking more than 2 blocks 2 16 tc s Example 1 2 3 4 1 a b c cd acd bcd d abd bd ad abcd abc bc ac ab R S T U IV 311 Mmamcai Enqmwmq Univerle oYKerde Only 2 weighing mechanisms are available each able to handle only 4 tc s The 23 is replicated tWIce Confound ABC Confound AB Machine 1 Machine 2 Machine 1 Machine 2 1 a 1 a ab b ab b ac c c ac bc abc abc bc A B C AC BC clean in both re lications AB from repl ABC from repl 1v 311 Memamcal Engineering quot Univerle oYKenmd Imagine that these blocks differ by constants in terms of the variable being measured a yields in the first block are too high or too low by R Similarly the other 3 blocks are too high or too low by amounts 8 T U respectively These letters play the role ofX in 2block confounding R S T U 0 by definition 1v 311 Memamcal Engineering quot Univerle oYKenmd Given the allocation of the 16 tc s to the smaller ACD D blocks shown above lengthy examination of all the Sign 0 bIDCk Sign 0 Wk 15 effects reveals that these unknown but constant treatment em treatment we and systematic block differences R S T 1 jr 3 j 1 confound estimates AB BCD and ACD of b T T estimates confounded at minimum l fewer than a j Z 3 of blocks but leave UNAFFECTED the 12 remaining ac T T b s S estimates in the 24 design ab fR R d U U ad T 1 This result is illustrated for ACD a confounded effect bd 3 3 and D a clean effect 11 Z 11 K acd 8 3 bcd T T abcd U U 31quot 5313395713 3 31quot Jf fy iiwm Summary ln estimating D block differences cancel ln HOW to lelde up the treatments to run estimating ACD block differences DO NOT cancel in smaller blocks should not be done the Rs S s T s and Us accumulate randomly Blocking involves sacri ces to be made losing one or more effects remember that some combinations are better than In fact we would estimate not ACD but ACD R2 82 T2 U2 others The ACD estimate is hopelessly confounded o In future we would like to examine how With block effecm to determine what effects are confounded IVE 311 Memamcal Eng qu IVE 311 Memamcal Enameumq Umversln oYKenmdry Umversln omenmm ME 311 Manual Experimentation Mechanical Engineering February 9 2006 REPORT WRITING All experiments conducted in ME 311 require a formal laboratory report The report should be written in such a way that anyone could duplicate the performed experiment and find the same results as the original experimenter The reports should be simple and clearly written Reports are due in week 4 of the experiment cycle 3 weeks after the experiment was started unless specified otherwise The report should communicate several ideas to the reader First the report should be neatly done The experimenter is in effect trying to convince the reader that the experiment was performed in a straightforward manner with great care and with full attention to detail A poorly written report might instead lead the reader to think that just as little care went into performing the experiment Second the report should be well organized The reader should be able to easily follow each step discussed in the text Third the report should contain accurate results This will require checking and rechecking the calculations until accuracy can be guaranteed Finally the report should be free of spelling and grammatical errors Long Report F ormut The following format is to be used for formal Laboratory Reports Title Page The title page should show the title of the experiment the date the experiment was performed experimenter s name and experimenter s partners39 names Use the title page on the ME 311 site Table of Contents Each page of the report must be numbered for this section Objective The objective is a clear concise statement explaining the purpose of the experiment This is one of the most important parts of the laboratory report because everything included in the report must somehow relate to the stated object The object can be as short as one sentence and it is usually written in the past tense Theory The theory section should contain a complete analytical development of any important equations pertinent to the experiment and how these equations are used in the reduction of data The theory section should be written textbook style Experimental Arrangement This section should contain a description of the setup with a schematic drawing of the experimental setup including all equipment used in a parts list with manufacturer information if any Show the function of each part when necessary for clarity Outline exactly how the experiment was run Results Includes a description the parameters examined and test matrix The results section should contain a formal analysis of the data with tables graphs etc Any presentation of data that serves the purpose of clearly showing the outcome of the experiment is sufficient Discussion and Conclusion This section should give an interpretation of the results explaining how the object of the experiment was accomplished If any analytical expression is to be verified calculate error and account for the sources Discuss this experiment with respect to its faults as well the as its strong points Suggest extensions of the experiment and improvements Also recommend any changes necessary to better accomplish the object Each experiment write up contains a number of questions These are to be answered or discussed in this section W 7 Exhlmyaphy 113mg 3 nfenmes a Make sun mm mm fw aws standald gmdzlmzs and s camsmm Appenmx u cum law he sham anable a Is dhya mp1 haw haw dz we mum a Cahhnnan cums at msmlmems um we used m h pe vrmance at m expelmum mm manufacmlu afthz mmmm mndzl and serial numhels m mmmx wm samzhmes supply cahhmmn curves a a cs at a a 0mm Thzm39y and Pmcednre we and cm um Mm Ths farm mm m 21 WM a mum thzary secnan wmlld m ma lung m shmud 31m mm m nwamasmcsmammmmmm m mmmwmpmmwm mm omevwnz mud anlu m many Instances n s maessary a campus a plm In mm m yaphlca y presem h mus Glaphs mm m dawn many fullnwmg a speci c mm Flgnle 1 Shaw an acceptahlz gaph pnpared mug a campm Th2 an many campmex pmyams um have yame capsulsz mclmmg Excel and mum Nevenhzlzss an acceptably dawn yaph has sevenl mum uf mm amanexamplz xsgvenhelaw Ohm s law Memrzmenu thuw 2 v z g quotD3 hezrv a um 39 Vnnmvmzm mu Flgnle 1 Samplz gule mm mm mammzms an Shawn by symhals mm a 1m mm m Hummus pnmcnan Smwla nn mm ale alsa planed wnhhms ME 311 Report Grade Name Report Team Section Report Grade Title page 5 Table of Contents 5 Objective 10 Theory 10 Procedure 15 Results 15 Discussion and Conclusions 15 References 5 Figures 10 Presentation including formatting clarity and grammar 5 Individual Team Appendices including raw data sample calculations and calibration curves 5 Total Comments Safety Rules When do I have access to the labs 39 Only during appointed lab times or by arrangement and when a lab assistant staff or faculty are present How should I conduct myself when in the lab 39 When in the presence of powered test machinery or when working with hazardous heavy or hot materials 39 Do not work alone in the lab 39 Do not wear bulky loose or trailing clothes 39 Use appropriate protective equipment gloves safety glasses aprons and face shields Never wear contacts when working with volatile materials 39 Always wear shoes that completely cover your feet and pants that completely cover your legs 39 Tie back long hair Remove metal bracelets or watch straps when working with electricity 39 Never lift objects that are heavier than you can safely handle Always use appropriate mechanical means for lifting carrying positioning or adjusting heavy equipment or supplies 39 Place all belongings out of the work area 39 Do not obstruct doorways 39 Do not play in the lab 39 Never eat drink or smoke in a lab Never taste or smell chemicals 39 Always wash your hands before leaving the lab 39 Know the location of and how to use the safety equipment 39 Know the hazards of the chemicals you are using Use the Material Safety Data Sheets MSDS provided in the lab or at httpehssiueedu 39 Dry wet hands and clothing before working with electricity Mop up all water spilled on the oor 39 Keep the lab clean 39 Treat all equipment with care 39 Be as careful for the safety of others as for yourself Think before you act General Precautions 39 Before equipment is made live the guards for all accessible moving mechanical components must be in place Beware of the effect of fluorescent lights which may cause rotating equipment to appear stationary 39 Always use goggle with the appropriate wavelength protection when operating lasers Remove shiny jewelry and contacts 39 Report faulty equipment to the teaching assistant and the professor immediately Do not use it until it is inspected and declared safe 39 Pay attention to the chemicals you are using Read the labels Use the fume hood if necessary especially with volatile chemicals and strong acids and bases Do not contaminate reagents Do not place items in the reagent bottles pour out an appropriate amount Do not place a stopper on the bench Do not return unused chemicals to the reagent bottle Always pour acid into water AW Never use an open ame when ammable chemicals are present To insert a glass tube or thermometer into a rubber stopper wet the tube with water wrap the tube in a cloth and push it gently into the stopper with a turning motion Place all broken glass in the glass container Place nonhazardous solid waste in waste baskets Place all hazardous waste in a properly labeled container Stop machinery or equipment capable of movement before cleaning or adjusting Also de energize or disengage the power source and if necessary mechanically block or lock out the moveable parts Use extreme caution when working with electricity under wet conditions Avoid if at all possible Make sure all electrical circuits components and equipment are properly grounded BEFORE using them Do not overload circuits Don t defeat or bypass the overload protection of a circuit If a circuit breaker or fuse trips or blows don t just reset or replace find the reason and correct the problem
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