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# Instrumentation, Measurements, and Statistics M E 345

Penn State

GPA 3.79

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## Popular in Mechanical Engineering

This 0 page Class Notes was uploaded by Chester Goldner III on Sunday November 1, 2015. The Class Notes belongs to M E 345 at Pennsylvania State University taught by John Cimbala in Fall. Since its upload, it has received 37 views. For similar materials see /class/233070/m-e-345-pennsylvania-state-university in Mechanical Engineering at Pennsylvania State University.

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Date Created: 11/01/15

M E 345 Fall 2009 Professor John M Cimbala Today we will Do a review example problem t PDF and chisquared PDF Review the pdf module Correlation and Trends Do some example problems linear correlation Review the pdf module Regression Analysis Do some example problems regression analysis Example Estimating population mean and population standard deviation Given 20 ball bearings are pulled from the assembly line and their diameters are measured The sample mean is 256 mm and the sample standard deviation is 0240 mm a To do Estimate the population mean and its con dence interval for 98 confidence level I b To do Estimate the population standard deviation and its con dence interval for 98 confidence level Solution M A39 to AFHS 9X7 4 1m 39 ov 97 om 5 w 0m 0 J 4 mm or Eurl quot T39NV ZI 3 A ill L s33f S39 a X Jcm a 154 2 one Mm U 4919 0 om J quot1 035 337 quotThlo of39 End 2 CHINV cams 34mg Hnw 099 mm I M S1 61w mm m Ont lt lt 0373 6quot 441 362 WP c A 1 k6 T 973 0319 Mm vall Example Linear correlation coef cient Given Matt measures both the shoe size and the weight of 18 football players He performs a linear regression analysis of shoe size 1 variable as a function of weight x variable He calculates rle 0582 To do To what confidence level can Matt state that a football player s shoe size is correlated with his weight Solution anquot hug e v Co ldsl 2 cwquot w 32314 mm MA 4 3997 M m 39r A Mi Example Linear correlation coef cient Given Several measurements are taken in a wind tunnel of pressure difference as a function of distance normal to the direction of flow over a body La iw a To do Calculate the linear correlation coefficient b To do To what confidence level can we state that there is a trend in the data Solution Q 8 C FE 90V 04 an 321 7 Wk 4 l 5 0659 MM Mh l l llul nun mmwr U a m 37 o W 7 umu my my Example Regression Analysis Given The same pressure vs distance measurements of the previous problem To do Perform a linear regression analysis 7 plot the bestfit straight line and compare the fitted curve to the data p01nts Solution See Excel spreadsheet 71 will show in class how to do the regression analysis in Excel ME345 Fall 2009 Professor John M Cimbala Today we will Do a review example problem outliers data pairs Review the pdf module Experimental Uncertainty Analysis 3 Do some example problems experimental uncertainty analysis Example Outliers data pair measurements Given Omar takes 12 data pair measurements temperature as a function of pressure He performs a regression analysis and plots the data and the leastsquares fit One of the data pairs looks a little suspect so he performs the standard statistical technique to determine if this data pair is an official outlier o The standard error from the regression analysis is 002440C Sly o The residual of the suspect data pair is 0046800 0 A plot f standardized residual vs pressure reveals that the standardized residual of the suspect at pair is Wt with its neighbors To do Ist ata pair an official outlier or not Solution HM v1 NAM C 39 Y 339 M Ml Jah a 5 a 5 Omsa c W LSlx S39s 0 O Lt11 Tu o um Cr n am emu B ecSxgtl I Inca144 H 7 quotcylnln f N0 4 k U A3 in V n39f MUJ r MW n l39tnquot km wt MWquot 00 W 0 1 4 Example Experimental uncertainty analysis Given The cutoff frequency of 39r torder filter is a i where a radian frequency radianss I R resistance measured to be R 1200 Q C capacitance measured to be C 0100 pf i To do to 95 confidence to 95 confidence Predict a to 95 confidence in standa 2 engineering format and with the proper number of signi cant digits S l t39 Gum 2 333 37 n w 00 quot39 39 RC no ow MG 5 J Unauthmlm 3 R C quot Ir Th thr Kra l w z w 1 L 3 1 a quot396 Km 2 w R R C Ln Wu 7 w h L39 oco 0 1 u w Q unfh QX0m S a E wu 0 a Emu4 ww 1 a 00 comammaz r WNW I 6AA SAJVr X 6013320 Mg 7 Example Experimental uncertainty analysis Example Given Quantities A and B are measured 200 times each The sample mean and sample standard deviation forA are E 520 and S A 0252 The sample mean and sample standard deviation for B are E 2232 and SE 2105 Quantity C is not measured directly but it is known that C A2 3B To do Report C in standard engineering format Solution Ru A A 1 392 10 t 2om 2 1110 IwA gt B th hint a 1kng m our W BN4 C C unbornwhy L 21A 9quot 3 AA AG 15 1 98 1A Sou J Do 3 76 c 8J9 Alkj 9 13113 i A e1 5W mm c M 3339 f f wt 3 E 1 a QA Vii av5855 2 632 1 3933 811339 7131 Vt EC a 9 Glimb u 2 10 quotBatman mum w w v5u C 390 395 71074 J Example Experimental uncertainty analysis Given Jan uses a thermocouple to measure temperature She performs a regression analysis on temperature T units 0C as a function of thermocouple voltage V units mV The bestfit straight line is T b aV where b 08152 C and a 231820CmV She takes her voltage data with a digital data acquisition system that has an uncertainty of 0124 mV To do When the voltage reading is V 5520 mV calculate the temperature with its appropriate uncertainties Solution WV T Li V NM uF WE f mPL l PWquot 4 h L yin 390 Km c um m d Am m1 mimic NT Nva onq mV3k1393l11 A Ev 1875 W V I hut Ml hmfwu quot hm mu huh39q Example Experimental uncertainty analysis Given In a uid mechanics experiment the change in pressure is AP 106 pV2 where AP change in pressure Nmz which is the same units as pascals Pa p density measured to be p 660 i 150 kgm3 to 95 confidence V velocity measured to be V 152 i 0028 ms to 95 confidence To do Write AP at these values of p and Vin standard engineering format Solution MN m our our on m ow Fm WWJE M g Pm M 51km Rollwm ow to yum PM AP My 2 19 E ME345 Fall 2009 Professor John M Cimbala Today we will Finish reviewing the pdf module Regression Analysis Do some example problems regression analysis Review the pdf module Outlier Points Do some example problems outlier points Example Regression Analysis Given Twenty data points of xy pairs with lots of scatter see Excel spreadsheet on website for the raw data 1 2 4 To do Perform regression analysis linear quadratic and cubic and compare how the tted curves t to the data points Solution See Excel spreadsheet I will show in class how to do the regression analysis in Excel b m P b h mu or T Mul h l K 1 Qumh mm w 5 L I Ml M 11 m39nvo gag 5 M S 7 w W NY 391 16quot N63 Example Outliers single set of data Given Four data points are measured sample mean 4425 7 7 sample standard deviation 63443 1 I To do Eliminate any official outliers one at atime Solution QA3I1 c 1 mm um 39 75 mech 34H lt C mi V4 quot J 3933 W 7 LU in i Q Nitfl gr ll 391 Lwlul I U 4qu f a U A Nam othr am m Example Outliers single set of measurements Given Janet takes l2 temperature measurements ranging from 230 C lowest reading togijoC highest reading 0 The sample mean of all 12 readings is 24880C o The sample standard deviation is LEE To do Determine which if any of the extreme readings lowest or highest is an outlier Solution I L 27 L613 39Z39J vhfm c 39 1 Q M mag J39 v3 11 4 V3 W quotWquot n M ft Q MM 8 Mt39c LE X m 4 quotc m Example Outliers single set of measurements Given Eleven measurements of pump efficiency are no hm I taken listed in increasing order below A N 5 8 72 74 74 76 77 80 80 82 85 and 22 3quot Wquot 39Fquot Rwy a To it Are there any official outliers Ifso remove them How many good measurements are left b To do Based on the measurements that are left estimate the population mean and its con dence interval to 95 con dence M 71quot i 23913 3993 1 1 D3 6 your Wu C To do Based on e m ft estimate 5 rule the population standar 39 39 u 39 nce interval to 95 con dence Zs 65 an X Solution 39 a I Jquot Irma X 1711 I 8510 4 Mai S v Q SI PMth 7 DIV1 S gtTJ TAquot AVUI 6 Law TS WHY R Pt l I HW MAIIW j 0 bl 7 i 7quot M J r 4quot m r PW30 8 high Lml CT l Slquot T39s N Bsb M rm J7 T l u39 Am An can quotIIMM 339 NW 19 2 No Make wmw Example Outliers data pairs Given Twenty data points of xy pairs are measured 37 42 66 To do Eliminate any official outliers one at a time Solution See Excel spreadsheet I will show in class how to do the analysis in Excel ReM q Wm on Mad WWW 3 No Mm Wcm 0M at 6 if my TCLIMJ 3 em my x almm h mt if MM pM39n ME345 Fall 2009 Professor John M Cimbala M E 345 Instrumentation Measurements and Statistics Today we will Introduce the course and instructor John M Cimbala 8632739 jmc6psuedu Brie y go over the course website at wwwmnepsuedume345 Review first pdf module Introduction to Mechanical Engineering Measurements Do some practice questions and example problems If time show some hints about plotting in Excel if time Practice Questions 1 How many significant digits are in each of these numbers Number Number of notation 603 600 7 7 50 001070 2 What is 20030 1 n m K 200 hquot ECLCLCC 3 3 30 3 3 v r ovr 54 0quotva 5 59 n w 1 3 km Example Signi cant digits Given 3 measurements with 3 different instruments i 134220 5 significant digits ii 02875 4 significant digits iii 29473 5 significant digits a To do Round each number to 3 significant digits or 3391 0 Emu A I Mt l 5 Solution b To do Add the 3 numbers and report the answer to the appropriate number of significant digits i 134220 5 significant digits 2 ii 02875 4 significant digits i L J 239 iii 29473 5 significant digits Solution 4 3ULI 39339S Lof Foul h h 103 umn 39 MI quot L h m o w vw MA 3 13730 c To do Multiply the first two numbers and report the answer to the appropriate number of significant digits i 134220 5 significant digits ii 02875 4 significant digits iii 29473 5 significant digits Solution 611 303315 m lgo x g MW hm S 8 L Ll It h 5 3quot INO Te I KLL a In th ampr Jub1 vmi OAwlu anJ ls Oohb uhsw lquot error M E 345 Fall 2009 Professor John M Cimbala Today we will Do a review example problem basic statistics Review the pdf module Histograms Do some example problems histograms Review the pdf module Probability Density Functions PDFs Do some example problems PDFs Example Basic statistics The sample mean is 44580 ms 4454 44x3 4462 W411 NW 445 3944 4 4456 4461 11 o LLlSK 33 I 14451 Pquot 445 4 4459 H or 4463 41 4458 43 digits a The systematic or bias error of the instrument based on these readings b The sample median c The sample mode V3 F quotAquot quot W VAquot Solution HWSIQ 63 1 ONX VI1 Vick H45quot 01 u M N bunch V Wonk d Rr Kw mmgt See also Excel spreadsheet on the website for this same problem Given The true exact speed in a wind tunnel is 4463 ms Ten velocity readings are taken 39 To do Calculate the following giving your answers to the appropriate number of significant Example Histograms Given The histogram shown here not normalized produced in Excel from 50 voltage measurements a How many data points have a voltage less than or equal to 6 V b Howmany I 39 voltage that lies between 6 V and 8 c How many data points have a voltage that lies between 3 V V d What is the probability 1 t that a given reading lies between 5 V 6 V 2 When we transform the vertical axis 7 fro ncy n mber of data points to x vertically normalized histogram what is the 7 m freque u value of x for the bin labeled 6 on the horizontal axis L r is6 L r 4 4 39 quot is13 Whenwetransform the vertically normalized histogramx into a PDF and then in o a normalized PDF what is the value of z for x 85 Solution Pm Eh in 01 s 070 3 PM 395 Ebr u may b hm ax g J 10 ex M H v EL To o1o a a CFb Q3nna 01m z 11 Ys m c 5 m ME345 Fall 2009 Professor John M Cimbala Today we will Do some review example problems The Gaussian PDF Review the pdf module The Central Limit Theorem CLT Do some example problems CLT Review the first half of the pdf module Other PDFs the Student s t PDF quot39 Om ouT o 39hnit Example Review and probability Given The temperature of an ice bath is measured numerous times with a digital thermometer The true temperature of the ice bath is 0000000 The sample mean temperature is T 00125 C The sample standard deviation of all the readings is 003410C i S We assume that the errors in the readings are purely random 2 a To do Write T in standard engineering format T 00125 i 0H 0C b To do Calculate the bias error also called systematic error c To do Calculate the probability that any random reading is greater than 0 C Solution 0 8w t c quot 1 moral 39 1 Z Xl l A XLEX TLir 1 0 071 0393 S39 0 ll AGFDM quot ow fab O39S Am WTgt 03963 39 S39om a 53m 3W4 Example Probability power requirement measurements Given Bev takes 2l measurements of the power requirement for an electronic instrument running in a steadystate mode We assume that the errors are purely random The sample mean is 3592 W and the sample standard deviation is 060 W To do a Considering the proper number of significant digits show how Bev should write the power in standard engineering format 95 confidence level ie P3592 i w 450 b Calculate the percentage of the readings that are expected to be less than 3592 W c Calculate the percentage of the readings that are expected to be greater than 3712 W J Estimate the number of readings that are expected to be greater than 3712 W Solution gt4quot 31114 31 Lu 2 T 45 2 a 060 Kt A W What a 1quot AW le 5940 3 Q 39 1 NM 2 04111 A Aw Uquot 00quot 0917 03773r Ema Q La 0mm my 2 le o 35 M11512 d 2 our Clea g N ehl mm Milbl N mquot 5 W on M 0oe3auk um gum v 1 ml luaug Squot A39 Q homr a fu J39mu M l but fake or 60cu5 nrf Ramses mm 357 mama was Q Example Estimating population standard deviation Given A company produces resistors by the thousands and Mark is in charge of quality control 0 He picks A resistors at random as sample 1 and calculates the mean f1 0 He picks 20 other resistors as 132 and calculates the meanizT 0 Mark continues to do this until sample2 and calculates the mean 25 The average of all the means is 1 2 2525 8235 kQ The standard deviation of all the means is 0397 m S7 0282 kQ To do Estimate the W Solution Uh H n 1 0 N 75 at thk 51 4ft 3 3 ovz 2 E Cc SIS 6 0131 H W AC Lu m M Won t r pquh nu39 639 Example Estimating population standard deviation Given Ron takes 50 pressure measurements and repeats this 19 more times for a total of 20 samples of 50 data points each He calculates the sample mean for each set sample of 50 measurements The standard deviation of the 20 sample means is 0150 kPalt 5 I To do Estimate the population standard deviation of all the measurements in units of kPa to 3 Slgmflcant d1g1ts Solution U Clxr Guam Lvlwil ltersquot C C S 01m lopb y T So cf mflz N11 4 Jun 3925 my M E 345 Fall 2009 Professor John M Cimbala Today we will Do a review example problem t PDF and chisquared PDF Review the pdf module Correlation and Trends Do some example problems linear correlation Review the pdf module Regression Analysis Do some example problems regression analysis Example Estimating population mean and population standard deviation Given 20 ball bearings are pulled from the assembly line and their diameters are measured The sample mean is 256 mm and the sample standard deviation is 0240 mm a To do Estimate the population mean and its con dence interval for 98 confidence level I b To do Estimate the population standard deviation and its con dence interval for 98 confidence level Solution M A39 to AFHS 9X7 4 1m 39 ov 97 om 5 w 0m 0 J 4 mm or Eurl quot T39NV ZI 3 A ill L s33f S39 a X Jcm a 154 2 one Mm U 4919 0 om J quot1 035 337 quotThlo of39 End 2 CHINV cams 34mg Hnw 099 mm I M S1 61w mm m Ont lt lt 0373 6quot 441 362 WP c A 1 k6 T 973 0319 Mm vall Example Linear correlation coef cient Given Matt measures both the shoe size and the weight of 18 football players He performs a linear regression analysis of shoe size 1 variable as a function of weight x variable He calculates rle 0582 To do To what confidence level can Matt state that a football player s shoe size is correlated with his weight Solution anquot hug e v Co ldsl 2 cwquot w 32314 mm MA 4 3997 M m 39r A Mi Example Linear correlation coef cient Given Several measurements are taken in a wind tunnel of pressure difference as a function of distance normal to the direction of flow over a body La iw a To do Calculate the linear correlation coefficient b To do To what confidence level can we state that there is a trend in the data Solution Q 8 C FE 90V 04 an 321 7 Wk 4 l 5 0659 MM Mh l l llul nun mmwr U a m 37 o W 7 umu my my Example Regression Analysis Given The same pressure vs distance measurements of the previous problem To do Perform a linear regression analysis 7 plot the bestfit straight line and compare the fitted curve to the data p01nts Solution See Excel spreadsheet 71 will show in class how to do the regression analysis in Excel M E 345 Fall 2009 Professor John M Cimbala Today we will Do a review example problem t PDF and chisquared PDF Review the pdf module Correlation and Trends Do some example problems linear correlation Review the pdf module Regression Analysis Do some example problems regression analysis Example Estimating population mean and population standard deviation Given 20 ball bearings are pulled from the assembly line and their diameters are measured The sample mean is 256 mm and the sample standard deviation is 0240 mm a To do Estimate the population mean and its con dence interval for 98 confidence level I b To do Estimate the population standard deviation and its con dence interval for 98 confidence level Solution M A39 to AFHS 9X7 4 1m 39 ov 97 001 5 w 0m 0 J J Tm or Eurl quot T39NV ZI 3 A ill L s33f S39 a X Jcm a 154 2 one Mm U 4919 A om J 39 035 337 quotThlo of39 39Ewa 2 CHINV cams 34mg Hnw 099 mm I M S1 61w mm m Ont lt lt 0373 6quot 441 362 WP Ck T E 1 9119 Mm X Irnl l Example Linear correlation coef cient Given Matt measures both the shoe size and the weight of 18 football players He performs a linear regression analysis of shoe size 1 variable as a function of weight x variable He calculates rle 0582 To do To what confidence level can Matt state that a football player s shoe size is correlated with his weight Solution anquot hug e v Co ldsl 2 cwquot w 32314 mm MA 4 3997 M m 39r A Mi Example Linear correlation coef cient Given Several measurements are taken in a wind tunnel of pressure difference as a function of distance normal to the direction of flow over a body La iw a To do Calculate the linear correlation coefficient b To do To what confidence level can we state that there is a trend in the data Solution Q 8 C FE 90V 067J ts e an 321 7 Wk 4 l 5 0659 MM Mh l l llul nun fm r U 4 W 37 o W 7 umu my by Example Regression Analysis Given The same pressure vs distance measurements of the previous problem To do Perform a linear regression analysis 7 plot the bestfit straight line and compare the fitted curve to the data points Solution See Excel spreadsheet 7 I will show in class how to do the regression analysis in Excel

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