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# Instrumentation, Measurements, and Statistics M E 345

Penn State

GPA 3.79

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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 16 views. For similar materials see /class/233070/m-e-345-pennsylvania-state-university in Mechanical Engineering at Pennsylvania State University.

## Reviews for Instrumentation, Measurements, and Statistics

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Date Created: 11/01/15

M E 345 Fall 2009 Professor John M Cimbala Today we will Do some review example problems predicting the mean using the t PDF Finish reviewing the pdf module Other PDFs the Chisquared PDF Do some example problems chisquared PDF mm M TIME Example Estimating the population mean Given A company produces resistors by the thousands and Gerry is in charge of quality control He picks 20 resistors at random as a sample and calculates the sample mean f 8240 kQ and sample standard deviation S 0314 kQ To do 39 10 a Estimate the population mean and the con dence interval of the population mean as a i value to standard 95 confidence level b Repeat for 99 confidence level Do you expect the con dence interval to be wider or narrower Solution 1 7 IS 9 M m ka 5L UL ohmy h yh hilv hobs S39 Q mu AF vm l3 Q 332 MA lWIl lo LI k0 m 10530 ll Tluv a 3f vaeoy 19 quot an M Example Estimating the population mean Five measurements are made of the force required to break an expensive Given component too costly to do more 3122 3206 3155 3141 and 3196 lbf The sample mean and sample standard deviation of these data are 3164 and 3592 lbf a To do level b To do interval to i20 lbf Solution W l39 5 M X klh 5 HL M Estimate the population mean and its con dence interval for 95 confidence Estimate how many samples would need to be taken to reduce the confidence an my M L e 3J2 W 1c 177 I 6h 33ltJvlt3zoa Iw G 317 W lmu Lm W l 111 v W39r39 6quot 6 m0 Jdvg e 1 t 31M 1 10 M J4 SM 4 n 13w 1 ink or no TIW Mama we ALIA 9 1quot 9qu m r 4 H QL gamma To int KS AF nI m 3 ern Lyo391 L33 HA1 E 41 5 1 1mm 1an r 5W 0 Dv PPnuml39b39ll I 7 7 N 8 1 Example Estimating population mean and standard deviation Given A quality control engineer pulls 10 resistors at random from an assembly line that makes 10kQ resistors and measures each esistance The measurements are given here n Measurement number 1 2 3 4 5 6 7 8 9 10 a To do level 1010 1008 1011 1009 1007 1005 1012 1011 1008 1006 Resistance kQ 3913 39 sample mean 10087 kQ sample standard deviation 002312 kQ Estimate the population mean and its con dence interval for 95 confidence b To do Estimate the population standard deviation and its con dence interval for 95 f con dence level c To do The company guidelines specify that the population standard deviation be less than 035 of the mean to 98 con dence Estimate how many additional resistors the quality control engineer needs to pull off the assembly line to measure Solution 4 th c w I 99 4 U 76quot HF N 140 329 Q 0mg 01quot I ow CAM Gr 6 r 5mm 71sz 00T t 0011 Ht 09 00 093 00 Ly 3 70 00m W1 1 037 quot1 m M 4111 V144 Mum 639 6quot Ll h 7 l out 7 g 111i 0013 1w 7Lquot 3901 lnj quot Sb 00m XLPILI 3 anquot bf 80153 lt Clt 00 111 MI lwvi mum A 6 J C 5 00m Ht La W WM 6 9 390 I mquot W0 03S A F a Mum M qok 7 60981 kn 003311 Plyfquot db 30 L us mm 1 e 3839 4953quot Vk C I agr n M puu d J V u 3 XLT 001 IL W6 hsw PMSMwm quotch mmmb NW9 m u om M New mu 5 v Amy as m 4 mm M but In our awn M E 345 Fall 2009 Professor John M Cimbala Today we will Do some review example problems hypothesis testing one sample Review the pdf module Two Samples Hypothesis Testing Do some example problems two samples hypothesis testing Example Hypothesis testing Given A manufacturer claims that a plastic part is at least 600 cm long You test the claim by performing a hypothesis test You pick 30 parts at random from the assembly line and carefully measure the length of each one You calculate y 6053 cm and S 0104 cm To do To what confidence level can we claim that the manufacturer s claim is true Solution Example Hypothesis testing Given A manufacturer claims that a plastic part is at least 600 cm long You test the claim by performing a hypothesis test You pick 30 parts at random from the assembly line and carefully measure the length of each one You calculate y 5983 cm and S 00670 cm To do Should we accept or reject the manufacturer s claim Also provide the confidence level for your decision Solution Example Hypothesis testing Given We buy a gadget that is supposed to increase the gas mileage of our car We take 6 trips without the gadget and 6 nearly identical trips with the gadget The results Without with 256 262 273 271 242 241 287 292 236 245 251 249 To do Determine if there is a statistically significant improvement increase in gas mileage Solution Example Hypothesis testing Given Continuation of previous example We buy a gadget that is supposed to increase the gas mileage of our car We take 6 trips without the gadget and 8 trips with the gadget We do not attempt to pair up the tests The results xA mpg Without gadget x3 mpg with gadget 256 262 273 271 242 241 287 292 236 245 251 249 265 258 To do Determine if there is a statistically significant improvement increase in gas mileage Solution M E 345 Fall 2009 Professor John M Cimbala Today we will Do some review example problems predicting the mean using the t PDF Finish reviewing the pdf module Other PDFs the Chisquared PDF Do some example problems chisquared PDF mm M TIME Example Estimating the population mean Given A company produces resistors by the thousands and Gerry is in charge of quality control He picks 20 resistors at random as a sample and calculates the sample mean f 8240 kQ and sample standard deviation S 0314 kQ To do 39 10 a Estimate the population mean and the con dence interval of the population mean as a i value to standard 95 confidence level b Repeat for 99 confidence level Do you expect the con dence interval to be wider or narrower Solution 1 7 IS 9 M m ka 5L UL ohmy h yh hilv hobs S39 Q mu AF vm l3 Q 332 MA lWIl lo LI k0 m 10530 ll Tluv a 3f vaeoy 19 quot an M Example Estimating the population mean Five measurements are made of the force required to break an expensive Given component too costly to do more 3122 3206 3155 3141 and 3196 lbf The sample mean and sample standard deviation of these data are 3164 and 3592 lbf a To do level b To do interval to i20 lbf Solution W l39 5 M X klh 5 HL M Estimate the population mean and its con dence interval for 95 confidence Estimate how many samples would need to be taken to reduce the confidence an my M L e 3J2 W 1c 177 I 6h 33ltJvlt3zoa Iw G 317 W lmu Lm W l 111 v W39r39 6quot 6 m0 Jdvg e 1 t 31M 1 10 M J4 SM 4 n 13w 1 ink or no TIW Mama we ALIA 9 1quot 9qu m r 4 H QL gamma To int KS AF nI m 3 ern Lyo391 L33 HA1 E 41 5 1 1mm 1an r 5W 0 Dv PPnuml39b39ll I 7 7 N 8 1 Example Estimating population mean and standard deviation Given A quality control engineer pulls 10 resistors at random from an assembly line that makes 10kQ resistors and measures each esistance The measurements are given here n Measurement number 1 2 3 4 5 6 7 8 9 10 a To do level 1010 1008 1011 1009 1007 1005 1012 1011 1008 1006 Resistance kQ 3913 39 sample mean 10087 kQ sample standard deviation 002312 kQ Estimate the population mean and its con dence interval for 95 confidence b To do Estimate the population standard deviation and its con dence interval for 95 f con dence level c To do The company guidelines specify that the population standard deviation be less than 035 of the mean to 98 con dence Estimate how many additional resistors the quality control engineer needs to pull off the assembly line to measure Solution 4 th c w I 99 4 U 76quot HF N 140 329 Q 0mg 01quot I ow CAM Gr 6 r 5mm 71sz 00T t 0011 Ht 09 00 093 00 Ly 3 70 00m W1 1 037 quot1 m M 4111 V144 Mum 639 6quot Ll h 7 l out 7 g 111i 0013 1w 7Lquot 3901 lnj quot Sb 00m XLPILI 3 anquot bf 80153 lt Clt 00 111 MI lwvi mum A 6 J C 5 00m Ht La W WM 6 9 390 I mquot W0 03S A F a Mum M qok 7 60981 kn 003311 Plyfquot db 30 L us mm 1 e 3839 4953quot Vk C I agr n M puu d J V u 3 XLT 001 IL W6 hsw PMSMwm quotch mmmb NW9 m u om M New mu 5 v Amy as m 4 mm M but In our awn 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 another example problem dimensional analysis Review the pdf module Errors and Calibration Do some example problems errors and calibration Review the pdf module Basic Statistics Do some example problems basic statistics Example Dimensional analysis pipe ow Given Consider fully developed laminar ow through a V D gt 11 very long round tube Volume ow rate V is a function of the tube s inner diameter D uid viscosity u and axial pressure gradient dPdx Pressure gradient dPdx To do If D is doubled holding u and dPdx fixed by what factor does V change Solution Use dimensional analysis to generate a nondimensional functional relationship v a Us A W lil lil Step2 kgES iLS t ltn 39 m Ml 6 J Pu 33 mm mm quotMLquot quot quot 39 q 3zx Step 3 Step 4 Pu 3 vm u l 39 l d 5 I WAX Step5 K b trig W39s V43 M or 2653 o C hf v M M 1 MM 3 M m 1 o gm bu t o 44 u 39 quotl Step 6 V AN I b I Answer If D is doubled holding u and dPdx fixed V goes up by a factor of 1 or Example Errors and calibration Given The actual true voltage is 46020 V 26 voltage readings are taken and the average voltage reading is 46015 V To do a Calculate the systematic bias error and the mean bias error for this set of measurements b Calculate the random precision error of a reading that is 46010 V Solution 8 UM Avquot 4A A lieu quotHug 1 t 0000 V 39 0 0 a Q f WW VAN g 391 annoy b a L MAW w 46019 1 on Example Errors and calibration Given The actual true temperature is 22100 C Six thermometer readings are taken 2215 2222 2209 2221 2218 and 2224 C To do Calculate the mean systematic error and mean bias error for this set of data and calculate the accuracy error inaccuracy and the precision error for each measurement T C Inaccuracy T Tm 0C Precision error Ti 0C 22157 11x 21100 1 on TLJE 39LZ81 new 1 o03 2222 on am 2209 E m 430 oo9 2221 em 003 2218 008 m 2224 I o m o06 Solution MM my 11I a 3921 33 r u 39 5 m N l39Nt V c uJ fvl W MGEJW quotquot cm H id 39I7f MSE V 0 in yaw 0 39 51 Am See also Excel spreadsheet on the website for this same problem Example Basic statistics Given Ten houses are sold in the State College area during a particular time period The selling prices are listed in increasing order rounded to the nearest 500 dollars 1 h l J lh fu h d Sunfwdc JV 1quot INC 51m 4v VF 39nu L 2 3 4 5 6 7 8 9 A O To do Calculate the mean mode and median and discuss Solution M 1 5 Hoo A 3 quot Unk n n NW Mun i Mi ox an Amw quotm C A Amid L ujc pm My L ov f 4 m If Imllv u in Metal man u LtHu39 Example Basic statistics Given The true exact temperature of a hot water tank is 8968 C Ten temperature readings are taken with a digital thermometer The sample mean is 8937 891 894 892 896 893 894 mm ms m w M W 895 6 LA 0 L 897 F5 7 894 891 To do Calculate the following giVing your answers to the appropriate number of significant digits a The systematic error of the instrument based on these readings b The sample median S c The sample mode a w mr Solution L m na L0 Ma 5 ME345 Fall 2009 Professor John M Cimbala Today we will Do a review example problem PDFs Review the pdf module The Gaussian 0139 Normal Probability Density Function Do some example problems Gaussian PDFs and probability estimates Example PDFs Given A sample consists of 1000 length measurements The sample mean is 5365 cm and the sample standard deviation is 125 cm 136 measurements lie between 550 lt x S 570 cm To do a Estimate the probability that a length measurement lies between 550 and 570 cm b Calculate the transformed variables 21 at x1 550 cm and 22 at x2 570 cm c Discuss why it is useful to transform from x to z Solution A m 2 5 U L 2 2 am M Z n low Q 50 A Xv 5m J3 i h 9 N S Hr 1L ZQX 39 39h M PM AN Cm Ps em a X X Eh p zf 0436 akaquot altha h Iquot W 3 9094616 Si Hon 2 lt h3L FGJE Q I I IF vi PM If mhr 3939 4 90F Lug GilML c 394quot n nk L WW mm 94 Example Con dence level Given Many voltage readings are taken from a power supply The sample mean voltage reading is 171012 V The sample standard deviation of all the readings is 0022 V We assume that the errors in the readings are purely random To do Write the voltage to 95 confidence level Solution 2C 9 13 SIX 6 I t1 LC 5 00 V my a 0mg 1 Example Probability exam scores Given In one of Professor Cimbala s midterm exams the mean was 736 out of 100 possible points and the standard deviation was 92 We assume that the distribution of exam scores is Gaussian The cutoff grade for a D is 60 points To do Predict the percentage of students who failed the exam score lt 60 points Solution a a MA GMMn m r 3 M An PMMIql Wu 1 tu Hn livid l J m X g 35 X x 1 6g 11 S 31 em 1 W I 08L A0 05 031359 00011 or a 311 Ass 39 IA MAM 1 173 a Lu q ALI 6313MB c Example Probability Given 100 velocity measurements are taken in a wind tunnel The sample mean velocity is V 5126 ms The sample standard deviation of all the readings is 00690 ms We assume that the errors in the readings are purely random a To do Calculate the probability that the velocity of a random measurement is in the range 5126 ms lt Vlt 5200 ms In other words calculate P5l26 lt Vlt 5200 b To do Calculate the probability that the velocity of a random measurement is in the range 5000 ms lt Vlt 5200 ms In other words calculate P5000 lt Vlt 5200 m Solution V M 60 m 17539 2 F 7 T 0 3 5211 rm 7 d 1311 006 F0 Fm Pm Aul MM 4 l 2 0 h Vi 9V V V Mm Mmm 02m Kemp 3532 m a km W W W Mm PM 1 RDA ADM L i We m w QM wN Fm ch nLe mum W My Mum ME345 Fall 2009 Professor John M Cimbala Today we will Do some more example problems significant digits Review the pdf module Dimensional Analysis Do some example problems dimensional analysis Example Signi cant digits Gas mileage calculations a Given You travel 2100 miles in your new car and use 700 gallons of gas To do Calculate your gas mileage in units of miles per gallon Give your answer to the appropriate number of significant digits Solution M PJ quot 300 my 10s L9H r b Given You estimate that your car gets 28 miles per gallon Gas costs 2899 per gallon To do How much does it cost to travel 455 miles Give your answer to the appropriate number of signi cant digits Cm Wry ltusa 341 909 Solution 1 X H7JOX7 Y Aw M7 or M 0 Given You fill up your tank drive 3165 miles and pay 4489 to fill up your tank again Gas costs 2799 per gallon To do Calculate your gas mileage in units of miles per gallon Give your answer to the appropriate number of significant digits Solution e 31 m39 2133 39V Paquot l A 973quot quot 1 9 AVku r UH M Example Signi cant digits Given Three quantities are measured a 755 b 6044 and c 10451 To do a Calculate a b giving your answer to the appropriate precision and number of significant digits Solution b Given the same three quantities a 755 b 6044 and c 10451 Calculate a b 0 giving your answer to the appropriate precision and number of signi cant digits Solution OK My on 4 Jun 40101 Example Primary dimensions shear stress force per unit length and power a Given In uid mechanics shear stress 239 is expressed in units of Nm2 To do Express the primary dimensions of r ie write an expression for T Solution F mLv N quot quot b Given Ray is conducting an experiment in which quantity a has dimensions of force per unit length A quot To do Express the primary dimensions of a ie write an expression for a Solution 2 F m J1 M a3 v z 39T 2 1 L t y 5c c Given Power W has the dimensions of energy per unit time To do Write the dimensions of power in terms of primary dimensions Solution Em 7 R u quot Vl wc PW Hat 339 FxL MLL V LI Niquot T Kk k U Example Dimensional analysis shaft power Given The output power W of a spinning shaft is a function of torque T and angular velocity 0 To do Express the relationship between W T and a in dimensionless form Solution Step1 E 399 quot EINgt 2 whyMl X7 Step3 Yy 03 LI wt JUL M39L t Lin39o 3 3390 J NV 9 313A 1 0 kiwiquot 3 7quot Step 4 ewlt MW l w f wk T OJ 0 A a K w T w mww39w Waist Kquotamp mquot H o l 0 390 m 2 m M 3 m q 52 quot La A L1 Lu Lu H 1 a 0 0 7 IL 0 Z i k 1 k 5c k w a Sm 2 T 4 mow Cons fM Y

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