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# INSTRUMENTATION & MEASUREMENT MECH 591

Wentworth Institute of Technology

GPA 3.84

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This 9 page Class Notes was uploaded by Kaden Johnston on Thursday October 29, 2015. The Class Notes belongs to MECH 591 at Wentworth Institute of Technology taught by Anthony Duva in Fall. Since its upload, it has received 19 views. For similar materials see /class/230953/mech-591-wentworth-institute-of-technology in Mechanical Engineering at Wentworth Institute of Technology.

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Date Created: 10/29/15

Summary of statistical analysis 1 Normal distribution If a random variable can be described by a normal distribution its density function is 1 x 2 fx 6m eXp 262 1 Where 6 is the standard deviation u is the mean value f x is called the probability density function X is magnitude of the random variable The probability of a lt x lt bwill be Pa lt x lt b 3fxdx zrlexp Wx 2 a 1 Gm 262 2 Standard normal distribution and the table The value of equation 2 is usually determined by a standard normal distribution table because the explicit solution is not available The standard normal distribution is a normal distribution with 6 1 and u0 The probability density function of the standard normal distribution is fz Em 3 Table l is the table of the standard normal 2 distribution which is the same as table in page 50 of the textbook The table gives the probability form 0 to 2 that is P0 lt Z lt 2 3 1 eXp 4 m 2 In order to use the standard normal distribution table we need to do following conversion Set 5 Then we have 21 22 7 6 The equation 2 will become b 1 x u2 22 1 x2 Paltxltb 2 ex 7 x iex i z JLCM27 pJ 262 quot21 V27 p 2 TLexp i dZ JLexp dz 2 gt2 gt0 0 27 2 0 27 2 2 1 JZIJ 2 22 2 1 u 1 u iex i dz iex i dz 2 gt0gtz J 27 pJ 2 J 27 pJ 2 J 2 1 izli 2 JzzJ 2 1 u 1 u iex 7dz iex 7dz 0gtz gtz 27 pJ 2 J 427 pJ 2 J 2 1 You can use equation 7 and table 1 to calculate the probability of a lt x lt b for a normal distribution 3 The con dence level when sample size is larger than 30 06 l C 8 Where C is the con dence level such 95 99 06 is the level of signi cance The bounds of measurement with con dence level C If the measured parameter complies with normal distribution and the sample number n is big enough gt30 the mean and standard deviation can be calculated based on the measurement x ltx ugt2 2 Z 9 n n According to the required con dence level C calculate the level of signi cance by using equation 8 Then use table 2 to determine Z n The actual value of the measured parameter inside the range 11 ZuZG Ll ZW2 G will have the con dence level C that is CPH ZuZGltXltHZuZG 10 The bounds of average of the measurement with confidence level C If the measured parameter complies with normal distribution and the sample number n is 30 use equation 1 l to calculate the mean and the standard deviation of the measurement 2 gamp 5x 11 n n According to the required con dence level C calculate the level of signi cance by using equation 8 Then use table 2 to determine Z n The actual value of the average of the measured parameter inside the range 7 S 7 S x ZaZ f lt Ll lt x ZaZ J2will have the con dence level C that is 7 S 7 S CPx zm2 x ltultxza27quot 12 n 5 Tab 2 the confidence intervals Con dence 85 90 95 99 level C Z H 1439 1645 1960 2575 Bounds for pt14396 pt16456 pt19606 pt25756 14396 u16456 u19606 u25756 u mean 6 Standard deviation Bound for 7 7 7 7 average x 71439 Sx x l645 Sx x l960 Sx x7 2575 Sx 5 xZ xZ xZ 7 S 7 S 7 S 7 S xl439 x xl645 x xl960 x x2575 x 1 1 1 1 f is the average of measurement S xis the standard deviation of the measurement n is the sample size of the 4 The con dence level when sample size is less than 30 If the sample number n is small lt30 we can not use normal distribution But we can use tdistribution which is another type of probability density function The bounds of measurement with con dence level C Use equation 1 l to calculate the mean and standard deviation of the measurement Use equation 8 to calculate the level of signi cance Then use table 3 t0 determine law In table 3 V is the degree of freedom For one random variable in this case vnl The actual value of the measured parameter inside the range f taZSX f taZSxwill have the con dence level C that is CPf ta2S 7cta2Sx 13 x 42 The bounds of average of the measurement with con dence level C Use equation 1 l to calculate the mean and standard deviation of the measurement Use equation 8 to calculate the level of signi cance Then use table 3 to determine law In table 3 v is the degree of freedom For one random variable in this case vn l The actual value of the average of the measurement inside the range S f t x and S S CP za2 quotltlt7cla2 x 14 J J S 2 x taZ will have the con dence level C that is n 5 The method of least squares If we think that there is a linear relationship between the independent variable X and dependent variable y And we get n sets of dataxi y i 12n The method of least squares can be used to determine this linear relationship ya13x 15 1 Zyizxiz inzxiyi 16 n2 x3 2 xi 2 IS n2x y Zx 22yquot 17 quot2 xi in The correlation coef cient r is a way to indicate the reliability of the t A 2 r2 52 Z9ltxgt y2 1 y 72y 19 n l A 52 ZZLvx y2 20 Where Xxi 1 5x If r is less than 09 the assumption that there was a linear relationship between X and y should be discarded The uncertainty for the tting line within the C con dence level intervals is S x4rt W 21 y t 12v J S 1 25xy2 2 22 yx 11 2 1 1 11 2 Use equation 8 to calculate the level of signi cance Then use table 3 to get toy2quot where vn2 The true slope within the C con dence intervals is Syx VS S 3fo lZx2 24 71 Use equation 8 to calculate the level of signi cance Then use table 3 to get toy2quot where vn2 13ith 23 Table 1 of the Standard Normal 2 Distribution TABLE 3 Student s l Distribulion Values of ram 0 V 010 1005 10025 100117 10005 v 1 3078 6314 12706 31821 63657 1 2 1886 2920 4303 6965 9925 2 3 1638 2353 3182 4541 5841 3 4 1533 2132 2776 3747 4604 4 5 1476 2015 2571 3365 4032 5 6 1440 1943 2447 3143 3707 6 7 1415 1395 2365 2998 3499 7 8 1397 1860 2306 2896 3355 8 9 1383 1833 2262 2821 3250 9 10 1372 1812 2228 2764 3169 10 11 1363 1796 2201 2718 3106 11 12 1356 1782 2179 2681 3055 12 13 1350 1771 2160 2650 3012 13 14 1345 1761 2145 2624 2977 14 15 1341 1753 2131 2602 2947 15 16 1337 1746 2120 2583 2921 16 17 1333 1740 2110 2567 2898 17 18 1330 1734 2101 2552 2878 18 19 1328 1729 2093 2539 2861 19 20 1325 1725 2086 2528 2845 20 21 1323 1721 2080 2518 2831 21 22 1321 1717 2074 2508 2819 22 23 1319 1714 2069 2500 2807 23 24 1318 1711 2064 2492 2797 24 25 1316 1798 2060 2485 2787 25 26 1315 1706 2056 2479 2779 26 27 1314 1703 2052 2473 2771 27 28 1313 1701 2048 2467 2763 28 29 1311 1699 2045 2462 2756 29 00 1282 1645 1960 2326 2576 00 Exercise 21 7 Analysis Track A B K C on a Waveform Chart untrl stopped by the user 1 2 3 45 Open ablank VIfrom the Gethng Started sereen Cnntrnls Mndarn Grzph Wzv2fnrm Chart Plaee a dlal eontxol on the front panel From the eontxols palette seleet w m t n w l Frequency Inquot to glv Go to the block dlagram ltCtxlEgt and e whrle loop down nghtrcllckto open the funetrons palette and selectExpress Execlltinn CnnuanhileLnnp llck and drag on the block dlagram to make the whrle loop the eorreet srze ame to thrs control pl a not already Notree that a stop button ls already eonneetedto the eondrhonal termrnal of the whrle loop For VI Frmth lwhw palette seleetExpremSlgnal Analysinsimulzte Signal and plaee rt on the ehoose Srmulate acqulslnon umlng quot clrek OK quot wnltttrd 39 h and Frequency measurements ln the Slngle Tone Measurements seehon Cllck OK the termrna1 so thatrtbeeomes the Wmng tool and ehexung onee on eaeh othe termrnals you wrsh to eonneet a Conneet the Sme outputtermrna1 ofthe Srmulate Srgna1 VIto the Srgna1squot rnput of the Tone Measurements VI Conneet the Sme ou utto the Waveform Chart create mdmators for the amphtude and frequency measurements by nghtr clmkmg on eaeh of the termrna1s ofthe Tone Measurements Express VI and se1eetrng reat2gtgtNllmeric Indicztnr Conneet the Frequency Inquot eontro1 to the Frequency termrna1 of the Srmulate Srgna1 VI 8 Return to the front an un VI Move the Frequency Inquot dwl and observe the frequency ofthe srgna1 Chek the stop button d 9 SaV t VI onee you are mshe as Enererse 2 1 a s mquot e e rmulated 10 close the VI Nmu hand eorner ofthe pa1ette Thrs wru taek down the p tte sothatrt doesn39t dASappear Thrs step wru be omrttedrn the fouowrng enererses but should be repe te The soluuon to thus exererse xs pnntedrn the back ofthrs manual

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