19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 2007

Size: px
Start display at page:

Download "19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 2007"

Transcription

1 19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, -7 SEPTEMBER 007 IMAGINE proect: urba measuremets of Lde ad Light ad calculatio of the associated ucertaities. PACS: rq Paviotti, Marco 1 ; Kephalopoulos, Styliaos 1 ; Joasso, Has 1 Europea Commissio, DG JRC; v. E. Fermi, 1, Ispra, 107, Italy; marco.paviotti@rc.it SP Techical Research Istitute of Swede, Box 857, SE Borås, SWEDEN; has.oasso@sp.se ABSTRACT This article describes the priciples of the ew measuremet method developed withi the IMAGINE Europea proect to determie L de ad L ight, as defied by the Europea Noise Directive 00/49/EC, by direct measuremet of the oise levels. The measuremet method was tested i a real ad complex urba eviromet icludig a maor road, a maor railway lie ad a idustrial site. A descriptio is give for the calculatio of the yearly averaged levels ad the ucertaity estimatio. Cocerig such log term idicators, estimatio of ucertaity is a rather complex task, especially if the yearly L de ad L ight are derived from measuremets performed over a short period of the year. The ucertaities cocer the microphoe positio, the source variatio, the meteorological variatios, the correctio for backgroud oise ad the soud level meter class 1 ucertaity. The example described here is based o a measuremet campaig performed over oe year i the city of Pisa (Italy).The aforemetioed measuremets would be typically applied to support the credibility of oise map calculatios towards the citizes ad to validate calculatios of oise maps i well-defied situatios. INTRODUCTION Followig the requiremets of the Evirometal Noise Directive 00/49/EC (END) [1], oise levels are to be mapped i Europe. The metrics used by this Europea Directive are L de ad L ight : I the END it is stated that L de ad L ight ca be either computed or measured. Therefore, withi the Europea IMAGINE proect [] work has bee performed to produce a protocol to measure these two values, repetative of the average year, as defied i the END. A lot of effort has bee spet to give ot oly a framework for the measuremet of the two idicators L de ad L ight, but also to attribute a overall ucertaity so that, depedig o the method used to perform the measuremet, it could be stated how accurate that value is. A cocrete example is give cocerig the oise measuremets performed of a maor road, however far from the road itself ad i a complex urba eviromet. BASIC PRINCIPLES The method to measure the Europea oise idicators L de ad L ight, makes it possible to separate the assessmet of the source variatio ad its ucertaity o oe had ad the trasfer fuctio betwee the source ad the receiver uder differet propagatio coditios, which depeds o the meteorological parameters, ad its ucertaity o the other had. The geeral equatios described i the END are used to calculate the L de ad L ight durig each combiatio of source ad propagatio coditio. Source ad propagatio are grouped i classes defied by level itervals that are specified by the team performig the measuremet. The choice of the itervals depeds o the variability of the source ad the propagatio coditios as well as o the possibility to have a large or a small amout of samples (geerally L pa ) to use for the evaluatio. Oce a combiatio of these itervals is set, L de ad L ight ca be calculated averagig the levels recorded durig these combiatios. Ucertaities are the associated with the overall log-term oise value obtaied, by itegratig the several separate short-term measuremets ad related ucertaities used to extrapolate the yearly average L de ad L ight. The method developed follows the requiremets of the GUM [3], which states that each sigificat source of error has to be idetified ad corrected for. The basic priciple is that, if the quatity L m is measured, which is a fuctio of the quatities x, the:

2 L = f x ) (Eq. 1) m ( If each quatity x has the stadard ucertaity σ the combied ucertaity is give by: where the sesitivity coefficiet c is give by: ( ) σ ( L m ) = c σ (Eq. ) 1 c f = (Eq. 3) x The measuremet ucertaity to be reported is the combied measuremet ucertaity associated with a chose coverage probability. By covetio, a coverage probability of 95% is usually chose, with a associated coverage factor of. This meas that the reported measuremet ucertaity becomes L m +σ. For evirometal oise measuremets f(x ) is complicated ad it is hardly feasible to formulate exact equatios for the fuctio f. The calculatio of ucertaity correlated to each parameter could be i ay case performed followig the approach suggested by Kephalopoulos et.al. [4] where it is explaied how oise levels could be associated to their corpodig ucertaities. Hece, i the followig, it is ecessary simplificatios to be made. Followig the priciples give i ISO 3745 [5], some importat error sources could be idetified: L = + δ + δ + δ + δ + δ (Eq. 4) true L m slm sou met loc where L true is the true (o errors) value durig the specified coditios for which we wat a measured value, L m is the measured value, δ slm is the error of the measuremet chai (soud level meter i the simplest case), δ sou is the error due to deviatios from the ideal operatig coditios of the source, δ met is the error due to meteorological coditios deviatig from the ideal coditios, δ loc is the error due to the selectio of receiver positio ad δ is the error due to idual oise. Equatio (4) is very simplified ad each source of error is a fuctio of several other sources of error. I priciple Eq. (4) could be applied o ay measuremet lastig from secods to years. I [6] the measuremets are divided ito short ad log-term measuremets. A shortterm measuremet may typically rage betwee 10 miutes ad a few hours whereas a typical log-term measuremet may rage betwee a moth ad a year. I may cases the measuremet ults should be maipulated to extrapolate them to other coditios, e.g., ormalizig to differet traffic flows, ad to use them for calculatig quatities like the L de. Let us cosider the followig specific case. I this L deotes the L pa for coditio, which lasts for p of the total time, whereas L. deotes the total L pa for the overall time iterval. We the get: L1 L L ( p 10 + p p ) L 10lg + = (Eq. 5) 1 10 If L 1,..,L are idepedet the sesitivity coefficiet c 1 is the give by L p 10 l(10) 0,1 p 10 = = 10lg( e) = L1 L L Li / 10 p 10 + p p 10 p 10 1 L c (Eq. 6) i L 19 th INTERNATIONAL CONGRESS ON ACOUSTICS ICA007MADRID

3 As p i = 1 these coefficiets are ot idepedet. Istead we write Eq. (5) i the form For c pi we get = L L L 1 10lg p110 p10... p 110 (1 pi ) 10 i= 1 1 L / 10 L (Eq. 7) L Li c pi = = 10lg( e) Li / 10 (Eq. 8) p p 10 i L i is determied with the stadard ucertaity u Li ad p i with the stadard ucertaity u pi. The stadard ucertaity u of L is the give by i 1 ul + i= 1 i i= 1 pi u = u i (Eq. 9) pi where the letter u istead of σ is used to specify that these ucertaities are estimated ucertaities. EXAMPLE OF UNCERTAINTY CALCULATION Maor road oise source The method developed, is based o the aforemetioed rules for the ucertaity calculatio, ad was tested o a assessmet positio located i a complex urba eviromet. The locatio was about 150 m away from a maor road i which were circulatig more tha three millio vehicles per year. The microphoe at the assessmet poit was placed at 4m height ad m away from the flat façade of a house. At this locatio, several other oise sources were most of the time simultaeously pet (e.g., fa oise of a earby idustrial pla, railway, local road, aircraft, people talkig), havig, whe active, approximately the same istataeous L eq as the specific source uder assessmet. It was therefore ecessary to discrimiate the cotributio of the maor road, ad evaluate the ucertaity of the overall measuremet performed. The oise from the maor road was measured to check the traffic variatios durig the day, the week ad the year. Two methods were used to quatify the traffic alog the maor road: the first by direct coutig ad classificatio of the vehicle pass-bys ad the secod by assessig the L 95 value usig a microphoe placed close to the road. Meteo classes The followig meteorological coditios were measured: wid directio, wid speed, temperature, temperature gradiet, T*, u*, 1/L (iverse Moi-Obuchov legth) gradiet. Based o these measuremets ad also followig the defiitio of meteorological classes relevat for oise propagatio developed i the HARMONOISE proect [7], four meteorological classes repetative of the possible propagatio coditios were established. These classes varyig betwee class M1 for stable meteorological coditios with favourable soud propagatio ad class M4 for very ustable meteorological coditios with ufavourable soud propagatio. It is essetial to uderlie the fact that a assumptio for the meteo classes was made. Havig measured the meteo classes over more tha four moths, ad for oe moth each seaso of the year, the iformatio about this subset of the meteo classes used was assumed to be repetative for the whole give year. To get the ucertaity due to the propagatio coditios, four possible meteorological situatios were distiguished, depedig o the curvature of the soud propagatio. About twety differet L pa,i,source were compared to the corpodig L pa,i,receiver ad, based o these measuremets, the followig meteo ucertaities were derived (Table 1). I the Table 1, class M1 corpods to the more favourable propagatig coditio whereas, class M4 to the ufavourable. Sice the eviromet cosidered i this study was a 19 th INTERNATIONAL CONGRESS ON ACOUSTICS ICA007MADRID 3

4 urba eviromet, free field meteo coditios were ideed ot eve closely related to the local meteo coditios betwee the buildigs ad above these [8]. The ucertaity values calculated cofirmed this hypothesis, sice all values were betwee 1.3 db ad.1 db despite of the fact that i geeral stable meteo coditios (class 1 i this study) should relate to a low ucertaity whereas ustable meteo coditio (class 4 i this study) should relate to a large ucertaity. Table 1. Ucertaities associated to meteo variatio for each meteo class (M1, M, M3, M4). Class M1 M M3 M4 σ met The meteo classes were the averaged durig the period ivestigated. Cosiderig the distributio of the samples (e.g., takig samples every day betwee :00 am ad 3:00 am, after oe moth there were 8 samples distributed betwee M1, M, M3 ad M4), the ucertaity cocerig the determiatio of the meteorological classes distributio could be foud usig the formula of (Eq. ). Operatig coditio classes To extrapolate the oise cotributio of the road, at first a classificatio of the road traffic was performed by moitorig the road traffic over 48h. This obviously itroduced a ucertaity, sice the extrapolatio to the aual average traffic from differet hourly time itervals should be estimated. To partially cope with this, the L 95 level was used as a idicator of the variatio of the road traffic alog the year. I other words, the traffic was moitored usig a microphoe, the the L 95 was used as a rough estimatio of the pece of vehicles. Usig several 15mi samples of the L 95, there was a sufficiet umber of elemets to perform a statistical evaluatio of the traffic coditios durig several momets of the day, durig differet days of the week ad durig differet seasos of the year, which could the be used to determie a σ for the road traffic flow fluctuatios. Five classes of traffic flow durig a sigle day were idetified, ad regarded as A or B durig the day, ad as C, D, E durig the ight. Usig the first techique (direct coutig of vehicles) o iformatio is available o the traffic volumes durig other days tha the oe used for the assessmet, however, coutig vehicles is ecessary to kow which is the expected ucertaity due to the specific types of vehicles durig the specific momet of the short-term measuremet. I other words, if oly oise levels are recorded, there is o iformatio whether or ot these levels are caused by the average vehicle pass bys or by a specific combiatio of acoustically exceptioal vehicles. Therefore, ot oly several levels are to be measured to calculate the oise levels at the assessmet positio, but also the umber of vehicles durig each recorded short-term L pa. For the situatio ivestigated, calculatios were based o 15 mi L pa records. I all differet periods subsequet samples were used to evaluate the ucertaity cosiderig the umber of vehicles each 15 mi, ultig i a selectio of five differet classes (Table ): Table. Ucertaities associated with source variatio for each traffic class. A B C D E σ sou This ucertaity is to be used if the traffic flow is measured directly, or take from local authority statistics, ad the extrapolatio of the L de is performed based o these umbers of vehicles ad traffic distributios. The same procedure was applied to estimate ucertaities comig ot oly from A, B, C, D, E classes for oe sigle day of a week, but also from classes which take ito cosideratio mothly or seasoal variatios by itegratig the differet traffic flows durig the t of the weekdays, Saturdays, Sudays, ad, o a two seaso (summer ad witer time) basis. Oce agai, these could directly be assessed o the basis of the L pa oise levels recorded i each period. I other words, over the weeks the oise measuremets performed (e.g., oe i September 005, oe i December 005, two i February 006, ad oe i Jue 006) to assess the road traffic oise source, the L pa were recorded for each A, B, C, D, E period, each day (therefore, 5 L pa every day). Over such periods the ucertaity is calculated 19 th INTERNATIONAL CONGRESS ON ACOUSTICS ICA007MADRID 4

5 based o the several i-th day samples (e.g., 9 weekdays implies 9 samples -Mo, Tue, Wed, etc.). Extrapolatio at the receiver by meas of measured trasfer fuctios I the situatio ivestigated, local oises were pet, such as cars passig by the local road, people chattig, birds sigig, besides the pece of three other maor evirometal oise sources (maor railway lie, idustrial source, maor airport). To evaluate the cotributio to the overall oise levels of the road traffic oly, a trasfer fuctio betwee the maor oise source ad the receiver was estimated uder several meteorological classes. Based o the trasfer fuctio it was possible to use oly the oise records (oise levels) whe the oise source uder assessmet (i.e., road traffic) was clearly distiguished from other oise sources. Besides estimatig the trasfer fuctio, a correctio should evertheless be made for the idual oise at the receiver, sice recordig a clearly distiguished oise did ot ecessarily mea havig it more tha 10 db (as typically cosidered i measuremets) louder tha all the other oise sources pet at the same time. What was regarded i other stadards as backgroud oise or extraeous oise or idual oise was all regarded here as idual oise. Residual oise is the oise produced by all sources but the specific source uder assessmet. Residual oise could therefore be cosidered a local oise occasioally produced, aother relevat mai oise source, or a frequet o-evirometal oise source (e.g., people chattig, birds sigig). Sice the short-term periods used i this study were selected durig a time period whe o other oise sources (such as railway, local road, aircraft, people, birds) were pet, i the time period used for the assessmet, the idual oise was limited oly to the fa from the earby idustrial pla. This fa oise was previously measured to be L pa, = 43.5 db ad it was a costat source. Sice the road traffic oise had a L pa o more tha 10dB higher tha the L pa of the fa (=L ), the correctio for idual oise should be used the. Therefore, the followig formula should be used: L' L = 10 log Subsequetly, ucertaity was calculated over the trasfer fuctio for each meteo class, ad the idual oise ucertaity was also cosidered. This latter ucertaity was derived usig the formula L (Eq. 10) c ( L' L ( L' L At the assessmet positio, the average LpA was L = 49.0 db, whereas LpA, = 43.5 db, hece, the sesitivity coefficiet is c L =c = Other specific ucertaities The distace betwee the source ad the receiver was about 50 m, therefore the ucertaity of atmospheric absorptio was also cosidered (although it was expected to be very low) ad assumed to be 0. db. For the soud level meters used (oe at the receiver poit, ad oe at the road side), the ucertaity attributed was u slm =0.5 db for each microphoe. The idual oise at the road was about 30 db(a) lower tha the road traffic oise, therefore o correctio was eeded ad the ucertaity of the idual oise was set u =0 db. Because of the positio m i frot of a façade, the ucertaity was take as u pos =0.5 db as derived from a study, performed i the cotext of the IMAGINE proect, to test the acoustical correctios for reflectios o a façade [9]. Calculatio of the overall ucertaity Basically, the sesitivity coefficiets were calculated usig the formula (Eq. 6), sice the oise levels used were idepedet values. 19 th INTERNATIONAL CONGRESS ON ACOUSTICS ICA007MADRID 5 ) = (Eq. 11) )

6 Subsequetly, the formula (Eq. 7) should be applied iteratively: the first time, while averagig over the day, eveig ad ight, the while averagig for the day of the week, the while averagig for the seasos ad fially while calculatig the L de usig the three periods (day, eveig, ight). It should be oted that u sou ad u met are the icluded i the calculatio of u de. For the measuremet campaig i this study, this was foud to be 0.7 db (u de is ot the overall ucertaity). Oce all the ucertaities were calculated, for this specific situatio, the overall ucertaity was: de pos slm1 slm atm u = u + c u + u + u + u + u (Eq. 1) ad, after substitutig with the values, we get u = 1. db. CONCLUSIONS I the pet article the method developed for the testig of the protocol cocerig the measuremet of L de ad L ight followig the requiremets of the END was peted, i the case of road oise source measuremets. Specifically, it was show how it is possible to measure the required values i complex situatios, obtaiig both the required oise levels ad the associated ucertaities. Oce this accomplished, the ults are satisfactory for commuicatig them to the populatio exposed as well as to the policy makers who eed to implemet oise reducig measu where appropriate. I the past the measuremets performed were ofte either icludig extraeous oise sources, or oly oe source, however this measuremet was repetative oly of a partial period of the year. The techique described here esu that log term oise values for a selected specific source could be obtaied i ay complex situatio which icludes periodical variatios ad coexistece of other oise sources, together with a statistically robust evaluatio of the ucertaity associated to the value calculated. I the past, ofte complais were arisig from the cosideratio that the measuremet performed was ot ecessarily repetative of a log term period close to the true yearly average. Adoptig i a iteratioal oise measuremet stadard which might be similar to the oe described by Joasso [6], the priciple of deliverig a ucertaity associated to ay measuremet performed to evaluate the L de ad L ight idicators, ca be clearly cosidered as a improvemet, sice this way is always possible, e.g. to state that a certai L de ad L ight is with 95% cofidece ot more tha the give L de +u ad L ight +u values. Also, i view of the oise mappig of the maor Europea cities, it will be possible to demostrate to the populatio as well as to the local authorities that the computed oise levels are a correct estimatio of the real value of L de ad L ight, by performig ust a few measuremets over short time periods i urba areas. Further implemetatio of these priciples i future oise measuremets will cotribute to the extesive testig ad evetual fial acceptace of these procedu by the iteratioal oise measuremet stadards. Refereces: [1] Directive 00/49/EC relatig to the assessmet ad maagemet of evirometal oise, 00 [] IMAGINE proect, "Improved Methods for the Assessmet of the Geeric Impact of Noise i the Eviromet", ( Cotract Number: SSPI-CT IMAGINE [3] ISO, Guide for the expsio of ucertaity i measuremet (GUM) [4] S. Kephalopoulos, M. Paviotti, D. Kauss ad M. Béregier, Ucertaities i Log-Term Road Noise Moitorig Icludig Meteorological Variatios, Noise Cotrol Eg. J. 55 (1), , 007. [5] EN ISO 3745 Acoustics - Determiatio of soud power levels of oise sources usig soud psure Precisio methods for aechoic ad hemi-aechoic rooms, 003 [6] H. Joasso, Determiatio of L de ad L ight usig measuremets, report delivered to IMAGINE proect (IMA SP10), 007 [7] D. Heima, O a meteorological classificatio for log term oise calculatios, report delivered to HARMONOISE proect (HAR5MO-0073-DLR01), 00. [8] M.V. Rotach, R. Vogt, C. Berhofer, E. Batchvarova, A. Christe, A. Clappier, B. Fedderse, S.-E. Gryig, G. Martucci, H. Meyer, V. Mitev, T.R. Oke, E. Parlow, H. Richer, M. Roth, Y.-A. Roulet, D. Ruffieux, J. A. Salmod, M. Schatyma, J.A. Voogt, BUBBLE a Urba Boudary Layer meteorology proect, Theor. Appl. Climatology 81, 31-61, 005 [9] G. Memoli, M. Paviotti, S. Kephalopoulos, G. Licitra, Effect o measured oise levels of the microphoe positio i frot of a façade, Iteroise006, Hoolulu (USA), 3-6 December th INTERNATIONAL CONGRESS ON ACOUSTICS ICA007MADRID 6

Expected mean in an environmental noise measurement and its related uncertainty

Expected mean in an environmental noise measurement and its related uncertainty Expected mea i a evirometal oise measuremet ad its related ucertaity M. Paviotti ad S. Kephalopoulos Europea Commissio, via e. fermi,, 00 Ispra, Italy marco.paviotti@jrc.it 85 I the cotext of the implemetatio

More information

Measurement uncertainty of the sound absorption

Measurement uncertainty of the sound absorption Measuremet ucertaity of the soud absorptio coefficiet Aa Izewska Buildig Research Istitute, Filtrowa Str., 00-6 Warsaw, Polad a.izewska@itb.pl 6887 The stadard ISO/IEC 705:005 o the competece of testig

More information

ANALYSIS OF EXPERIMENTAL ERRORS

ANALYSIS OF EXPERIMENTAL ERRORS ANALYSIS OF EXPERIMENTAL ERRORS All physical measuremets ecoutered i the verificatio of physics theories ad cocepts are subject to ucertaities that deped o the measurig istrumets used ad the coditios uder

More information

A statistical method to determine sample size to estimate characteristic value of soil parameters

A statistical method to determine sample size to estimate characteristic value of soil parameters A statistical method to determie sample size to estimate characteristic value of soil parameters Y. Hojo, B. Setiawa 2 ad M. Suzuki 3 Abstract Sample size is a importat factor to be cosidered i determiig

More information

Properties and Hypothesis Testing

Properties and Hypothesis Testing Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Cross-sectioal data. 2. Time series data.

More information

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals

More information

Provläsningsexemplar / Preview TECHNICAL REPORT INTERNATIONAL SPECIAL COMMITTEE ON RADIO INTERFERENCE

Provläsningsexemplar / Preview TECHNICAL REPORT INTERNATIONAL SPECIAL COMMITTEE ON RADIO INTERFERENCE TECHNICAL REPORT CISPR 16-4-3 2004 AMENDMENT 1 2006-10 INTERNATIONAL SPECIAL COMMITTEE ON RADIO INTERFERENCE Amedmet 1 Specificatio for radio disturbace ad immuity measurig apparatus ad methods Part 4-3:

More information

The target reliability and design working life

The target reliability and design working life Safety ad Security Egieerig IV 161 The target reliability ad desig workig life M. Holický Kloker Istitute, CTU i Prague, Czech Republic Abstract Desig workig life ad target reliability levels recommeded

More information

If, for instance, we were required to test whether the population mean μ could be equal to a certain value μ

If, for instance, we were required to test whether the population mean μ could be equal to a certain value μ STATISTICAL INFERENCE INTRODUCTION Statistical iferece is that brach of Statistics i which oe typically makes a statemet about a populatio based upo the results of a sample. I oesample testig, we essetially

More information

1 Inferential Methods for Correlation and Regression Analysis

1 Inferential Methods for Correlation and Regression Analysis 1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet

More information

TRACEABILITY SYSTEM OF ROCKWELL HARDNESS C SCALE IN JAPAN

TRACEABILITY SYSTEM OF ROCKWELL HARDNESS C SCALE IN JAPAN HARDMEKO 004 Hardess Measuremets Theory ad Applicatio i Laboratories ad Idustries - November, 004, Washigto, D.C., USA TRACEABILITY SYSTEM OF ROCKWELL HARDNESS C SCALE IN JAPAN Koichiro HATTORI, Satoshi

More information

Soo King Lim Figure 1: Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Figure 7:

Soo King Lim Figure 1: Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Figure 7: 0 Multivariate Cotrol Chart 3 Multivariate Normal Distributio 5 Estimatio of the Mea ad Covariace Matrix 6 Hotellig s Cotrol Chart 6 Hotellig s Square 8 Average Value of k Subgroups 0 Example 3 3 Value

More information

6.3 Testing Series With Positive Terms

6.3 Testing Series With Positive Terms 6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial

More information

PH 425 Quantum Measurement and Spin Winter SPINS Lab 1

PH 425 Quantum Measurement and Spin Winter SPINS Lab 1 PH 425 Quatum Measuremet ad Spi Witer 23 SPIS Lab Measure the spi projectio S z alog the z-axis This is the experimet that is ready to go whe you start the program, as show below Each atom is measured

More information

Activity 3: Length Measurements with the Four-Sided Meter Stick

Activity 3: Length Measurements with the Four-Sided Meter Stick Activity 3: Legth Measuremets with the Four-Sided Meter Stick OBJECTIVE: The purpose of this experimet is to study errors ad the propagatio of errors whe experimetal data derived usig a four-sided meter

More information

Analysis of Experimental Measurements

Analysis of Experimental Measurements Aalysis of Experimetal Measuremets Thik carefully about the process of makig a measuremet. A measuremet is a compariso betwee some ukow physical quatity ad a stadard of that physical quatity. As a example,

More information

Chapter 6 Sampling Distributions

Chapter 6 Sampling Distributions Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to

More information

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 018/019 DR. ANTHONY BROWN 8. Statistics 8.1. Measures of Cetre: Mea, Media ad Mode. If we have a series of umbers the

More information

There is no straightforward approach for choosing the warmup period l.

There is no straightforward approach for choosing the warmup period l. B. Maddah INDE 504 Discrete-Evet Simulatio Output Aalysis () Statistical Aalysis for Steady-State Parameters I a otermiatig simulatio, the iterest is i estimatig the log ru steady state measures of performace.

More information

Analysis of Experimental Data

Analysis of Experimental Data Aalysis of Experimetal Data 6544597.0479 ± 0.000005 g Quatitative Ucertaity Accuracy vs. Precisio Whe we make a measuremet i the laboratory, we eed to kow how good it is. We wat our measuremets to be both

More information

Estimation of a population proportion March 23,

Estimation of a population proportion March 23, 1 Social Studies 201 Notes for March 23, 2005 Estimatio of a populatio proportio Sectio 8.5, p. 521. For the most part, we have dealt with meas ad stadard deviatios this semester. This sectio of the otes

More information

SNAP Centre Workshop. Basic Algebraic Manipulation

SNAP Centre Workshop. Basic Algebraic Manipulation SNAP Cetre Workshop Basic Algebraic Maipulatio 8 Simplifyig Algebraic Expressios Whe a expressio is writte i the most compact maer possible, it is cosidered to be simplified. Not Simplified: x(x + 4x)

More information

Discrete Mathematics for CS Spring 2008 David Wagner Note 22

Discrete Mathematics for CS Spring 2008 David Wagner Note 22 CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 22 I.I.D. Radom Variables Estimatig the bias of a coi Questio: We wat to estimate the proportio p of Democrats i the US populatio, by takig

More information

Linear Regression Demystified

Linear Regression Demystified Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to

More information

GUIDELINES ON REPRESENTATIVE SAMPLING

GUIDELINES ON REPRESENTATIVE SAMPLING DRUGS WORKING GROUP VALIDATION OF THE GUIDELINES ON REPRESENTATIVE SAMPLING DOCUMENT TYPE : REF. CODE: ISSUE NO: ISSUE DATE: VALIDATION REPORT DWG-SGL-001 002 08 DECEMBER 2012 Ref code: DWG-SGL-001 Issue

More information

IP Reference guide for integer programming formulations.

IP Reference guide for integer programming formulations. IP Referece guide for iteger programmig formulatios. by James B. Orli for 15.053 ad 15.058 This documet is iteded as a compact (or relatively compact) guide to the formulatio of iteger programs. For more

More information

Measures of Spread: Variance and Standard Deviation

Measures of Spread: Variance and Standard Deviation Lesso 1-6 Measures of Spread: Variace ad Stadard Deviatio BIG IDEA Variace ad stadard deviatio deped o the mea of a set of umbers. Calculatig these measures of spread depeds o whether the set is a sample

More information

Chapter 8: Estimating with Confidence

Chapter 8: Estimating with Confidence Chapter 8: Estimatig with Cofidece Sectio 8.2 The Practice of Statistics, 4 th editio For AP* STARNES, YATES, MOORE Chapter 8 Estimatig with Cofidece 8.1 Cofidece Itervals: The Basics 8.2 8.3 Estimatig

More information

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 9

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 9 Hypothesis testig PSYCHOLOGICAL RESEARCH (PYC 34-C Lecture 9 Statistical iferece is that brach of Statistics i which oe typically makes a statemet about a populatio based upo the results of a sample. I

More information

11 Correlation and Regression

11 Correlation and Regression 11 Correlatio Regressio 11.1 Multivariate Data Ofte we look at data where several variables are recorded for the same idividuals or samplig uits. For example, at a coastal weather statio, we might record

More information

Expectation and Variance of a random variable

Expectation and Variance of a random variable Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio

More information

Statistical Intervals for a Single Sample

Statistical Intervals for a Single Sample 3/5/06 Applied Statistics ad Probability for Egieers Sixth Editio Douglas C. Motgomery George C. Ruger Chapter 8 Statistical Itervals for a Sigle Sample 8 CHAPTER OUTLINE 8- Cofidece Iterval o the Mea

More information

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + 62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of

More information

Accuracy of prediction methods for the improvement of impact sound pressure levels using floor coverings

Accuracy of prediction methods for the improvement of impact sound pressure levels using floor coverings Accuracy of predictio methods for the improvemet of impact soud pressure levels usig floor coverigs Daiel GRIFFIN 1 1 Marshall Day Acoustics, Australia ABSTRACT The improvemet of impact soud pressure levels

More information

Simulation. Two Rule For Inverting A Distribution Function

Simulation. Two Rule For Inverting A Distribution Function Simulatio Two Rule For Ivertig A Distributio Fuctio Rule 1. If F(x) = u is costat o a iterval [x 1, x 2 ), the the uiform value u is mapped oto x 2 through the iversio process. Rule 2. If there is a jump

More information

Number of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day

Number of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the

More information

577. Estimation of surface roughness using high frequency vibrations

577. Estimation of surface roughness using high frequency vibrations 577. Estimatio of surface roughess usig high frequecy vibratios V. Augutis, M. Sauoris, Kauas Uiversity of Techology Electroics ad Measuremets Systems Departmet Studetu str. 5-443, LT-5368 Kauas, Lithuaia

More information

Overview. p 2. Chapter 9. Pooled Estimate of. q = 1 p. Notation for Two Proportions. Inferences about Two Proportions. Assumptions

Overview. p 2. Chapter 9. Pooled Estimate of. q = 1 p. Notation for Two Proportions. Inferences about Two Proportions. Assumptions Chapter 9 Slide Ifereces from Two Samples 9- Overview 9- Ifereces about Two Proportios 9- Ifereces about Two Meas: Idepedet Samples 9-4 Ifereces about Matched Pairs 9-5 Comparig Variatio i Two Samples

More information

Data Analysis and Statistical Methods Statistics 651

Data Analysis and Statistical Methods Statistics 651 Data Aalysis ad Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasii/teachig.html Suhasii Subba Rao Review of testig: Example The admistrator of a ursig home wats to do a time ad motio

More information

ST 305: Exam 3 ( ) = P(A)P(B A) ( ) = P(A) + P(B) ( ) = 1 P( A) ( ) = P(A) P(B) σ X 2 = σ a+bx. σ ˆp. σ X +Y. σ X Y. σ X. σ Y. σ n.

ST 305: Exam 3 ( ) = P(A)P(B A) ( ) = P(A) + P(B) ( ) = 1 P( A) ( ) = P(A) P(B) σ X 2 = σ a+bx. σ ˆp. σ X +Y. σ X Y. σ X. σ Y. σ n. ST 305: Exam 3 By hadig i this completed exam, I state that I have either give or received assistace from aother perso durig the exam period. I have used o resources other tha the exam itself ad the basic

More information

MATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4

MATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4 MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable.

More information

Resampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n.

Resampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n. Jauary 1, 2019 Resamplig Methods Motivatio We have so may estimators with the property θ θ d N 0, σ 2 We ca also write θ a N θ, σ 2 /, where a meas approximately distributed as Oce we have a cosistet estimator

More information

6 Sample Size Calculations

6 Sample Size Calculations 6 Sample Size Calculatios Oe of the major resposibilities of a cliical trial statisticia is to aid the ivestigators i determiig the sample size required to coduct a study The most commo procedure for determiig

More information

The standard deviation of the mean

The standard deviation of the mean Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider

More information

Chapter 12 - Quality Cotrol Example: The process of llig 12 ouce cas of Dr. Pepper is beig moitored. The compay does ot wat to uderll the cas. Hece, a target llig rate of 12.1-12.5 ouces was established.

More information

REGRESSION (Physics 1210 Notes, Partial Modified Appendix A)

REGRESSION (Physics 1210 Notes, Partial Modified Appendix A) REGRESSION (Physics 0 Notes, Partial Modified Appedix A) HOW TO PERFORM A LINEAR REGRESSION Cosider the followig data poits ad their graph (Table I ad Figure ): X Y 0 3 5 3 7 4 9 5 Table : Example Data

More information

Lecture 5. Materials Covered: Chapter 6 Suggested Exercises: 6.7, 6.9, 6.17, 6.20, 6.21, 6.41, 6.49, 6.52, 6.53, 6.62, 6.63.

Lecture 5. Materials Covered: Chapter 6 Suggested Exercises: 6.7, 6.9, 6.17, 6.20, 6.21, 6.41, 6.49, 6.52, 6.53, 6.62, 6.63. STT 315, Summer 006 Lecture 5 Materials Covered: Chapter 6 Suggested Exercises: 67, 69, 617, 60, 61, 641, 649, 65, 653, 66, 663 1 Defiitios Cofidece Iterval: A cofidece iterval is a iterval believed to

More information

Random Variables, Sampling and Estimation

Random Variables, Sampling and Estimation Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig

More information

THE KALMAN FILTER RAUL ROJAS

THE KALMAN FILTER RAUL ROJAS THE KALMAN FILTER RAUL ROJAS Abstract. This paper provides a getle itroductio to the Kalma filter, a umerical method that ca be used for sesor fusio or for calculatio of trajectories. First, we cosider

More information

Accuracy assessment methods and challenges

Accuracy assessment methods and challenges Accuracy assessmet methods ad challeges Giles M. Foody School of Geography Uiversity of Nottigham giles.foody@ottigham.ac.uk Backgroud Need for accuracy assessmet established. Cosiderable progress ow see

More information

Lecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting

Lecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would

More information

EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY

EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA, 016 MODULE : Statistical Iferece Time allowed: Three hours Cadidates should aswer FIVE questios. All questios carry equal marks. The umber

More information

CEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering

CEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio

More information

Inferential Statistics. Inference Process. Inferential Statistics and Probability a Holistic Approach. Inference Process.

Inferential Statistics. Inference Process. Inferential Statistics and Probability a Holistic Approach. Inference Process. Iferetial Statistics ad Probability a Holistic Approach Iferece Process Chapter 8 Poit Estimatio ad Cofidece Itervals This Course Material by Maurice Geraghty is licesed uder a Creative Commos Attributio-ShareAlike

More information

2 1. The r.s., of size n2, from population 2 will be. 2 and 2. 2) The two populations are independent. This implies that all of the n1 n2

2 1. The r.s., of size n2, from population 2 will be. 2 and 2. 2) The two populations are independent. This implies that all of the n1 n2 Chapter 8 Comparig Two Treatmets Iferece about Two Populatio Meas We wat to compare the meas of two populatios to see whether they differ. There are two situatios to cosider, as show i the followig examples:

More information

It is always the case that unions, intersections, complements, and set differences are preserved by the inverse image of a function.

It is always the case that unions, intersections, complements, and set differences are preserved by the inverse image of a function. MATH 532 Measurable Fuctios Dr. Neal, WKU Throughout, let ( X, F, µ) be a measure space ad let (!, F, P ) deote the special case of a probability space. We shall ow begi to study real-valued fuctios defied

More information

CS284A: Representations and Algorithms in Molecular Biology

CS284A: Representations and Algorithms in Molecular Biology CS284A: Represetatios ad Algorithms i Molecular Biology Scribe Notes o Lectures 3 & 4: Motif Discovery via Eumeratio & Motif Represetatio Usig Positio Weight Matrix Joshua Gervi Based o presetatios by

More information

Seunghee Ye Ma 8: Week 5 Oct 28

Seunghee Ye Ma 8: Week 5 Oct 28 Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value

More information

Statistical inference: example 1. Inferential Statistics

Statistical inference: example 1. Inferential Statistics Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either

More information

1 Introduction to reducing variance in Monte Carlo simulations

1 Introduction to reducing variance in Monte Carlo simulations Copyright c 010 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a ukow mea µ = E(X) of a distributio by

More information

Random Walks on Discrete and Continuous Circles. by Jeffrey S. Rosenthal School of Mathematics, University of Minnesota, Minneapolis, MN, U.S.A.

Random Walks on Discrete and Continuous Circles. by Jeffrey S. Rosenthal School of Mathematics, University of Minnesota, Minneapolis, MN, U.S.A. Radom Walks o Discrete ad Cotiuous Circles by Jeffrey S. Rosethal School of Mathematics, Uiversity of Miesota, Mieapolis, MN, U.S.A. 55455 (Appeared i Joural of Applied Probability 30 (1993), 780 789.)

More information

Sequences and Series of Functions

Sequences and Series of Functions Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges

More information

1 Review of Probability & Statistics

1 Review of Probability & Statistics 1 Review of Probability & Statistics a. I a group of 000 people, it has bee reported that there are: 61 smokers 670 over 5 960 people who imbibe (drik alcohol) 86 smokers who imbibe 90 imbibers over 5

More information

7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals

7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals 7-1 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7- Sectio 1. Samplig Distributio 7-3 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses

More information

Castiel, Supernatural, Season 6, Episode 18

Castiel, Supernatural, Season 6, Episode 18 13 Differetial Equatios the aswer to your questio ca best be epressed as a series of partial differetial equatios... Castiel, Superatural, Seaso 6, Episode 18 A differetial equatio is a mathematical equatio

More information

Lecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting

Lecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would

More information

Section 5.1 The Basics of Counting

Section 5.1 The Basics of Counting 1 Sectio 5.1 The Basics of Coutig Combiatorics, the study of arragemets of objects, is a importat part of discrete mathematics. I this chapter, we will lear basic techiques of coutig which has a lot of

More information

Statistical Analysis on Uncertainty for Autocorrelated Measurements and its Applications to Key Comparisons

Statistical Analysis on Uncertainty for Autocorrelated Measurements and its Applications to Key Comparisons Statistical Aalysis o Ucertaity for Autocorrelated Measuremets ad its Applicatios to Key Comparisos Nie Fa Zhag Natioal Istitute of Stadards ad Techology Gaithersburg, MD 0899, USA Outlies. Itroductio.

More information

Statistics 511 Additional Materials

Statistics 511 Additional Materials Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability

More information

Output Analysis (2, Chapters 10 &11 Law)

Output Analysis (2, Chapters 10 &11 Law) B. Maddah ENMG 6 Simulatio Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should be doe

More information

Chapter If n is odd, the median is the exact middle number If n is even, the median is the average of the two middle numbers

Chapter If n is odd, the median is the exact middle number If n is even, the median is the average of the two middle numbers Chapter 4 4-1 orth Seattle Commuity College BUS10 Busiess Statistics Chapter 4 Descriptive Statistics Summary Defiitios Cetral tedecy: The extet to which the data values group aroud a cetral value. Variatio:

More information

ECE 901 Lecture 12: Complexity Regularization and the Squared Loss

ECE 901 Lecture 12: Complexity Regularization and the Squared Loss ECE 90 Lecture : Complexity Regularizatio ad the Squared Loss R. Nowak 5/7/009 I the previous lectures we made use of the Cheroff/Hoeffdig bouds for our aalysis of classifier errors. Hoeffdig s iequality

More information

DISTRIBUTION LAW Okunev I.V.

DISTRIBUTION LAW Okunev I.V. 1 DISTRIBUTION LAW Okuev I.V. Distributio law belogs to a umber of the most complicated theoretical laws of mathematics. But it is also a very importat practical law. Nothig ca help uderstad complicated

More information

Probability, Expectation Value and Uncertainty

Probability, Expectation Value and Uncertainty Chapter 1 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such

More information

t distribution [34] : used to test a mean against an hypothesized value (H 0 : µ = µ 0 ) or the difference

t distribution [34] : used to test a mean against an hypothesized value (H 0 : µ = µ 0 ) or the difference EXST30 Backgroud material Page From the textbook The Statistical Sleuth Mea [0]: I your text the word mea deotes a populatio mea (µ) while the work average deotes a sample average ( ). Variace [0]: The

More information

Estimation for Complete Data

Estimation for Complete Data Estimatio for Complete Data complete data: there is o loss of iformatio durig study. complete idividual complete data= grouped data A complete idividual data is the oe i which the complete iformatio of

More information

The Random Walk For Dummies

The Random Walk For Dummies The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli

More information

x a x a Lecture 2 Series (See Chapter 1 in Boas)

x a x a Lecture 2 Series (See Chapter 1 in Boas) Lecture Series (See Chapter i Boas) A basic ad very powerful (if pedestria, recall we are lazy AD smart) way to solve ay differetial (or itegral) equatio is via a series expasio of the correspodig solutio

More information

Assessment of extreme discharges of the Vltava River in Prague

Assessment of extreme discharges of the Vltava River in Prague Flood Recovery, Iovatio ad Respose I 05 Assessmet of extreme discharges of the Vltava River i Prague M. Holický, K. Jug & M. Sýkora Kloker Istitute, Czech Techical Uiversity i Prague, Czech Republic Abstract

More information

Linear regression. Daniel Hsu (COMS 4771) (y i x T i β)2 2πσ. 2 2σ 2. 1 n. (x T i β y i ) 2. 1 ˆβ arg min. β R n d

Linear regression. Daniel Hsu (COMS 4771) (y i x T i β)2 2πσ. 2 2σ 2. 1 n. (x T i β y i ) 2. 1 ˆβ arg min. β R n d Liear regressio Daiel Hsu (COMS 477) Maximum likelihood estimatio Oe of the simplest liear regressio models is the followig: (X, Y ),..., (X, Y ), (X, Y ) are iid radom pairs takig values i R d R, ad Y

More information

AP Statistics Review Ch. 8

AP Statistics Review Ch. 8 AP Statistics Review Ch. 8 Name 1. Each figure below displays the samplig distributio of a statistic used to estimate a parameter. The true value of the populatio parameter is marked o each samplig distributio.

More information

The Poisson Distribution

The Poisson Distribution MATH 382 The Poisso Distributio Dr. Neal, WKU Oe of the importat distributios i probabilistic modelig is the Poisso Process X t that couts the umber of occurreces over a period of t uits of time. This

More information

Discrete probability distributions

Discrete probability distributions Discrete probability distributios I the chapter o probability we used the classical method to calculate the probability of various values of a radom variable. I some cases, however, we may be able to develop

More information

Optimally Sparse SVMs

Optimally Sparse SVMs A. Proof of Lemma 3. We here prove a lower boud o the umber of support vectors to achieve geeralizatio bouds of the form which we cosider. Importatly, this result holds ot oly for liear classifiers, but

More information

Module 1 Fundamentals in statistics

Module 1 Fundamentals in statistics Normal Distributio Repeated observatios that differ because of experimetal error ofte vary about some cetral value i a roughly symmetrical distributio i which small deviatios occur much more frequetly

More information

Algebra of Least Squares

Algebra of Least Squares October 19, 2018 Algebra of Least Squares Geometry of Least Squares Recall that out data is like a table [Y X] where Y collects observatios o the depedet variable Y ad X collects observatios o the k-dimesioal

More information

April 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE

April 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE April 18, 2017 CONFIDENCE INTERVALS AND HYPOTHESIS TESTING, UNDERGRADUATE MATH 526 STYLE TERRY SOO Abstract These otes are adapted from whe I taught Math 526 ad meat to give a quick itroductio to cofidece

More information

STA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:

STA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to: STA 2023 Module 10 Comparig Two Proportios Learig Objectives Upo completig this module, you should be able to: 1. Perform large-sample ifereces (hypothesis test ad cofidece itervals) to compare two populatio

More information

Error & Uncertainty. Error. More on errors. Uncertainty. Page # The error is the difference between a TRUE value, x, and a MEASURED value, x i :

Error & Uncertainty. Error. More on errors. Uncertainty. Page # The error is the difference between a TRUE value, x, and a MEASURED value, x i : Error Error & Ucertaity The error is the differece betwee a TRUE value,, ad a MEASURED value, i : E = i There is o error-free measuremet. The sigificace of a measuremet caot be judged uless the associate

More information

Math 257: Finite difference methods

Math 257: Finite difference methods Math 257: Fiite differece methods 1 Fiite Differeces Remember the defiitio of a derivative f f(x + ) f(x) (x) = lim 0 Also recall Taylor s formula: (1) f(x + ) = f(x) + f (x) + 2 f (x) + 3 f (3) (x) +...

More information

Lecture 1 Probability and Statistics

Lecture 1 Probability and Statistics Wikipedia: Lecture 1 Probability ad Statistics Bejami Disraeli, British statesma ad literary figure (1804 1881): There are three kids of lies: lies, damed lies, ad statistics. popularized i US by Mark

More information

II. Descriptive Statistics D. Linear Correlation and Regression. 1. Linear Correlation

II. Descriptive Statistics D. Linear Correlation and Regression. 1. Linear Correlation II. Descriptive Statistics D. Liear Correlatio ad Regressio I this sectio Liear Correlatio Cause ad Effect Liear Regressio 1. Liear Correlatio Quatifyig Liear Correlatio The Pearso product-momet correlatio

More information

CSE 191, Class Note 05: Counting Methods Computer Sci & Eng Dept SUNY Buffalo

CSE 191, Class Note 05: Counting Methods Computer Sci & Eng Dept SUNY Buffalo Coutig Methods CSE 191, Class Note 05: Coutig Methods Computer Sci & Eg Dept SUNY Buffalo c Xi He (Uiversity at Buffalo CSE 191 Discrete Structures 1 / 48 Need for Coutig The problem of coutig the umber

More information

Testing Statistical Hypotheses for Compare. Means with Vague Data

Testing Statistical Hypotheses for Compare. Means with Vague Data Iteratioal Mathematical Forum 5 o. 3 65-6 Testig Statistical Hypotheses for Compare Meas with Vague Data E. Baloui Jamkhaeh ad A. adi Ghara Departmet of Statistics Islamic Azad iversity Ghaemshahr Brach

More information

10-701/ Machine Learning Mid-term Exam Solution

10-701/ Machine Learning Mid-term Exam Solution 0-70/5-78 Machie Learig Mid-term Exam Solutio Your Name: Your Adrew ID: True or False (Give oe setece explaatio) (20%). (F) For a cotiuous radom variable x ad its probability distributio fuctio p(x), it

More information

EECS564 Estimation, Filtering, and Detection Hwk 2 Solns. Winter p θ (z) = (2θz + 1 θ), 0 z 1

EECS564 Estimation, Filtering, and Detection Hwk 2 Solns. Winter p θ (z) = (2θz + 1 θ), 0 z 1 EECS564 Estimatio, Filterig, ad Detectio Hwk 2 Sols. Witer 25 4. Let Z be a sigle observatio havig desity fuctio where. p (z) = (2z + ), z (a) Assumig that is a oradom parameter, fid ad plot the maximum

More information

ECON 3150/4150, Spring term Lecture 3

ECON 3150/4150, Spring term Lecture 3 Itroductio Fidig the best fit by regressio Residuals ad R-sq Regressio ad causality Summary ad ext step ECON 3150/4150, Sprig term 2014. Lecture 3 Ragar Nymoe Uiversity of Oslo 21 Jauary 2014 1 / 30 Itroductio

More information

Polynomial Functions and Their Graphs

Polynomial Functions and Their Graphs Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively

More information

µ and π p i.e. Point Estimation x And, more generally, the population proportion is approximately equal to a sample proportion

µ and π p i.e. Point Estimation x And, more generally, the population proportion is approximately equal to a sample proportion Poit Estimatio Poit estimatio is the rather simplistic (ad obvious) process of usig the kow value of a sample statistic as a approximatio to the ukow value of a populatio parameter. So we could for example

More information

Department of Mathematics

Department of Mathematics Departmet of Mathematics Ma 3/103 KC Border Itroductio to Probability ad Statistics Witer 2017 Lecture 19: Estimatio II Relevat textbook passages: Larse Marx [1]: Sectios 5.2 5.7 19.1 The method of momets

More information