Quantitative analysis requires : sound knowledge of chemistry : possibility of interferences WHY do we need to use STATISTICS in Anal. Chem.?

Size: px
Start display at page:

Download "Quantitative analysis requires : sound knowledge of chemistry : possibility of interferences WHY do we need to use STATISTICS in Anal. Chem.?"

Transcription

1 Ch 4. Statstcs 4.1 Quattatve aalyss requres : soud kowledge of chemstry : possblty of terfereces WHY do we eed to use STATISTICS Aal. Chem.? ucertaty ests. wll we accept ucertaty always? f ot, from how wll we dsregard the data? by statstcal treatmet Radom Evets follows Gaussa Dstrbuto 4-1 Gaussa Dstrbuto 4. test of the lfe tmes of 4768 lght bulbs 1) mea value & stadard devato * mea : : or average

2 4-1 Gaussa Dstrbuto (Cot.) 4.3 * stadard dev. : s : measures how closely the data are clustered aroud the mea s ( ) 1-1 : degrees of freedom for a fte set of data: s (mea) (mu, (sgma, or popular mea) popular stadard devato) : var ace 4.4

3 4-1 Gaussa Dstrbuto (Cot.) 4.5 ) std.dev. & probablty Gaussa curve tells the broadess of Gaussa curve a gaussa curve area uder 1 = 68.3 % = 95.5 % 3 = 99.7 % 1 ( ) ep( y ) 4-1 Gaussa Dstrbuto (Cot.) 4.6 3) std.dev. of mea more measuremets more cofdet o average 1 (early the true value) ucertaty decreases by : = umber of meas. stadard devato of mea = : s = std.dev. * relatve stadard devato = (RSD) or to percetage = s = C.V. precso of mea = average devato of mea = d ( d ) s s 100

4 4- Cofdece Itervals 4.7 1) cofdece terval : a epresso statg that true mea,, s lkely to le wth a certa dstace our measuremets, s (stead of, ) True mea () s lkely to le wth a certa rage from Cofdece tervals t s 4.8 E. The cotet of carbohydrate a glycoprote (a prote wth sugars attached to t) s determed to be 1.6, 11.9, 13.0, 1.7, ad 1.5 g per 100 g of prote replcated aalyss. Fd the 50% ad 90% cofdece tervals for the carbohydrate cotet. mea = 1.5, std = 0.4

5 4-3 Comparso of meas wth Studet's t (from dfferet measuremets) 4.9 : tool for epressg cofdece terval for comparg results from other epermetal tech. Normally, 95% cofdece level : Two results do ot dffer from each other IF there s 95% chace that our cocluso s correct. Case 1. t test : measured result wth kow value 4.10 : whe we test a ew aalytcal method, we wat to see f t agrees to a kow value. e) N cotet; kow value : % (from std. Materal) measured value : 0.039, 0.03, , % The 95% cofdece terval? ths terval does't cover , thus, measured value are dfferet from kow val. Not wth the radom error boudary. (t mples there ests systematc errors)

6 1. <t-test> You are developg a procedure for determg traces of copper 4.11 bologcal materals usg a wet dgesto followed by measuremets by atomc absorpto spectrophotometry. I order to test the valdty of the method, you obta a NIST orchard leaves stadard referece materal ad aalyze ths materal. Fve replcas are sampled ad aalyzed, ad the mea of the results s foud to be ppm wth a stadard devato of 0.7ppm. The lsted value s 11.7ppm. Does your method gves a statstcally correct value at the 95% cofdece level? Case. t test: comparg replcate measuremets (test of two sets of measuremets) : test the two techques are statstcally the SAME or NOT 4.1 for two sets of data, 1, measuremets t 1 S pooled 1 1 S pooled ( 1) ( j ) s1 ( 1 1) s ( 1 1 1) If t cal > t table (wth 95%) ths dfferece s sgfcat (out of radom error rage) there ests systematc error

7 4.13 E) The average mass of troge from ar Table 4-3 s = g, wth a stadard devato of s 1 = , (for 1 =7 measuremets). The average mass from chemcal sources s =.9947 g, wth a stadard devato of s = (for =8 measuremets). <t-test> A ew gravmetrc method s developed for ro (II) whch the ro 4.14 s precptated crystalle form wth a orgaocarbo "cage" compoud. The accuracy of the method s checked by aalyzg the ro a ore sample ad comparg wth the results usg the stadard precptato wth ammoa ad weghg of FeO3. The results, reported as % Fe for each aalyss, were as follows. Test method Referece Method 0.10% 18.89% =19.65% =19.4% Is there a dfferece betwee the two methods?

8 Case 3; Comparg dvdual dffereces 4.15 Two dfferet methods o several dfferet samples (o duplcato) Cholesterol cotet (g/l) Plasma sample Method A Method B Dfferece (d ) d =+0.06 t cal d S d s d ( d d ) 1 Is my red blood cell cout hgh today? 4.16 Red cell couts o fve ormal days : 5.1, 5.3, 4.8, 5.4, ad cells/l =5.16 s=0.3 Today s value = cells/l t cal today' s cout S d What s the probablty of fdg t=4.8 for 4 degrees of freedom? See table 4.: at 4 degrees of freedom, 4.8 les betwee 98 & 99% There s less tha a % probablty of observg a cout of cells/l o ormal days. reasoable to coclude that today s cout s elevated.

9 4-4 Comparso of st.dev. wth the F test 4.17 F test ---- check two std.devs are sgfcatly dfferet each other. S F calc If F S calc > F table the sgfcat Grubbs test for a outler 4.18 durg measuremets of mass lost of zc, we eed to dscard some questoable data 10., 10.8, 11.6, 9.9, 9.4, 7.8, 10.0, 9., 11.3, 9.5, 10.6, 11.6 If G calc > G tab, the rejected.

10 4-7. Method of Least Squares Fdg the BEST STRAIGHT LINE ; correlato betwee data pots 1) Method of Least Squares y = m + b m: slope, b: y-tercept each data --- (, y ) vertcal devato = d = y -y = y -(m + b) 4-7. Method of Least Squares 4.0 we wat to MINIMIZE d (whether postve or eg.) -- drect summato of each d? o good method of mamum lkelhood : Assume a gaussa dstrbuto wth std.dev.. for the observatos about the actual value y( ) at = the probablty P P 1 ep 1 y y mamze the probablty? mmze the sum the epoetal

11 4-7. Method of Least Squares 4.1 d d = (y -y) = (y -m -b) mmzg (assume ) m b METHOD OF LEAST SQUARES m b ( y ) ( ) ( ) ( ) y (y) ( ) ( ) y 4-7. Method of Least Squares 4. ) How relable are least-squares parameters? estmate UNCERTAINTY slope & tercept std. dev. of y (d ) s y y degrees of freedom ( - ) m b ( ( y ) ( y ( ) ( ) ) )

12 4-8. Calbrato Curves 4.3 Std. Soluto : solutos wth kow cocetratos How to buld calbrato? 1. prepare a seres of std. Solutos (varyg coc.) measure absorbace.. subtract the absorbace of blak soluto 4-8. Calbrato Curves Plot the absorbaces vs. Cocetrato the do least squares.

13 4-8. Calbrato Curves 4.5 Ucertaty Propagato Calbrato curve m : slope Depeds o # of calbrato pots. Lowest error data from the ceter of calbrato Homework F, 13, 14, 16, 0, 33, Addtoal Problems Set

14 Addtoal Problems Set The followg replcate calcum determatos o a blood sample usg AAS ad a ew colormetrc method were reported. Is there a sgfcat dfferece the precso of the two methods? AAS (mg/dl) 10.9, 10.1, 10.6, 11., 9.7, 10.0 Colormetrc (mg/dl) 9., 10.5,9.7, 11.5,11.6, 9.3, 10.1, 11.. Studets measured the cocetrato of HCl a soluto by ttratos usg dfferet dcators to fd the ed pot. Is the dfferece betwee dcators 1 ad sgfcat at the 95% cofdece level? Aswer the same questo for dcator ad Idcator 1. Bromothymol blue. Methyl red 3. Bromocresol gree Mea HCl cocetrato (M) (+std.dev.) Number of Measuremets

15 A Stadard Referece Materal s certfed to cota 94.6 ppm of a orgac cotamat sol. Your aalyss gves values of 98.6, 98.4, 97., 94.6, ad 96. ppm. Do your results dffer from the epected results at the 95% cofdece level? If you made oe more measuremet ad foud 94.5, would your cocluso chage?

ESS Line Fitting

ESS Line Fitting ESS 5 014 17. Le Fttg A very commo problem data aalyss s lookg for relatoshpetwee dfferet parameters ad fttg les or surfaces to data. The smplest example s fttg a straght le ad we wll dscuss that here

More information

Lecture 7. Confidence Intervals and Hypothesis Tests in the Simple CLR Model

Lecture 7. Confidence Intervals and Hypothesis Tests in the Simple CLR Model Lecture 7. Cofdece Itervals ad Hypothess Tests the Smple CLR Model I lecture 6 we troduced the Classcal Lear Regresso (CLR) model that s the radom expermet of whch the data Y,,, K, are the outcomes. The

More information

Analyzing Two-Dimensional Data. Analyzing Two-Dimensional Data

Analyzing Two-Dimensional Data. Analyzing Two-Dimensional Data /7/06 Aalzg Two-Dmesoal Data The most commo aaltcal measuremets volve the determato of a ukow cocetrato based o the respose of a aaltcal procedure (usuall strumetal). Such a measuremet requres calbrato,

More information

Mean is only appropriate for interval or ratio scales, not ordinal or nominal.

Mean is only appropriate for interval or ratio scales, not ordinal or nominal. Mea Same as ordary average Sum all the data values ad dvde by the sample sze. x = ( x + x +... + x Usg summato otato, we wrte ths as x = x = x = = ) x Mea s oly approprate for terval or rato scales, ot

More information

Continuous Distributions

Continuous Distributions 7//3 Cotuous Dstrbutos Radom Varables of the Cotuous Type Desty Curve Percet Desty fucto, f (x) A smooth curve that ft the dstrbuto 3 4 5 6 7 8 9 Test scores Desty Curve Percet Probablty Desty Fucto, f

More information

Chapter 8: Statistical Analysis of Simulated Data

Chapter 8: Statistical Analysis of Simulated Data Marquette Uversty MSCS600 Chapter 8: Statstcal Aalyss of Smulated Data Dael B. Rowe, Ph.D. Departmet of Mathematcs, Statstcs, ad Computer Scece Copyrght 08 by Marquette Uversty MSCS600 Ageda 8. The Sample

More information

best estimate (mean) for X uncertainty or error in the measurement (systematic, random or statistical) best

best estimate (mean) for X uncertainty or error in the measurement (systematic, random or statistical) best Error Aalyss Preamble Wheever a measuremet s made, the result followg from that measuremet s always subject to ucertaty The ucertaty ca be reduced by makg several measuremets of the same quatty or by mprovg

More information

STA 108 Applied Linear Models: Regression Analysis Spring Solution for Homework #1

STA 108 Applied Linear Models: Regression Analysis Spring Solution for Homework #1 STA 08 Appled Lear Models: Regresso Aalyss Sprg 0 Soluto for Homework #. Let Y the dollar cost per year, X the umber of vsts per year. The the mathematcal relato betwee X ad Y s: Y 300 + X. Ths s a fuctoal

More information

Chapter 8. Inferences about More Than Two Population Central Values

Chapter 8. Inferences about More Than Two Population Central Values Chapter 8. Ifereces about More Tha Two Populato Cetral Values Case tudy: Effect of Tmg of the Treatmet of Port-We tas wth Lasers ) To vestgate whether treatmet at a youg age would yeld better results tha

More information

Lecture Notes Types of economic variables

Lecture Notes Types of economic variables Lecture Notes 3 1. Types of ecoomc varables () Cotuous varable takes o a cotuum the sample space, such as all pots o a le or all real umbers Example: GDP, Polluto cocetrato, etc. () Dscrete varables fte

More information

A Study of the Reproducibility of Measurements with HUR Leg Extension/Curl Research Line

A Study of the Reproducibility of Measurements with HUR Leg Extension/Curl Research Line HUR Techcal Report 000--9 verso.05 / Frak Borg (borgbros@ett.f) A Study of the Reproducblty of Measuremets wth HUR Leg Eteso/Curl Research Le A mportat property of measuremets s that the results should

More information

: At least two means differ SST

: At least two means differ SST Formula Card for Eam 3 STA33 ANOVA F-Test: Completely Radomzed Desg ( total umber of observatos, k = Number of treatmets,& T = total for treatmet ) Step : Epress the Clam Step : The ypotheses: :... 0 A

More information

ENGI 3423 Simple Linear Regression Page 12-01

ENGI 3423 Simple Linear Regression Page 12-01 ENGI 343 mple Lear Regresso Page - mple Lear Regresso ometmes a expermet s set up where the expermeter has cotrol over the values of oe or more varables X ad measures the resultg values of aother varable

More information

Lecture 1 Review of Fundamental Statistical Concepts

Lecture 1 Review of Fundamental Statistical Concepts Lecture Revew of Fudametal Statstcal Cocepts Measures of Cetral Tedecy ad Dsperso A word about otato for ths class: Idvduals a populato are desgated, where the dex rages from to N, ad N s the total umber

More information

Lecture 3. Sampling, sampling distributions, and parameter estimation

Lecture 3. Sampling, sampling distributions, and parameter estimation Lecture 3 Samplg, samplg dstrbutos, ad parameter estmato Samplg Defto Populato s defed as the collecto of all the possble observatos of terest. The collecto of observatos we take from the populato s called

More information

Simple Linear Regression

Simple Linear Regression Statstcal Methods I (EST 75) Page 139 Smple Lear Regresso Smple regresso applcatos are used to ft a model descrbg a lear relatoshp betwee two varables. The aspects of least squares regresso ad correlato

More information

Econometric Methods. Review of Estimation

Econometric Methods. Review of Estimation Ecoometrc Methods Revew of Estmato Estmatg the populato mea Radom samplg Pot ad terval estmators Lear estmators Ubased estmators Lear Ubased Estmators (LUEs) Effcecy (mmum varace) ad Best Lear Ubased Estmators

More information

= 1. UCLA STAT 13 Introduction to Statistical Methods for the Life and Health Sciences. Parameters and Statistics. Measures of Centrality

= 1. UCLA STAT 13 Introduction to Statistical Methods for the Life and Health Sciences. Parameters and Statistics. Measures of Centrality UCLA STAT Itroducto to Statstcal Methods for the Lfe ad Health Sceces Istructor: Ivo Dov, Asst. Prof. of Statstcs ad Neurology Teachg Assstats: Fred Phoa, Krste Johso, Mg Zheg & Matlda Hseh Uversty of

More information

Simulation Output Analysis

Simulation Output Analysis Smulato Output Aalyss Summary Examples Parameter Estmato Sample Mea ad Varace Pot ad Iterval Estmato ermatg ad o-ermatg Smulato Mea Square Errors Example: Sgle Server Queueg System x(t) S 4 S 4 S 3 S 5

More information

STA302/1001-Fall 2008 Midterm Test October 21, 2008

STA302/1001-Fall 2008 Midterm Test October 21, 2008 STA3/-Fall 8 Mdterm Test October, 8 Last Name: Frst Name: Studet Number: Erolled (Crcle oe) STA3 STA INSTRUCTIONS Tme allowed: hour 45 mutes Ads allowed: A o-programmable calculator A table of values from

More information

CHAPTER VI Statistical Analysis of Experimental Data

CHAPTER VI Statistical Analysis of Experimental Data Chapter VI Statstcal Aalyss of Expermetal Data CHAPTER VI Statstcal Aalyss of Expermetal Data Measuremets do ot lead to a uque value. Ths s a result of the multtude of errors (maly radom errors) that ca

More information

Summary of the lecture in Biostatistics

Summary of the lecture in Biostatistics Summary of the lecture Bostatstcs Probablty Desty Fucto For a cotuos radom varable, a probablty desty fucto s a fucto such that: 0 dx a b) b a dx A probablty desty fucto provdes a smple descrpto of the

More information

Measures of Dispersion

Measures of Dispersion Chapter 8 Measures of Dsperso Defto of Measures of Dsperso (page 31) A measure of dsperso s a descrptve summary measure that helps us characterze the data set terms of how vared the observatos are from

More information

ENGI 4421 Propagation of Error Page 8-01

ENGI 4421 Propagation of Error Page 8-01 ENGI 441 Propagato of Error Page 8-01 Propagato of Error [Navd Chapter 3; ot Devore] Ay realstc measuremet procedure cotas error. Ay calculatos based o that measuremet wll therefore also cota a error.

More information

MEASURES OF DISPERSION

MEASURES OF DISPERSION MEASURES OF DISPERSION Measure of Cetral Tedecy: Measures of Cetral Tedecy ad Dsperso ) Mathematcal Average: a) Arthmetc mea (A.M.) b) Geometrc mea (G.M.) c) Harmoc mea (H.M.) ) Averages of Posto: a) Meda

More information

12.2 Estimating Model parameters Assumptions: ox and y are related according to the simple linear regression model

12.2 Estimating Model parameters Assumptions: ox and y are related according to the simple linear regression model 1. Estmatg Model parameters Assumptos: ox ad y are related accordg to the smple lear regresso model (The lear regresso model s the model that says that x ad y are related a lear fasho, but the observed

More information

Chapter 13, Part A Analysis of Variance and Experimental Design. Introduction to Analysis of Variance. Introduction to Analysis of Variance

Chapter 13, Part A Analysis of Variance and Experimental Design. Introduction to Analysis of Variance. Introduction to Analysis of Variance Chapter, Part A Aalyss of Varace ad Epermetal Desg Itroducto to Aalyss of Varace Aalyss of Varace: Testg for the Equalty of Populato Meas Multple Comparso Procedures Itroducto to Aalyss of Varace Aalyss

More information

b. There appears to be a positive relationship between X and Y; that is, as X increases, so does Y.

b. There appears to be a positive relationship between X and Y; that is, as X increases, so does Y. .46. a. The frst varable (X) s the frst umber the par ad s plotted o the horzotal axs, whle the secod varable (Y) s the secod umber the par ad s plotted o the vertcal axs. The scatterplot s show the fgure

More information

Class 13,14 June 17, 19, 2015

Class 13,14 June 17, 19, 2015 Class 3,4 Jue 7, 9, 05 Pla for Class3,4:. Samplg dstrbuto of sample mea. The Cetral Lmt Theorem (CLT). Cofdece terval for ukow mea.. Samplg Dstrbuto for Sample mea. Methods used are based o CLT ( Cetral

More information

2.28 The Wall Street Journal is probably referring to the average number of cubes used per glass measured for some population that they have chosen.

2.28 The Wall Street Journal is probably referring to the average number of cubes used per glass measured for some population that they have chosen. .5 x 54.5 a. x 7. 786 7 b. The raked observatos are: 7.4, 7.5, 7.7, 7.8, 7.9, 8.0, 8.. Sce the sample sze 7 s odd, the meda s the (+)/ 4 th raked observato, or meda 7.8 c. The cosumer would more lkely

More information

Section l h l Stem=Tens. 8l Leaf=Ones. 8h l 03. 9h 58

Section l h l Stem=Tens. 8l Leaf=Ones. 8h l 03. 9h 58 Secto.. 6l 34 6h 667899 7l 44 7h Stem=Tes 8l 344 Leaf=Oes 8h 5557899 9l 3 9h 58 Ths dsplay brgs out the gap the data: There are o scores the hgh 7's. 6. a. beams cylders 9 5 8 88533 6 6 98877643 7 488

More information

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Exam: ECON430 Statstcs Date of exam: Frday, December 8, 07 Grades are gve: Jauary 4, 08 Tme for exam: 0900 am 00 oo The problem set covers 5 pages Resources allowed:

More information

f f... f 1 n n (ii) Median : It is the value of the middle-most observation(s).

f f... f 1 n n (ii) Median : It is the value of the middle-most observation(s). CHAPTER STATISTICS Pots to Remember :. Facts or fgures, collected wth a defte pupose, are called Data.. Statstcs s the area of study dealg wth the collecto, presetato, aalyss ad terpretato of data.. The

More information

Chapter 11 The Analysis of Variance

Chapter 11 The Analysis of Variance Chapter The Aalyss of Varace. Oe Factor Aalyss of Varace. Radomzed Bloc Desgs (ot for ths course) NIPRL . Oe Factor Aalyss of Varace.. Oe Factor Layouts (/4) Suppose that a expermeter s terested populatos

More information

Random Variables and Probability Distributions

Random Variables and Probability Distributions Radom Varables ad Probablty Dstrbutos * If X : S R s a dscrete radom varable wth rage {x, x, x 3,. } the r = P (X = xr ) = * Let X : S R be a dscrete radom varable wth rage {x, x, x 3,.}.If x r P(X = x

More information

Example: Multiple linear regression. Least squares regression. Repetition: Simple linear regression. Tron Anders Moger

Example: Multiple linear regression. Least squares regression. Repetition: Simple linear regression. Tron Anders Moger Example: Multple lear regresso 5000,00 4000,00 Tro Aders Moger 0.0.007 brthweght 3000,00 000,00 000,00 0,00 50,00 00,00 50,00 00,00 50,00 weght pouds Repetto: Smple lear regresso We defe a model Y = β0

More information

( ) = ( ) ( ) Chapter 13 Asymptotic Theory and Stochastic Regressors. Stochastic regressors model

( ) = ( ) ( ) Chapter 13 Asymptotic Theory and Stochastic Regressors. Stochastic regressors model Chapter 3 Asmptotc Theor ad Stochastc Regressors The ature of eplaator varable s assumed to be o-stochastc or fed repeated samples a regresso aalss Such a assumpto s approprate for those epermets whch

More information

Simple Linear Regression

Simple Linear Regression Correlato ad Smple Lear Regresso Berl Che Departmet of Computer Scece & Iformato Egeerg Natoal Tawa Normal Uversty Referece:. W. Navd. Statstcs for Egeerg ad Scetsts. Chapter 7 (7.-7.3) & Teachg Materal

More information

Median as a Weighted Arithmetic Mean of All Sample Observations

Median as a Weighted Arithmetic Mean of All Sample Observations Meda as a Weghted Arthmetc Mea of All Sample Observatos SK Mshra Dept. of Ecoomcs NEHU, Shllog (Ida). Itroducto: Iumerably may textbooks Statstcs explctly meto that oe of the weakesses (or propertes) of

More information

Multiple Regression. More than 2 variables! Grade on Final. Multiple Regression 11/21/2012. Exam 2 Grades. Exam 2 Re-grades

Multiple Regression. More than 2 variables! Grade on Final. Multiple Regression 11/21/2012. Exam 2 Grades. Exam 2 Re-grades STAT 101 Dr. Kar Lock Morga 11/20/12 Exam 2 Grades Multple Regresso SECTIONS 9.2, 10.1, 10.2 Multple explaatory varables (10.1) Parttog varablty R 2, ANOVA (9.2) Codtos resdual plot (10.2) Trasformatos

More information

Objectives of Multiple Regression

Objectives of Multiple Regression Obectves of Multple Regresso Establsh the lear equato that best predcts values of a depedet varable Y usg more tha oe eplaator varable from a large set of potetal predctors {,,... k }. Fd that subset of

More information

Evaluation of uncertainty in measurements

Evaluation of uncertainty in measurements Evaluato of ucertaty measuremets Laboratory of Physcs I Faculty of Physcs Warsaw Uversty of Techology Warszawa, 05 Itroducto The am of the measuremet s to determe the measured value. Thus, the measuremet

More information

Chapter 13 Student Lecture Notes 13-1

Chapter 13 Student Lecture Notes 13-1 Chapter 3 Studet Lecture Notes 3- Basc Busess Statstcs (9 th Edto) Chapter 3 Smple Lear Regresso 4 Pretce-Hall, Ic. Chap 3- Chapter Topcs Types of Regresso Models Determg the Smple Lear Regresso Equato

More information

Multiple Choice Test. Chapter Adequacy of Models for Regression

Multiple Choice Test. Chapter Adequacy of Models for Regression Multple Choce Test Chapter 06.0 Adequac of Models for Regresso. For a lear regresso model to be cosdered adequate, the percetage of scaled resduals that eed to be the rage [-,] s greater tha or equal to

More information

Correlation and Simple Linear Regression

Correlation and Simple Linear Regression Correlato ad Smple Lear Regresso Berl Che Departmet of Computer Scece & Iformato Egeerg Natoal Tawa Normal Uverst Referece:. W. Navd. Statstcs for Egeerg ad Scetsts. Chapter 7 (7.-7.3) & Teachg Materal

More information

Statistics Descriptive and Inferential Statistics. Instructor: Daisuke Nagakura

Statistics Descriptive and Inferential Statistics. Instructor: Daisuke Nagakura Statstcs Descrptve ad Iferetal Statstcs Istructor: Dasuke Nagakura (agakura@z7.keo.jp) 1 Today s topc Today, I talk about two categores of statstcal aalyses, descrptve statstcs ad feretal statstcs, ad

More information

1. The weight of six Golden Retrievers is 66, 61, 70, 67, 92 and 66 pounds. The weight of six Labrador Retrievers is 54, 60, 72, 78, 84 and 67.

1. The weight of six Golden Retrievers is 66, 61, 70, 67, 92 and 66 pounds. The weight of six Labrador Retrievers is 54, 60, 72, 78, 84 and 67. Ecoomcs 3 Itroducto to Ecoometrcs Sprg 004 Professor Dobk Name Studet ID Frst Mdterm Exam You must aswer all the questos. The exam s closed book ad closed otes. You may use your calculators but please

More information

is the score of the 1 st student, x

is the score of the 1 st student, x 8 Chapter Collectg, Dsplayg, ad Aalyzg your Data. Descrptve Statstcs Sectos explaed how to choose a sample, how to collect ad orgaze data from the sample, ad how to dsplay your data. I ths secto, you wll

More information

ENGI 4421 Joint Probability Distributions Page Joint Probability Distributions [Navidi sections 2.5 and 2.6; Devore sections

ENGI 4421 Joint Probability Distributions Page Joint Probability Distributions [Navidi sections 2.5 and 2.6; Devore sections ENGI 441 Jot Probablty Dstrbutos Page 7-01 Jot Probablty Dstrbutos [Navd sectos.5 ad.6; Devore sectos 5.1-5.] The jot probablty mass fucto of two dscrete radom quattes, s, P ad p x y x y The margal probablty

More information

Module 7. Lecture 7: Statistical parameter estimation

Module 7. Lecture 7: Statistical parameter estimation Lecture 7: Statstcal parameter estmato Parameter Estmato Methods of Parameter Estmato 1) Method of Matchg Pots ) Method of Momets 3) Mamum Lkelhood method Populato Parameter Sample Parameter Ubased estmato

More information

Chapter Statistics Background of Regression Analysis

Chapter Statistics Background of Regression Analysis Chapter 06.0 Statstcs Backgroud of Regresso Aalyss After readg ths chapter, you should be able to:. revew the statstcs backgroud eeded for learg regresso, ad. kow a bref hstory of regresso. Revew of Statstcal

More information

Chapter Business Statistics: A First Course Fifth Edition. Learning Objectives. Correlation vs. Regression. In this chapter, you learn:

Chapter Business Statistics: A First Course Fifth Edition. Learning Objectives. Correlation vs. Regression. In this chapter, you learn: Chapter 3 3- Busess Statstcs: A Frst Course Ffth Edto Chapter 2 Correlato ad Smple Lear Regresso Busess Statstcs: A Frst Course, 5e 29 Pretce-Hall, Ic. Chap 2- Learg Objectves I ths chapter, you lear:

More information

Chapter 5 Properties of a Random Sample

Chapter 5 Properties of a Random Sample Lecture 6 o BST 63: Statstcal Theory I Ku Zhag, /0/008 Revew for the prevous lecture Cocepts: t-dstrbuto, F-dstrbuto Theorems: Dstrbutos of sample mea ad sample varace, relatoshp betwee sample mea ad sample

More information

STATISTICAL INFERENCE

STATISTICAL INFERENCE (STATISTICS) STATISTICAL INFERENCE COMPLEMENTARY COURSE B.Sc. MATHEMATICS III SEMESTER ( Admsso) UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CALICUT UNIVERSITY P.O., MALAPPURAM, KERALA, INDIA -

More information

IFYMB002 Mathematics Business Appendix C Formula Booklet

IFYMB002 Mathematics Business Appendix C Formula Booklet Iteratoal Foudato Year (IFY IFYMB00 Mathematcs Busess Apped C Formula Booklet Related Documet: IFY Mathematcs Busess Syllabus 07/8 IFYMB00 Maths Busess Apped C Formula Booklet Cotets lease ote that the

More information

THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA

THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA THE ROYAL STATISTICAL SOCIETY EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA PAPER II STATISTICAL THEORY & METHODS The Socety provdes these solutos to assst caddates preparg for the examatos future years ad for

More information

SPECIAL CONSIDERATIONS FOR VOLUMETRIC Z-TEST FOR PROPORTIONS

SPECIAL CONSIDERATIONS FOR VOLUMETRIC Z-TEST FOR PROPORTIONS SPECIAL CONSIDERAIONS FOR VOLUMERIC Z-ES FOR PROPORIONS Oe s stctve reacto to the questo of whether two percetages are sgfcatly dfferet from each other s to treat them as f they were proportos whch the

More information

Line Fitting and Regression

Line Fitting and Regression Marquette Uverst MSCS6 Le Fttg ad Regresso Dael B. Rowe, Ph.D. Professor Departmet of Mathematcs, Statstcs, ad Computer Scece Coprght 8 b Marquette Uverst Least Squares Regresso MSCS6 For LSR we have pots

More information

C. Statistics. X = n geometric the n th root of the product of numerical data ln X GM = or ln GM = X 2. X n X 1

C. Statistics. X = n geometric the n th root of the product of numerical data ln X GM = or ln GM = X 2. X n X 1 C. Statstcs a. Descrbe the stages the desg of a clcal tral, takg to accout the: research questos ad hypothess, lterature revew, statstcal advce, choce of study protocol, ethcal ssues, data collecto ad

More information

Statistics MINITAB - Lab 5

Statistics MINITAB - Lab 5 Statstcs 10010 MINITAB - Lab 5 PART I: The Correlato Coeffcet Qute ofte statstcs we are preseted wth data that suggests that a lear relatoshp exsts betwee two varables. For example the plot below s of

More information

Module 7: Probability and Statistics

Module 7: Probability and Statistics Lecture 4: Goodess of ft tests. Itroducto Module 7: Probablty ad Statstcs I the prevous two lectures, the cocepts, steps ad applcatos of Hypotheses testg were dscussed. Hypotheses testg may be used to

More information

STA 105-M BASIC STATISTICS (This is a multiple choice paper.)

STA 105-M BASIC STATISTICS (This is a multiple choice paper.) DCDM BUSINESS SCHOOL September Mock Eamatos STA 0-M BASIC STATISTICS (Ths s a multple choce paper.) Tme: hours 0 mutes INSTRUCTIONS TO CANDIDATES Do ot ope ths questo paper utl you have bee told to do

More information

REVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION

REVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION REVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION I lear regreo, we coder the frequecy dtrbuto of oe varable (Y) at each of everal level of a ecod varable (X). Y kow a the depedet varable. The

More information

Statistics: Unlocking the Power of Data Lock 5

Statistics: Unlocking the Power of Data Lock 5 STAT 0 Dr. Kar Lock Morga Exam 2 Grades: I- Class Multple Regresso SECTIONS 9.2, 0., 0.2 Multple explaatory varables (0.) Parttog varablty R 2, ANOVA (9.2) Codtos resdual plot (0.2) Exam 2 Re- grades Re-

More information

Multivariate Transformation of Variables and Maximum Likelihood Estimation

Multivariate Transformation of Variables and Maximum Likelihood Estimation Marquette Uversty Multvarate Trasformato of Varables ad Maxmum Lkelhood Estmato Dael B. Rowe, Ph.D. Assocate Professor Departmet of Mathematcs, Statstcs, ad Computer Scece Copyrght 03 by Marquette Uversty

More information

STATISTICS 13. Lecture 5 Apr 7, 2010

STATISTICS 13. Lecture 5 Apr 7, 2010 STATISTICS 13 Leture 5 Apr 7, 010 Revew Shape of the data -Bell shaped -Skewed -Bmodal Measures of eter Arthmet Mea Meda Mode Effets of outlers ad skewess Measures of Varablt A quattatve measure that desrbes

More information

LECTURE - 4 SIMPLE RANDOM SAMPLING DR. SHALABH DEPARTMENT OF MATHEMATICS AND STATISTICS INDIAN INSTITUTE OF TECHNOLOGY KANPUR

LECTURE - 4 SIMPLE RANDOM SAMPLING DR. SHALABH DEPARTMENT OF MATHEMATICS AND STATISTICS INDIAN INSTITUTE OF TECHNOLOGY KANPUR amplg Theory MODULE II LECTURE - 4 IMPLE RADOM AMPLIG DR. HALABH DEPARTMET OF MATHEMATIC AD TATITIC IDIA ITITUTE OF TECHOLOGY KAPUR Estmato of populato mea ad populato varace Oe of the ma objectves after

More information

Some Applications of the Resampling Methods in Computational Physics

Some Applications of the Resampling Methods in Computational Physics Iteratoal Joural of Mathematcs Treds ad Techoloy Volume 6 February 04 Some Applcatos of the Resampl Methods Computatoal Physcs Sotraq Marko #, Lorec Ekoom * # Physcs Departmet, Uversty of Korca, Albaa,

More information

Ordinary Least Squares Regression. Simple Regression. Algebra and Assumptions.

Ordinary Least Squares Regression. Simple Regression. Algebra and Assumptions. Ordary Least Squares egresso. Smple egresso. Algebra ad Assumptos. I ths part of the course we are gog to study a techque for aalysg the lear relatoshp betwee two varables Y ad X. We have pars of observatos

More information

PROPERTIES OF GOOD ESTIMATORS

PROPERTIES OF GOOD ESTIMATORS ESTIMATION INTRODUCTION Estmato s the statstcal process of fdg a appromate value for a populato parameter. A populato parameter s a characterstc of the dstrbuto of a populato such as the populato mea,

More information

GOALS The Samples Why Sample the Population? What is a Probability Sample? Four Most Commonly Used Probability Sampling Methods

GOALS The Samples Why Sample the Population? What is a Probability Sample? Four Most Commonly Used Probability Sampling Methods GOLS. Epla why a sample s the oly feasble way to lear about a populato.. Descrbe methods to select a sample. 3. Defe ad costruct a samplg dstrbuto of the sample mea. 4. Epla the cetral lmt theorem. 5.

More information

i 2 σ ) i = 1,2,...,n , and = 3.01 = 4.01

i 2 σ ) i = 1,2,...,n , and = 3.01 = 4.01 ECO 745, Homework 6 Le Cabrera. Assume that the followg data come from the lear model: ε ε ~ N, σ,,..., -6. -.5 7. 6.9 -. -. -.9. -..6.4.. -.6 -.7.7 Fd the mamum lkelhood estmates of,, ad σ ε s.6. 4. ε

More information

Handout #1. Title: Foundations of Econometrics. POPULATION vs. SAMPLE

Handout #1. Title: Foundations of Econometrics. POPULATION vs. SAMPLE Hadout #1 Ttle: Foudatos of Ecoometrcs Course: Eco 367 Fall/015 Istructor: Dr. I-Mg Chu POPULATION vs. SAMPLE From the Bureau of Labor web ste (http://www.bls.gov), we ca fd the uemploymet rate for each

More information

Regresso What s a Model? 1. Ofte Descrbe Relatoshp betwee Varables 2. Types - Determstc Models (o radomess) - Probablstc Models (wth radomess) EPI 809/Sprg 2008 9 Determstc Models 1. Hypothesze

More information

Comparison of Parameters of Lognormal Distribution Based On the Classical and Posterior Estimates

Comparison of Parameters of Lognormal Distribution Based On the Classical and Posterior Estimates Joural of Moder Appled Statstcal Methods Volume Issue Artcle 8 --03 Comparso of Parameters of Logormal Dstrbuto Based O the Classcal ad Posteror Estmates Raja Sulta Uversty of Kashmr, Sragar, Ida, hamzasulta8@yahoo.com

More information

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Postpoed exam: ECON430 Statstcs Date of exam: Jauary 0, 0 Tme for exam: 09:00 a.m. :00 oo The problem set covers 5 pages Resources allowed: All wrtte ad prted

More information

Multiple Linear Regression Analysis

Multiple Linear Regression Analysis LINEA EGESSION ANALYSIS MODULE III Lecture - 4 Multple Lear egresso Aalyss Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur Cofdece terval estmato The cofdece tervals multple

More information

Descriptive Statistics

Descriptive Statistics Page Techcal Math II Descrptve Statstcs Descrptve Statstcs Descrptve statstcs s the body of methods used to represet ad summarze sets of data. A descrpto of how a set of measuremets (for eample, people

More information

Chapter 4: Elements of Statistics

Chapter 4: Elements of Statistics Chapter : lemets of tatstcs - Itroducto The amplg Problem Ubased stmators -&3 amplg Theory --The ample Mea ad ace amplg Theorem - amplg Dstrbutos ad Cofdece Itervals tudet s T-Dstrbuto -5 Hypothess Testg

More information

Simple Linear Regression - Scalar Form

Simple Linear Regression - Scalar Form Smple Lear Regresso - Scalar Form Q.. Model Y X,..., p..a. Derve the ormal equatos that mmze Q. p..b. Solve for the ordary least squares estmators, p..c. Derve E, V, E, V, COV, p..d. Derve the mea ad varace

More information

Discrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand DIS 10b

Discrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand DIS 10b CS 70 Dscrete Mathematcs ad Probablty Theory Fall 206 Sesha ad Walrad DIS 0b. Wll I Get My Package? Seaky delvery guy of some compay s out delverg packages to customers. Not oly does he had a radom package

More information

1. A real number x is represented approximately by , and we are told that the relative error is 0.1 %. What is x? Note: There are two answers.

1. A real number x is represented approximately by , and we are told that the relative error is 0.1 %. What is x? Note: There are two answers. PROBLEMS A real umber s represeted appromately by 63, ad we are told that the relatve error s % What s? Note: There are two aswers Ht : Recall that % relatve error s What s the relatve error volved roudg

More information

Maximum Likelihood Estimation

Maximum Likelihood Estimation Marquette Uverst Maxmum Lkelhood Estmato Dael B. Rowe, Ph.D. Professor Departmet of Mathematcs, Statstcs, ad Computer Scece Coprght 08 b Marquette Uverst Maxmum Lkelhood Estmato We have bee sag that ~

More information

hp calculators HP 30S Statistics Averages and Standard Deviations Average and Standard Deviation Practice Finding Averages and Standard Deviations

hp calculators HP 30S Statistics Averages and Standard Deviations Average and Standard Deviation Practice Finding Averages and Standard Deviations HP 30S Statstcs Averages ad Stadard Devatos Average ad Stadard Devato Practce Fdg Averages ad Stadard Devatos HP 30S Statstcs Averages ad Stadard Devatos Average ad stadard devato The HP 30S provdes several

More information

Linear Regression. Hsiao-Lung Chan Dept Electrical Engineering Chang Gung University, Taiwan

Linear Regression. Hsiao-Lung Chan Dept Electrical Engineering Chang Gung University, Taiwan Lear Regresso Hsao-Lug Cha Dept Electrcal Egeerg Chag Gug Uverst, Tawa chahl@mal.cgu.edu.tw Curve fttg Least-squares regresso Data ehbt a sgfcat degree of error or scatter A curve for the tred of the data

More information

Uncertainty, Data, and Judgment

Uncertainty, Data, and Judgment Ucertaty, Data, ad Judgmet Sesso 06 Structure of the Course Topc Sesso Probablty -5 Estmato 6-8 Hypothess Testg 9-10 Regresso 11-16 1 Mcrosoft AND Itel (50-50) You vest $,500 MSFT ad $,500 INTC X = Aual

More information

Linear Regression. Hsiao-Lung Chan Dept Electrical Engineering Chang Gung University, Taiwan

Linear Regression. Hsiao-Lung Chan Dept Electrical Engineering Chang Gung University, Taiwan Lear Regresso Hsao-Lug Cha Dept Electrcal Egeerg Chag Gug Uverst, Tawa chahl@mal.cgu.edu.tw Curve fttg Least-squares regresso Data ehbt a sgfcat degree of error or scatter A curve for the tred of the data

More information

Fitting models to data.

Fitting models to data. Fttg models to data. Prevous lectures dscussed model geerato. Start wth physcal pcture or dagram of what s happeg Make lst of assumptos (e.g., cell drug uptake s by dffuso; covecto ca be eglected) Wrte

More information

X ε ) = 0, or equivalently, lim

X ε ) = 0, or equivalently, lim Revew for the prevous lecture Cocepts: order statstcs Theorems: Dstrbutos of order statstcs Examples: How to get the dstrbuto of order statstcs Chapter 5 Propertes of a Radom Sample Secto 55 Covergece

More information

(Monte Carlo) Resampling Technique in Validity Testing and Reliability Testing

(Monte Carlo) Resampling Technique in Validity Testing and Reliability Testing Iteratoal Joural of Computer Applcatos (0975 8887) (Mote Carlo) Resamplg Techque Valdty Testg ad Relablty Testg Ad Setawa Departmet of Mathematcs, Faculty of Scece ad Mathematcs, Satya Wacaa Chrsta Uversty

More information

Reaction Time VS. Drug Percentage Subject Amount of Drug Times % Reaction Time in Seconds 1 Mary John Carl Sara William 5 4

Reaction Time VS. Drug Percentage Subject Amount of Drug Times % Reaction Time in Seconds 1 Mary John Carl Sara William 5 4 CHAPTER Smple Lear Regreo EXAMPLE A expermet volvg fve ubject coducted to determe the relatohp betwee the percetage of a certa drug the bloodtream ad the legth of tme t take the ubject to react to a tmulu.

More information

CLASS NOTES. for. PBAF 528: Quantitative Methods II SPRING Instructor: Jean Swanson. Daniel J. Evans School of Public Affairs

CLASS NOTES. for. PBAF 528: Quantitative Methods II SPRING Instructor: Jean Swanson. Daniel J. Evans School of Public Affairs CLASS NOTES for PBAF 58: Quattatve Methods II SPRING 005 Istructor: Jea Swaso Dael J. Evas School of Publc Affars Uversty of Washgto Ackowledgemet: The structor wshes to thak Rachel Klet, Assstat Professor,

More information

Arithmetic Mean Suppose there is only a finite number N of items in the system of interest. Then the population arithmetic mean is

Arithmetic Mean Suppose there is only a finite number N of items in the system of interest. Then the population arithmetic mean is Topc : Probablty Theory Module : Descrptve Statstcs Measures of Locato Descrptve statstcs are measures of locato ad shape that perta to probablty dstrbutos The prmary measures of locato are the arthmetc

More information

Statistics. Correlational. Dr. Ayman Eldeib. Simple Linear Regression and Correlation. SBE 304: Linear Regression & Correlation 1/3/2018

Statistics. Correlational. Dr. Ayman Eldeib. Simple Linear Regression and Correlation. SBE 304: Linear Regression & Correlation 1/3/2018 /3/08 Sstems & Bomedcal Egeerg Departmet SBE 304: Bo-Statstcs Smple Lear Regresso ad Correlato Dr. Ama Eldeb Fall 07 Descrptve Orgasg, summarsg & descrbg data Statstcs Correlatoal Relatoshps Iferetal Geeralsg

More information

PGE 310: Formulation and Solution in Geosystems Engineering. Dr. Balhoff. Interpolation

PGE 310: Formulation and Solution in Geosystems Engineering. Dr. Balhoff. Interpolation PGE 30: Formulato ad Soluto Geosystems Egeerg Dr. Balhoff Iterpolato Numercal Methods wth MATLAB, Recktewald, Chapter 0 ad Numercal Methods for Egeers, Chapra ad Caale, 5 th Ed., Part Fve, Chapter 8 ad

More information

UNIVERSITY OF EAST ANGLIA. Main Series UG Examination

UNIVERSITY OF EAST ANGLIA. Main Series UG Examination UNIVERSITY OF EAST ANGLIA School of Ecoomcs Ma Seres UG Examato 03-4 INTRODUCTORY MATHEMATICS AND STATISTICS FOR ECONOMISTS ECO-400Y Tme allowed: 3 hours Aswer ALL questos from both Sectos. Aswer EACH

More information

Bayes Estimator for Exponential Distribution with Extension of Jeffery Prior Information

Bayes Estimator for Exponential Distribution with Extension of Jeffery Prior Information Malaysa Joural of Mathematcal Sceces (): 97- (9) Bayes Estmator for Expoetal Dstrbuto wth Exteso of Jeffery Pror Iformato Hadeel Salm Al-Kutub ad Noor Akma Ibrahm Isttute for Mathematcal Research, Uverst

More information

Lecture 8: Linear Regression

Lecture 8: Linear Regression Lecture 8: Lear egresso May 4, GENOME 56, Sprg Goals Develop basc cocepts of lear regresso from a probablstc framework Estmatg parameters ad hypothess testg wth lear models Lear regresso Su I Lee, CSE

More information

Special Instructions / Useful Data

Special Instructions / Useful Data JAM 6 Set of all real umbers P A..d. B, p Posso Specal Istructos / Useful Data x,, :,,, x x Probablty of a evet A Idepedetly ad detcally dstrbuted Bomal dstrbuto wth parameters ad p Posso dstrbuto wth

More information

STK3100 and STK4100 Autumn 2018

STK3100 and STK4100 Autumn 2018 SK3 ad SK4 Autum 8 Geeralzed lear models Part III Covers the followg materal from chaters 4 ad 5: Cofdece tervals by vertg tests Cosder a model wth a sgle arameter β We may obta a ( α% cofdece terval for

More information