Statistics Chapter 4
|
|
- Karin Lamb
- 5 years ago
- Views:
Transcription
1 Statstcs Chapter 4 "There are three knds of les: les, damned les, and statstcs." Benjamn Dsrael, 1895 (Brtsh statesman) Gaussan Dstrbuton, 4-1 If a measurement s repeated many tmes a statstcal treatment of the data can provde an ndcaton of the relablty of the results. If the errors assocated wth the measurement are completely random then the Central Lmt Theorem assures that the data wll follow a mathematcal form called a Gaussan dstrbuton (bell-shaped curve). In the lmt of an nfnte number of measurements the hstogram below becomes the populaton dstrbuton denoted by the sold lne. The hstogram on the rght was drawn to have the same mean, standard devaton, and area as the smooth curve. Ths s not true for a fnte number of measurements. Only n the nfnte lmt wll they bethe same. The populaton dstrbuton (nfnte n) scharacterzed by a populaton mean (average - center of the symmetrc dstrbuton), µ, and a populaton standard devaton (measure of the wdth of the dstrbuton), σ. Another useful measure s the square of the standard devaton, σ 2,known as the varance. µ = lm 1 n n n Σ x =1 σ 2 = lm 1 n n n Σ(x µ) 2 =1 AGaussan curve, y Gaussan can be expressed n terms of these varables. y Gaussan = (x µ) 1 2 σ 2π e 2σ 2
2 Obtanng the Probablty -2- As all measurements contan expermental error no result s completely certan. However, by a judcous use of statstcs one can assocate a probablty wth the result. To convert the Gaussan dstrbuton nto a probablty densty a new varable z s ntroduced defned as z = (x µ) /σ. Then the Gaussan curve becomes the normal Gaussan error curve or the z dstrbuton y = f (z) = f (0) = f (1) = 1 2π e z2 /2 1 2π 1 2π e = = (x = µ ± σ are nflecton ponts) Integraton of the populaton dstrbuton between two lmts as from x = µ zσ to x = µ + zσ gves the probablty of obtanng a value of x between these lmts. The ntegral lmts that gve 95% of the populaton dstrbuton area (a probablty of 0.95) are 95% confdence lmts. The symmetrc Gaussan dstrbuton requres that these lmts be equdstant from µ. The ntegral of agaussan between fnte lmts s not analytc and the area under the curve sgven ntables. (0z0 below s really the absolute value of z, z ) The sample data s not nfnte! How can one characterze ts relablty? Frst one needs to obtan the mean and standard devaton for fnte data. The fnte dstrbuton s characterzed by a sample mean (sample average), < x >, and a sample standard devaton, s. Agan, another useful measure s the square of the standard devaton s 2 known as the sample varance. The quantty n 1below scalled the degrees of freedom. < x > = Σ x n Σ(x < x >) 2 s = n 1
3 -3- EX 1. For the data on the bulbs gven nthe hstogram where < x > = hr and s = hr a) What fracton of bulbs s expected to have a lfetme greater than hr? b) What fracton of bulbs s expected to have a lfetme between and hr? Comparson of Standard Devatons wth F Test, 4-2 To examne whether two standard devatons are statstcally dfferent determne F calculated = s 2 1/s 2 2 where F 1. If F calculated > F table the dfference s sgnfcant and the two measurements are statstcally dfferent. Hypothess testng based upon assumng that the null hypothess s true at a certan level of probablty, generally chosen to be 5%. The hypothess s accepted f the probablty for t beng true s > 5% and rejected f the probablty for t beng true s < 5%. Null hypothess for the F test: two sets of measurements taken from populatons wth the same populaton standard devaton; all dfferences arse from only random varatons n measurement accept: F calculated < F table => standard devatons are not statstcally dfferent reject: F calculated > F table => standard devatons are statstcally dfferent
4 -4- Confdence Intervals, 4-3 (nferences based on small samples) From a lmted number of measurements one wants an estmate of the uncertanty of the measurement. One can use a confdence nterval confdence nterval = < x > ± ts n whch mples that the true populaton mean wll be found wthn a range of st/ n of the sample mean wth a confdence level (level of certanty) specfed by the partcular t chosen f one were to repeat the n measurements many tmes. Then a 95% confdence nterval would nclude the true populaton mean n 95% of these sets of n measurements. Note that ths mples that the uncertanty n the sample mean s reduced by more measurements by a factor of 1/ n. Null hypothess for the t test: two sets of measurements taken from populatons wth the same mean; all dfferences arse from only random varatons n measurement accept: t calculated < t table => means are not statstcally dfferent reject: t calculated > t table => means are statstcally dfferent EX 2. The percentage of an addtve n gasolne was measured sx tmes wth the followng results: 0.13, 0.12, 0.16, 0.17, 0.20 and 0.11%. Fnd the 90% and 99% confdence ntervals for the percentage of the addtve.
5 Comparson of Means wth Student s t, 4-4 (always use 95% confdence ntervals) -5- Case 1: Comparson to a Known or Standard Value EX 3. AStandard Reference Materal s certfed to contan 94.6 ppm of an organc contamnant n sol. You analyze the reference compound fve tmes obtanng < x > = , s = Do your results dffer from the expected result at the 95% confdence level? Case 2: Comparson of Replcate Measurements (apply F test frst) For ths comparson one frst obtans a pooled standard devaton then uses t to calculate a value of t whch s compared wth t n Student s t. Ifthe calculated t s greater than the 95% confdence level value of t the two replcates are consdered dfferent. For the two sets of data wth n 1 and n 2 measurements wth means < x 1 >and < x 2 >and standard devatons s 1 and s 2 F calculated < F table F calculated > F table t calculated = <x 1 > < x 2 > s pooled n 1n 2 s s pooled = 2 1 (n 1 1) + s 2 2 (n 2 1) n 1 + n 2 2 <x 1 > < x 2 > n 1 + n 2 s 2 1 /n 1 + s 2 2 /n 2 degrees of freedom = (s 2 1/n 1 + s 2 2/n 2 ) 2 (s 2 1 /n 1) 2 n (s2 2 /n 2) 2 n 2 1 EX 4. Atranee n a medcal lab wll be released to work on her own when her results agree wth those of an experenced worker at the 95% confdence level. Consderng the results for blood urea ntrogen analyss gven below, should the tranee be released to work alone? tranee < x > = mg/dl s = mg/dl n = 6samples experenced worker < x > = mg/dl s = mg/dl n = 5samples
6 Case 3: Comparson of Indvdual Dfferences wth Pared t Test -6- Each sample s measured once by each method and the dfferences d,average value of the dfferences < d >, and standard devaton of the dfferences s d determned n order to calculate t. s d = Σ(d < d >) 2 n 1 t calculated = <d > s d n EX 5. The T content (wt%) of fve dfferent ore samples (each wth dfferent T content) was measured by each of two methods. Do the two technques gve results that are sgnfcantly dfferent at the 95% confdence level? Sample Method 1 Method 2 d d < d > (d < d >) 2 A B C D E Grubbs Test for Outlers, 4-6 (use 95% confdence) To determne whether a partcular data pont can be excluded based upon ts questonable veracty, form the Grubbs statstc, G G calculated = x questonable < x > s If G calculated > G table then the pont can be excluded wth the chosen confdence level (here 95%). The mean and standard devaton wll need to be recalculated. Hnt: generally do not exclude a data pont unless you are certan that an error occurred n ts measurement. Never exclude more than one pont. Always use a value of G of at least a 95% confdence level. NOTE: For the F, t, and G statstcs f the calculated value s less than the table value the null hypothess s true, you do nothng!
7 -7- Method of Lnear Least Squares, 4-7 For a set of n data ponts (x, y ) one wants to fnd the "best" straght lne through the data: y y=mx+b Each y devates from the lne d = y y = y (mx + b) where y s the value when x = x.tomnmze the devatons from lnearty rrespectve of ther sgn one consders the square of the devatons x d 2 = (y mx b) 2 = y 2 2mx y + m 2 x 2 + 2mbx 2by + b 2 In the method of least squares one mnmzes the sum of the squares of all the devatons: SSE = Σ d 2 = Σ y 2 2m Σ x y + m 2 Σ x 2 + 2mb Σ x 2b Σ y + The values of m and b are found whch mnmze SSE m SSE = b Σ b 2 b SSE = m
8 -8- Lnear Regresson Equatons equaton n Harrs varant (n sample spreadsheet) slope, m (4-16) n Σ x y Σ x Σ y n Σ x 2 (Σ x ) 2 Σ(x < x >)(y < y >) Σ(x < x >) 2 y-ntercept, b (4-17) Σ x 2 Σ y Σ x Σ x y n Σ x 2 (Σ x ) 2 Σ y m Σ x n varance of the regresson, s 2 y (standard error) (4-20) Σ(y mx b) 2 n 2 varance of the slope, s 2 m (standard error) (4-21) n n 2 Σ(y mx b) 2 n Σ x 2 (Σ x ) 2 Σ(y mx b) 2 (n 2)Σ(x < x >) 2 varance of the ntercept, s 2 b (standard error) (4-22) Σ(y mx b) 2 Σ x 2 (n 2)[n Σ x 2 (Σ x ) 2 ] Σ x 2 Σ(y mx b) 2 n(n 2)Σ(x < x >) 2 correlaton coeffcent, R (5-2) Σ(x < x >)(y < y >) Σ x y Σ x Σ y /n Σ(x < x >) 2 Σ(y < y >) 2 Σ(x < x >) 2 Σ(y < y >) 2 Abbrevatons used n spreadsheet n addton to SSE (smlar expressons n y) Sx = Σ x SSx = Σ x 2 SSDx = Σ(x < x >) 2 Sxy = Σ x y SDxSDy = Σ(x < x >)(y < y >)
9 -9- Calbraton Curves, 4-8 standard solutons blank solutons questonable: (0.392) NOTE: Propagaton of uncertanty for a calbraton curve follows Eq. 4-27
Answers Problem Set 2 Chem 314A Williamsen Spring 2000
Answers Problem Set Chem 314A Wllamsen Sprng 000 1) Gve me the followng crtcal values from the statstcal tables. a) z-statstc,-sded test, 99.7% confdence lmt ±3 b) t-statstc (Case I), 1-sded test, 95%
More informationStatistics for Managers Using Microsoft Excel/SPSS Chapter 13 The Simple Linear Regression Model and Correlation
Statstcs for Managers Usng Mcrosoft Excel/SPSS Chapter 13 The Smple Lnear Regresson Model and Correlaton 1999 Prentce-Hall, Inc. Chap. 13-1 Chapter Topcs Types of Regresson Models Determnng the Smple Lnear
More informationLecture 16 Statistical Analysis in Biomaterials Research (Part II)
3.051J/0.340J 1 Lecture 16 Statstcal Analyss n Bomaterals Research (Part II) C. F Dstrbuton Allows comparson of varablty of behavor between populatons usng test of hypothess: σ x = σ x amed for Brtsh statstcan
More informationDepartment of Quantitative Methods & Information Systems. Time Series and Their Components QMIS 320. Chapter 6
Department of Quanttatve Methods & Informaton Systems Tme Seres and Ther Components QMIS 30 Chapter 6 Fall 00 Dr. Mohammad Zanal These sldes were modfed from ther orgnal source for educatonal purpose only.
More informationStatistics for Economics & Business
Statstcs for Economcs & Busness Smple Lnear Regresson Learnng Objectves In ths chapter, you learn: How to use regresson analyss to predct the value of a dependent varable based on an ndependent varable
More informationEconomics 130. Lecture 4 Simple Linear Regression Continued
Economcs 130 Lecture 4 Contnued Readngs for Week 4 Text, Chapter and 3. We contnue wth addressng our second ssue + add n how we evaluate these relatonshps: Where do we get data to do ths analyss? How do
More informationSystematic Error Illustration of Bias. Sources of Systematic Errors. Effects of Systematic Errors 9/23/2009. Instrument Errors Method Errors Personal
9/3/009 Sstematc Error Illustraton of Bas Sources of Sstematc Errors Instrument Errors Method Errors Personal Prejudce Preconceved noton of true value umber bas Prefer 0/5 Small over large Even over odd
More information1. Inference on Regression Parameters a. Finding Mean, s.d and covariance amongst estimates. 2. Confidence Intervals and Working Hotelling Bands
Content. Inference on Regresson Parameters a. Fndng Mean, s.d and covarance amongst estmates.. Confdence Intervals and Workng Hotellng Bands 3. Cochran s Theorem 4. General Lnear Testng 5. Measures of
More informationCorrelation and Regression. Correlation 9.1. Correlation. Chapter 9
Chapter 9 Correlaton and Regresson 9. Correlaton Correlaton A correlaton s a relatonshp between two varables. The data can be represented b the ordered pars (, ) where s the ndependent (or eplanator) varable,
More informationwhere I = (n x n) diagonal identity matrix with diagonal elements = 1 and off-diagonal elements = 0; and σ 2 e = variance of (Y X).
11.4.1 Estmaton of Multple Regresson Coeffcents In multple lnear regresson, we essentally solve n equatons for the p unnown parameters. hus n must e equal to or greater than p and n practce n should e
More informationEcon107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)
I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes
More informationU-Pb Geochronology Practical: Background
U-Pb Geochronology Practcal: Background Basc Concepts: accuracy: measure of the dfference between an expermental measurement and the true value precson: measure of the reproducblty of the expermental result
More informationChapter 11: Simple Linear Regression and Correlation
Chapter 11: Smple Lnear Regresson and Correlaton 11-1 Emprcal Models 11-2 Smple Lnear Regresson 11-3 Propertes of the Least Squares Estmators 11-4 Hypothess Test n Smple Lnear Regresson 11-4.1 Use of t-tests
More informationUNIVERSITY OF TORONTO Faculty of Arts and Science. December 2005 Examinations STA437H1F/STA1005HF. Duration - 3 hours
UNIVERSITY OF TORONTO Faculty of Arts and Scence December 005 Examnatons STA47HF/STA005HF Duraton - hours AIDS ALLOWED: (to be suppled by the student) Non-programmable calculator One handwrtten 8.5'' x
More informationx = , so that calculated
Stat 4, secton Sngle Factor ANOVA notes by Tm Plachowsk n chapter 8 we conducted hypothess tests n whch we compared a sngle sample s mean or proporton to some hypotheszed value Chapter 9 expanded ths to
More informationChapter 13: Multiple Regression
Chapter 13: Multple Regresson 13.1 Developng the multple-regresson Model The general model can be descrbed as: It smplfes for two ndependent varables: The sample ft parameter b 0, b 1, and b are used to
More informationStatistics II Final Exam 26/6/18
Statstcs II Fnal Exam 26/6/18 Academc Year 2017/18 Solutons Exam duraton: 2 h 30 mn 1. (3 ponts) A town hall s conductng a study to determne the amount of leftover food produced by the restaurants n the
More informationComparison of Regression Lines
STATGRAPHICS Rev. 9/13/2013 Comparson of Regresson Lnes Summary... 1 Data Input... 3 Analyss Summary... 4 Plot of Ftted Model... 6 Condtonal Sums of Squares... 6 Analyss Optons... 7 Forecasts... 8 Confdence
More information/ n ) are compared. The logic is: if the two
STAT C141, Sprng 2005 Lecture 13 Two sample tests One sample tests: examples of goodness of ft tests, where we are testng whether our data supports predctons. Two sample tests: called as tests of ndependence
More information[The following data appear in Wooldridge Q2.3.] The table below contains the ACT score and college GPA for eight college students.
PPOL 59-3 Problem Set Exercses n Smple Regresson Due n class /8/7 In ths problem set, you are asked to compute varous statstcs by hand to gve you a better sense of the mechancs of the Pearson correlaton
More informationChemometrics. Unit 2: Regression Analysis
Chemometrcs Unt : Regresson Analyss The problem of predctng the average value of one varable n terms of the known values of other varables s the problem of regresson. In carryng out a regresson analyss
More informationPHYS 450 Spring semester Lecture 02: Dealing with Experimental Uncertainties. Ron Reifenberger Birck Nanotechnology Center Purdue University
PHYS 45 Sprng semester 7 Lecture : Dealng wth Expermental Uncertantes Ron Refenberger Brck anotechnology Center Purdue Unversty Lecture Introductory Comments Expermental errors (really expermental uncertantes)
More informationChapter 14 Simple Linear Regression
Chapter 4 Smple Lnear Regresson Chapter 4 - Smple Lnear Regresson Manageral decsons often are based on the relatonshp between two or more varables. Regresson analss can be used to develop an equaton showng
More informationLecture 4 Hypothesis Testing
Lecture 4 Hypothess Testng We may wsh to test pror hypotheses about the coeffcents we estmate. We can use the estmates to test whether the data rejects our hypothess. An example mght be that we wsh to
More information2016 Wiley. Study Session 2: Ethical and Professional Standards Application
6 Wley Study Sesson : Ethcal and Professonal Standards Applcaton LESSON : CORRECTION ANALYSIS Readng 9: Correlaton and Regresson LOS 9a: Calculate and nterpret a sample covarance and a sample correlaton
More information28. SIMPLE LINEAR REGRESSION III
8. SIMPLE LINEAR REGRESSION III Ftted Values and Resduals US Domestc Beers: Calores vs. % Alcohol To each observed x, there corresponds a y-value on the ftted lne, y ˆ = βˆ + βˆ x. The are called ftted
More informationHere is the rationale: If X and y have a strong positive relationship to one another, then ( x x) will tend to be positive when ( y y)
Secton 1.5 Correlaton In the prevous sectons, we looked at regresson and the value r was a measurement of how much of the varaton n y can be attrbuted to the lnear relatonshp between y and x. In ths secton,
More informationStatistics MINITAB - Lab 2
Statstcs 20080 MINITAB - Lab 2 1. Smple Lnear Regresson In smple lnear regresson we attempt to model a lnear relatonshp between two varables wth a straght lne and make statstcal nferences concernng that
More informationLecture 6: Introduction to Linear Regression
Lecture 6: Introducton to Lnear Regresson An Manchakul amancha@jhsph.edu 24 Aprl 27 Lnear regresson: man dea Lnear regresson can be used to study an outcome as a lnear functon of a predctor Example: 6
More information18. SIMPLE LINEAR REGRESSION III
8. SIMPLE LINEAR REGRESSION III US Domestc Beers: Calores vs. % Alcohol Ftted Values and Resduals To each observed x, there corresponds a y-value on the ftted lne, y ˆ ˆ = α + x. The are called ftted values.
More informationBasic Business Statistics, 10/e
Chapter 13 13-1 Basc Busness Statstcs 11 th Edton Chapter 13 Smple Lnear Regresson Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc. Chap 13-1 Learnng Objectves In ths chapter, you learn: How to use regresson
More informationNANYANG TECHNOLOGICAL UNIVERSITY SEMESTER I EXAMINATION MTH352/MH3510 Regression Analysis
NANYANG TECHNOLOGICAL UNIVERSITY SEMESTER I EXAMINATION 014-015 MTH35/MH3510 Regresson Analyss December 014 TIME ALLOWED: HOURS INSTRUCTIONS TO CANDIDATES 1. Ths examnaton paper contans FOUR (4) questons
More informationFirst Year Examination Department of Statistics, University of Florida
Frst Year Examnaton Department of Statstcs, Unversty of Florda May 7, 010, 8:00 am - 1:00 noon Instructons: 1. You have four hours to answer questons n ths examnaton.. You must show your work to receve
More informationLearning Objectives for Chapter 11
Chapter : Lnear Regresson and Correlaton Methods Hldebrand, Ott and Gray Basc Statstcal Ideas for Managers Second Edton Learnng Objectves for Chapter Usng the scatterplot n regresson analyss Usng the method
More informationSTAT 511 FINAL EXAM NAME Spring 2001
STAT 5 FINAL EXAM NAME Sprng Instructons: Ths s a closed book exam. No notes or books are allowed. ou may use a calculator but you are not allowed to store notes or formulas n the calculator. Please wrte
More informationx yi In chapter 14, we want to perform inference (i.e. calculate confidence intervals and perform tests of significance) in this setting.
The Practce of Statstcs, nd ed. Chapter 14 Inference for Regresson Introducton In chapter 3 we used a least-squares regresson lne (LSRL) to represent a lnear relatonshp etween two quanttatve explanator
More informationSTATISTICS QUESTIONS. Step by Step Solutions.
STATISTICS QUESTIONS Step by Step Solutons www.mathcracker.com 9//016 Problem 1: A researcher s nterested n the effects of famly sze on delnquency for a group of offenders and examnes famles wth one to
More informationStatistics for Business and Economics
Statstcs for Busness and Economcs Chapter 11 Smple Regresson Copyrght 010 Pearson Educaton, Inc. Publshng as Prentce Hall Ch. 11-1 11.1 Overvew of Lnear Models n An equaton can be ft to show the best lnear
More informationResource Allocation and Decision Analysis (ECON 8010) Spring 2014 Foundations of Regression Analysis
Resource Allocaton and Decson Analss (ECON 800) Sprng 04 Foundatons of Regresson Analss Readng: Regresson Analss (ECON 800 Coursepak, Page 3) Defntons and Concepts: Regresson Analss statstcal technques
More informationSTAT 3008 Applied Regression Analysis
STAT 3008 Appled Regresson Analyss Tutoral : Smple Lnear Regresson LAI Chun He Department of Statstcs, The Chnese Unversty of Hong Kong 1 Model Assumpton To quantfy the relatonshp between two factors,
More information4 Analysis of Variance (ANOVA) 5 ANOVA. 5.1 Introduction. 5.2 Fixed Effects ANOVA
4 Analyss of Varance (ANOVA) 5 ANOVA 51 Introducton ANOVA ANOVA s a way to estmate and test the means of multple populatons We wll start wth one-way ANOVA If the populatons ncluded n the study are selected
More informationLinear Regression Analysis: Terminology and Notation
ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page ) Lnear Regresson Analyss: Termnology and Notaton Consder the generc verson of the smple (two-varable) lnear regresson model. It s represented
More informationLecture 15 Statistical Analysis in Biomaterials Research
3.05/BE.340 Lecture 5 tatstcal Analyss n Bomaterals Research. Ratonale: Why s statstcal analyss needed n bomaterals research? Many error sources n measurements on lvng systems! Eamples of Measured Values
More informationIntroduction to Regression
Introducton to Regresson Dr Tom Ilvento Department of Food and Resource Economcs Overvew The last part of the course wll focus on Regresson Analyss Ths s one of the more powerful statstcal technques Provdes
More informationBOOTSTRAP METHOD FOR TESTING OF EQUALITY OF SEVERAL MEANS. M. Krishna Reddy, B. Naveen Kumar and Y. Ramu
BOOTSTRAP METHOD FOR TESTING OF EQUALITY OF SEVERAL MEANS M. Krshna Reddy, B. Naveen Kumar and Y. Ramu Department of Statstcs, Osmana Unversty, Hyderabad -500 007, Inda. nanbyrozu@gmal.com, ramu0@gmal.com
More informationSTAT 3340 Assignment 1 solutions. 1. Find the equation of the line which passes through the points (1,1) and (4,5).
(out of 15 ponts) STAT 3340 Assgnment 1 solutons (10) (10) 1. Fnd the equaton of the lne whch passes through the ponts (1,1) and (4,5). β 1 = (5 1)/(4 1) = 4/3 equaton for the lne s y y 0 = β 1 (x x 0
More informationStatistical Inference. 2.3 Summary Statistics Measures of Center and Spread. parameters ( population characteristics )
Ismor Fscher, 8//008 Stat 54 / -8.3 Summary Statstcs Measures of Center and Spread Dstrbuton of dscrete contnuous POPULATION Random Varable, numercal True center =??? True spread =???? parameters ( populaton
More informationThe SAS program I used to obtain the analyses for my answers is given below.
Homework 1 Answer sheet Page 1 The SAS program I used to obtan the analyses for my answers s gven below. dm'log;clear;output;clear'; *************************************************************; *** EXST7034
More informationStatistical Evaluation of WATFLOOD
tatstcal Evaluaton of WATFLD By: Angela MacLean, Dept. of Cvl & Envronmental Engneerng, Unversty of Waterloo, n. ctober, 005 The statstcs program assocated wth WATFLD uses spl.csv fle that s produced wth
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationSampling Theory MODULE V LECTURE - 17 RATIO AND PRODUCT METHODS OF ESTIMATION
Samplng Theory MODULE V LECTURE - 7 RATIO AND PRODUCT METHODS OF ESTIMATION DR. SHALABH DEPARTMENT OF MATHEMATICS AND STATISTICS INDIAN INSTITUTE OF TECHNOLOG KANPUR Propertes of separate rato estmator:
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationAnalytical Chemistry Calibration Curve Handout
I. Quck-and Drty Excel Tutoral Analytcal Chemstry Calbraton Curve Handout For those of you wth lttle experence wth Excel, I ve provded some key technques that should help you use the program both for problem
More informationECONOMICS 351*-A Mid-Term Exam -- Fall Term 2000 Page 1 of 13 pages. QUEEN'S UNIVERSITY AT KINGSTON Department of Economics
ECOOMICS 35*-A Md-Term Exam -- Fall Term 000 Page of 3 pages QUEE'S UIVERSITY AT KIGSTO Department of Economcs ECOOMICS 35* - Secton A Introductory Econometrcs Fall Term 000 MID-TERM EAM ASWERS MG Abbott
More informationis the calculated value of the dependent variable at point i. The best parameters have values that minimize the squares of the errors
Multple Lnear and Polynomal Regresson wth Statstcal Analyss Gven a set of data of measured (or observed) values of a dependent varable: y versus n ndependent varables x 1, x, x n, multple lnear regresson
More informationInterval Estimation in the Classical Normal Linear Regression Model. 1. Introduction
ECONOMICS 35* -- NOTE 7 ECON 35* -- NOTE 7 Interval Estmaton n the Classcal Normal Lnear Regresson Model Ths note outlnes the basc elements of nterval estmaton n the Classcal Normal Lnear Regresson Model
More informationLecture Notes for STATISTICAL METHODS FOR BUSINESS II BMGT 212. Chapters 14, 15 & 16. Professor Ahmadi, Ph.D. Department of Management
Lecture Notes for STATISTICAL METHODS FOR BUSINESS II BMGT 1 Chapters 14, 15 & 16 Professor Ahmad, Ph.D. Department of Management Revsed August 005 Chapter 14 Formulas Smple Lnear Regresson Model: y =
More informationDepartment of Statistics University of Toronto STA305H1S / 1004 HS Design and Analysis of Experiments Term Test - Winter Solution
Department of Statstcs Unversty of Toronto STA35HS / HS Desgn and Analyss of Experments Term Test - Wnter - Soluton February, Last Name: Frst Name: Student Number: Instructons: Tme: hours. Ads: a non-programmable
More informationNegative Binomial Regression
STATGRAPHICS Rev. 9/16/2013 Negatve Bnomal Regresson Summary... 1 Data Input... 3 Statstcal Model... 3 Analyss Summary... 4 Analyss Optons... 7 Plot of Ftted Model... 8 Observed Versus Predcted... 10 Predctons...
More informationLecture 9: Linear regression: centering, hypothesis testing, multiple covariates, and confounding
Recall: man dea of lnear regresson Lecture 9: Lnear regresson: centerng, hypothess testng, multple covarates, and confoundng Sandy Eckel seckel@jhsph.edu 6 May 8 Lnear regresson can be used to study an
More informationLecture 9: Linear regression: centering, hypothesis testing, multiple covariates, and confounding
Lecture 9: Lnear regresson: centerng, hypothess testng, multple covarates, and confoundng Sandy Eckel seckel@jhsph.edu 6 May 008 Recall: man dea of lnear regresson Lnear regresson can be used to study
More informationANOVA. The Observations y ij
ANOVA Stands for ANalyss Of VArance But t s a test of dfferences n means The dea: The Observatons y j Treatment group = 1 = 2 = k y 11 y 21 y k,1 y 12 y 22 y k,2 y 1, n1 y 2, n2 y k, nk means: m 1 m 2
More informationTopic- 11 The Analysis of Variance
Topc- 11 The Analyss of Varance Expermental Desgn The samplng plan or expermental desgn determnes the way that a sample s selected. In an observatonal study, the expermenter observes data that already
More informationx i1 =1 for all i (the constant ).
Chapter 5 The Multple Regresson Model Consder an economc model where the dependent varable s a functon of K explanatory varables. The economc model has the form: y = f ( x,x,..., ) xk Approxmate ths by
More informationBasic Statistical Analysis and Yield Calculations
October 17, 007 Basc Statstcal Analyss and Yeld Calculatons Dr. José Ernesto Rayas Sánchez 1 Outlne Sources of desgn-performance uncertanty Desgn and development processes Desgn for manufacturablty A general
More informationLaboratory 3: Method of Least Squares
Laboratory 3: Method of Least Squares Introducton Consder the graph of expermental data n Fgure 1. In ths experment x s the ndependent varable and y the dependent varable. Clearly they are correlated wth
More informationEconometrics of Panel Data
Econometrcs of Panel Data Jakub Mućk Meetng # 8 Jakub Mućk Econometrcs of Panel Data Meetng # 8 1 / 17 Outlne 1 Heterogenety n the slope coeffcents 2 Seemngly Unrelated Regresson (SUR) 3 Swamy s random
More informationSee Book Chapter 11 2 nd Edition (Chapter 10 1 st Edition)
Count Data Models See Book Chapter 11 2 nd Edton (Chapter 10 1 st Edton) Count data consst of non-negatve nteger values Examples: number of drver route changes per week, the number of trp departure changes
More informationJanuary Examinations 2015
24/5 Canddates Only January Examnatons 25 DO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR STUDENT CANDIDATE NO.. Department Module Code Module Ttle Exam Duraton (n words)
More informationDefinition. Measures of Dispersion. Measures of Dispersion. Definition. The Range. Measures of Dispersion 3/24/2014
Measures of Dsperson Defenton Range Interquartle Range Varance and Standard Devaton Defnton Measures of dsperson are descrptve statstcs that descrbe how smlar a set of scores are to each other The more
More informationPubH 7405: REGRESSION ANALYSIS. SLR: INFERENCES, Part II
PubH 7405: REGRESSION ANALSIS SLR: INFERENCES, Part II We cover te topc of nference n two sessons; te frst sesson focused on nferences concernng te slope and te ntercept; ts s a contnuaton on estmatng
More informationLaboratory 1c: Method of Least Squares
Lab 1c, Least Squares Laboratory 1c: Method of Least Squares Introducton Consder the graph of expermental data n Fgure 1. In ths experment x s the ndependent varable and y the dependent varable. Clearly
More informationMeasurement Uncertainties Reference
Measurement Uncertantes Reference Introducton We all ntutvely now that no epermental measurement can be perfect. It s possble to mae ths dea quanttatve. It can be stated ths way: the result of an ndvdual
More informationStatistics Spring MIT Department of Nuclear Engineering
Statstcs.04 Sprng 00.04 S00 Statstcs/Probablty Analyss of eperments Measurement error Measurement process systematc vs. random errors Nose propertes of sgnals and mages quantum lmted mages.04 S00 Probablty
More informationChapter 9: Statistical Inference and the Relationship between Two Variables
Chapter 9: Statstcal Inference and the Relatonshp between Two Varables Key Words The Regresson Model The Sample Regresson Equaton The Pearson Correlaton Coeffcent Learnng Outcomes After studyng ths chapter,
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 31 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 6. Rdge regresson The OLSE s the best lnear unbased
More informationj) = 1 (note sigma notation) ii. Continuous random variable (e.g. Normal distribution) 1. density function: f ( x) 0 and f ( x) dx = 1
Random varables Measure of central tendences and varablty (means and varances) Jont densty functons and ndependence Measures of assocaton (covarance and correlaton) Interestng result Condtonal dstrbutons
More informationCathy Walker March 5, 2010
Cathy Walker March 5, 010 Part : Problem Set 1. What s the level of measurement for the followng varables? a) SAT scores b) Number of tests or quzzes n statstcal course c) Acres of land devoted to corn
More informationTHE ROYAL STATISTICAL SOCIETY 2006 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE
THE ROYAL STATISTICAL SOCIETY 6 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE PAPER I STATISTICAL THEORY The Socety provdes these solutons to assst canddates preparng for the eamnatons n future years and for
More informationMethods of Detecting Outliers in A Regression Analysis Model.
Methods of Detectng Outlers n A Regresson Analyss Model. Ogu, A. I. *, Inyama, S. C+, Achugamonu, P. C++ *Department of Statstcs, Imo State Unversty,Owerr +Department of Mathematcs, Federal Unversty of
More informationPolynomial Regression Models
LINEAR REGRESSION ANALYSIS MODULE XII Lecture - 6 Polynomal Regresson Models Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Test of sgnfcance To test the sgnfcance
More informationSIMPLE LINEAR REGRESSION
Smple Lnear Regresson and Correlaton Introducton Prevousl, our attenton has been focused on one varable whch we desgnated b x. Frequentl, t s desrable to learn somethng about the relatonshp between two
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Experment-I MODULE VII LECTURE - 3 ANALYSIS OF COVARIANCE Dr Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Any scentfc experment s performed
More informationA Robust Method for Calculating the Correlation Coefficient
A Robust Method for Calculatng the Correlaton Coeffcent E.B. Nven and C. V. Deutsch Relatonshps between prmary and secondary data are frequently quantfed usng the correlaton coeffcent; however, the tradtonal
More informationCHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng
More informationDO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR. Introductory Econometrics 1 hour 30 minutes
25/6 Canddates Only January Examnatons 26 Student Number: Desk Number:...... DO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR Department Module Code Module Ttle Exam Duraton
More informationChapter 2 - The Simple Linear Regression Model S =0. e i is a random error. S β2 β. This is a minimization problem. Solution is a calculus exercise.
Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where y + = β + β e for =,..., y and are observable varables e s a random error How can an estmaton rule be constructed for the
More informatione i is a random error
Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where + β + β e for,..., and are observable varables e s a random error How can an estmaton rule be constructed for the unknown
More informationUncertainty in measurements of power and energy on power networks
Uncertanty n measurements of power and energy on power networks E. Manov, N. Kolev Department of Measurement and Instrumentaton, Techncal Unversty Sofa, bul. Klment Ohrdsk No8, bl., 000 Sofa, Bulgara Tel./fax:
More informationY = β 0 + β 1 X 1 + β 2 X β k X k + ε
Chapter 3 Secton 3.1 Model Assumptons: Multple Regresson Model Predcton Equaton Std. Devaton of Error Correlaton Matrx Smple Lnear Regresson: 1.) Lnearty.) Constant Varance 3.) Independent Errors 4.) Normalty
More informationLecture 3 Stat102, Spring 2007
Lecture 3 Stat0, Sprng 007 Chapter 3. 3.: Introducton to regresson analyss Lnear regresson as a descrptve technque The least-squares equatons Chapter 3.3 Samplng dstrbuton of b 0, b. Contnued n net lecture
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationHydrological statistics. Hydrological statistics and extremes
5--0 Stochastc Hydrology Hydrologcal statstcs and extremes Marc F.P. Berkens Professor of Hydrology Faculty of Geoscences Hydrologcal statstcs Mostly concernes wth the statstcal analyss of hydrologcal
More informationProperties of Least Squares
Week 3 3.1 Smple Lnear Regresson Model 3. Propertes of Least Squares Estmators Y Y β 1 + β X + u weekly famly expendtures X weekly famly ncome For a gven level of x, the expected level of food expendtures
More informationSome basic statistics and curve fitting techniques
Some basc statstcs and curve fttng technques Statstcs s the dscplne concerned wth the study of varablty, wth the study of uncertanty, and wth the study of decsonmakng n the face of uncertanty (Lndsay et
More informationStatistics for Managers Using Microsoft Excel/SPSS Chapter 14 Multiple Regression Models
Statstcs for Managers Usng Mcrosoft Excel/SPSS Chapter 14 Multple Regresson Models 1999 Prentce-Hall, Inc. Chap. 14-1 Chapter Topcs The Multple Regresson Model Contrbuton of Indvdual Independent Varables
More informationCHAPTER IV RESEARCH FINDING AND DISCUSSIONS
CHAPTER IV RESEARCH FINDING AND DISCUSSIONS A. Descrpton of Research Fndng. The Implementaton of Learnng Havng ganed the whole needed data, the researcher then dd analyss whch refers to the statstcal data
More informationa. (All your answers should be in the letter!
Econ 301 Blkent Unversty Taskn Econometrcs Department of Economcs Md Term Exam I November 8, 015 Name For each hypothess testng n the exam complete the followng steps: Indcate the test statstc, ts crtcal
More information7.1. Single classification analysis of variance (ANOVA) Why not use multiple 2-sample 2. When to use ANOVA
Sngle classfcaton analyss of varance (ANOVA) When to use ANOVA ANOVA models and parttonng sums of squares ANOVA: hypothess testng ANOVA: assumptons A non-parametrc alternatve: Kruskal-Walls ANOVA Power
More informationCHAPTER 8. Exercise Solutions
CHAPTER 8 Exercse Solutons 77 Chapter 8, Exercse Solutons, Prncples of Econometrcs, 3e 78 EXERCISE 8. When = N N N ( x x) ( x x) ( x x) = = = N = = = N N N ( x ) ( ) ( ) ( x x ) x x x x x = = = = Chapter
More information