Department of Statistics University of Toronto STA305H1S / 1004 HS Design and Analysis of Experiments Term Test  Winter Solution


 Derrick Sullivan
 1 years ago
 Views:
Transcription
1 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 nonprogrammable calculator and a twosded 8.5'' '' ad sheet. If you do not understand a queston, or are havng some other dffculty, do not hestate to ask your nstructor or T.A for clarfcaton. There are 3 pages ncludng ths page. The last pages are statstcal tables. Please check that you are not mssng any page. Show all your work and answer n the space provded, n nk. Pencl may be used, but then remarks wll not be allowed. Use back of pages for rough work. Total pont: Good luck!!! Queston 3 Total Max 6 3 Score
2 Queston A study was conducted to test the mpact of 3 fertlzers on crop yeld. The 3 fertlzers were appled to 7 plots of land n a random fashon such that each fertlzer was appled to 9 plots. a) ( mark) Name the expermental desgn that was used n plannng ths study. Ths s a completely randomzed desgn. b) ( marks) Were randomzaton and replcaton used n ths experment? If yes, explan how? Randomzaton was used snce the fertlzers (treatments) were randomly assgned to each plot of land (expermental unt). Replcaton was used snce each fertlzer (treatment) was appled to 9 plots of land (expermental unts). c) ( marks) What statstcal model would you use for ths study desgn? The model s onefactor fxed effect model descrbed by the followng equaton: Y = + τ + ε, =,,3, j =,...,9 Where s the overall mean yeld τ s the effect of the th fertlzer ε s the random or expermental error d) (3 marks) What assumptons dd you make n part (c)? We assume that ε are..d. wth dstrbuton that s N(, σ ). e) (5 marks) Set up one set of orthogonal contrasts that mght be used n ths study. Snce there are three treatments a set of orthogonal contrast wll contan two orthogonal contrasts. An example of such a set s:
3 + 3 Ψ = and Ψ = 3 Checkng for orthogonalty 3 Sum c d  c d f) (3 marks) For each contrast n (d), state the null and alternatve hypotheses to whch the contrast corresponds. The frst contrast test whether the mean crop yeld wth the frst fertlzer s the same as the average mean crop yeld wth the other two fertlzers. The hypotheses are: H H a + 3 : + 3 : = The second contrast test whether the mean crop yeld wth the second fertlzer s the same as the mean crop yeld wth the thrd fertlzers. The hypotheses are: H H a : 3 = : 3 g) ( marks) Suppose that the 7 plots were selected at random from a bg feld contanng many more plots of land. How would your answers to parts (a), (c) and (d) change? Explan! If the plots of land were selected at random the desgn wll stll be a completely randomzed desgn and the model wll stll be onefactor fxed effect model. So the answers to parts (a), (c) and (d) wll not change. However, f the fertlzers were selected at random from say all avalable fertlzers n the market. Then the model wll be onefactor random effect model. The equaton of the model wll be the same; but, there wll be two addtonal assumptons: () τ are statstcally ndependent of the ε N, σ τ () the τ are..d ( ) 3
4 Queston Four dfferent desgns for a dgtal computer crcut are beng studed to compare the amount of nose present. The results are shown n the table bellow: Crcut Desgn Nose Observed Mean Std Dev SS Total = 99, SS Treat = a) (5 marks) Explan what knd of expermental desgn was used n ths experment. Are the effects of the factor random or fxed? What s the statstcal model you would use to analyze ths data? Ths s a completely randomzed desgn. The effects of the factor (crcut desgn) are fxed snce they were specfed by the expermenter rather than beng selected at random. The statstcal model we would use to analyze ths data s a onefactor fxed effect model descrbed by the equaton: Y = + τ + ε, =,,3,, j =,...,5 Where s the overall mean yeld τ s the effect of the th crcut desgn ε s the random or expermental error b) (9 marks) Construct the ANOVA table for ths experment. Fnd the Pvalues. The ANOVA table produced by SAS s gven below: Source DF SS MS Frato Pvalue Treatment 3.78 <. Error Total 9 99
5 c) (3 marks) Do the three crcut desgns have the same mean nose observed? Use α = %. The hypotheses of nterests here are: H H a : τ = τ = τ 3 = τ = : at least oneτ The test statstc obtaned from the ANOVA table produced n part (b)s: F obs =. 78 whch has an F(3,6) dstrbuton. Snce the Pvalue <. < α =. we reject the null hypothess and conclude that we have sgnfcant evdence that the three crcut desgns do not have the same mean nose observed. d) (5 marks) It was suspected before the experment that crcut desgns and are smlar n the nose present. Test ths hypothess usng a ttest and α = 5%. The hypotheses to test here are: H H a : τ = τ : τ τ or H H a : τ τ = : τ τ Further, we know that an unbased estmate of the dfference between two treatment effects s Y Y j and that Y + Y j ~ N τ τ j, σ. r rj Therefore, the test statstcs s: t = Y MS Y j E + r rj whch has a t(na) dstrbuton Substtutng all the values we get t obs = =. wth df = The Pvalue can be estmated as follows: ( t( 6) > t ) = P( t( 6) >.). P value = > P obs Snce Pvalue s very large we cannot reject H and we able to conclude that that crcut desgns and are not dfferent n the nose present. 5
6 e) (5 marks) How many dfferent orthogonal contrasts you can create smultaneously n ths experment? Create two contrasts, one to test the queston n part (d), the other to test whether the mean response for crcut desgn 3 s the same as for the average for crcut desgn and. Are these contrasts orthogonal? Snce the factor (crcut desgn) has a = levels, a  = 3 dfferent orthogonal contrasts can be created smultaneously. The contrast to test the queston n part (d) s: Ψ =. The contrast to test whether the mean response for crcut desgn 3 s the same as for + the average for crcut desgn and s: Ψ = 3. Checkng for orthogonalty 3 Sum c  d c d So yes, these contrasts are orthogonal. f) (8 marks) Calculate SS for both contrasts n part (e). Test the hypothess regardng these two contrasts usng Ftest. How do they compare wth part (c)? Contrast : the hypotheses are H = vs H :. The sum of square of the contrast s: : a SS contrast a ( cy ) ( ) = = a c / = r = ( / 5) + ( / 5) SSContrast /. The test statstcs s: F obs = = =. 37 wth df = (, 6) MS 8.35 The Pvalue can then be estmated as follow: E ( F(,6) > F ) = P( F(,6) >.37). P value = P obs >. So cannot reject H. =. Note, the F statstcs n ths case s smply the square of the t statstcs from part (d). Contrast : the hypotheses are: ( + )/ = vs H : ( + )/ The sum of square of the contrast s: SS = H. : 3 a 3 contrast The test statstcs s: F obs = = wth df = (, 6) 8.35 P value = P F,6 > 6.53 <.. The Pvalue can then be estmated as follow: ( ( ) ) So we reject H. Combnaton of these two results s n agreement wth (c), meanng that not all means are equal (two may be). 6
7 g) (3 marks) How can you calculate the SS for the contrast comparng crcut desgn wth the average of the other three wthout usng the formula for SS contrast? Do t! The contrast comparng crcut desgn wth the average of the other three s orthogonal to the two contrasts n part (e), therefore formng a set of orthogonal contrasts. For any set of orthogonal contrasts we have that SS. Treat = SScontrast + SScontrast + + SScontrast a Therefore, the SS of ths contrast s SS 3 SS SS = = 69.7 contrast = Treat contrast SScontrast Below are plot of the resduals versus the ftted values and a normal quantle plot of the resduals for the model used to analyze the data above. z ftted z Normal Percent l es 7
8 h) (5 marks) What are the assumptons of the model used to construct the ANOVA table n part (c)? Comment on the valdty of these assumptons. The assumptons are: The model form s as specfed n part (a), that s E(Y ) = + τ. The resduals, ε, are..d. N(, σ ). Based on the resdual plots above, t looks lke all of these assumptons are vald for ths data. ) (3 marks) Based on the plots above are there any outlers n ths data? Explan. Yes, t looks lke there s one outler n ths data. It appears n the plot of the resduals versus ftted value as the rghtmost and lowest pont (.e., large negatve resdual). In addton, t appears on the normal quantle plot as the lowst pont on the left. Lookng at the data we see that the value of 5 th observaton taken on crcut desgn s 8 whch s much smaller than the rest of the observatons, suggestng agan that ths may be an outler. Queston 3 An experment was conducted to study the lfe (n hours) of two dfferent brands of batteres (brand A and B) n three dfferent devces (rado, camera and portable DVD player). A completely randomzed twofactor factoral experment was conducted. Some SAS outputs used to analyze the data from ths experment are gven below: 8
9 a) (5 marks) What statstcal model was used to analyze ths data? Wrte the model and descrbe each term n the model n the context of ths study. Lst all assumptons requred for the model. Ths s a twofactor fxedeffect model. Its equaton s: Y k = + α + β j + γ + ε k where: Overall mean α Battery effect (factor A) of level =, β j Devce effect (factor B) of level j =,, 3 γ Interacton effect of batter (factor A) level and devce (factor B) level j Expermental error ε The assumptons of the model are: ε k are..d. N(, σ ) In order to obtan unbased estmators, we requre that: a = b j= a = α = β = j b γ = γ j= = 9
10 b) ( marks) Create the ANOVA table that was used n the analyss of ths data usng the results from the SAS output above, ncludng Pvalues. The ANOVA table s: Source DF SS MS Frato Pvalue Factor A Factor B Interacton A B <..63 Error Total 3.85 c) (5 marks) Plot an nteracton plot usng the cell means as gven n the output above. Use dfferent symbols/colors for the dfferent battery brand. What do you learn from ths plot? Here s an nteracton plot produced by MINITAB Interacton Plot (data means) Battery A B 8. Mean Camera DVD Devce Rado From the plots, t appears that there s no nteracton between battery type and devce. Further, for every devce the mean lfetme for battery B appears to be larger than that of batter A suggestng that there mght be a battery effect. Fnally, t looks lke batteres used n DVDs have the smallest lfetme whle batteres used n Rados last the longest. Ths, n turn, suggests that there mght be a devce effect.
11 d) (9 marks) Do battery brand and devce type nteract? Is there any dfference n lfe tme of the two battery brands? Does devce type have any effect on battery lfe tme? Answer these three questons, f approprate, usng sgnfcant level 5%. State each queston n terms of the model and state your conclusons n plan language n the context of ths experment. Frst we need to test for nteracton. The hypotheses of nterest are: H : γ =, for all, j vs H a : at least one γ. From the ANOVA table we get the test statstcs F obs =.8, wth Pvalue =.63, therefore we cannot reject the null hypothess and we conclude that there s no sgnfcant nteracton between battery type and devce type. Snce there s no nteracton we can proceed to test for man effects of battery and devce. Man effect of battery: The hypotheses f nterest are: H : α =, for =, vs H a : at least one α. From the ANOVA table we get the test statstcs F obs = 9.33, wth Pvalue =.. Hence, we can reject the null hypothess at α = 5% and we conclude that there s a sgnfcant effect of battery type on lfetme. Note, that ths s moderate evdence of sgnfcance as we would not be able to reject the null hypothess at α = %. Man effect of devce: The hypotheses f nterest are: H : β =, for =,, 3 vs H a : at least one β. From the ANOVA table we get the test statstcs F obs = 3.75, wth Pvalue<.. Hence, we have strong evdence to reject the null hypothess and we conclude that there s a sgnfcant effect of devce type on lfetme. e) ( marks) If the researchers assumed (from experence) before the experment that battery brand and devce type don t nteract, how would ths affect the model used to analyze ths data? Wrte the model and descrbe each term n the context of ths study. In ths case the model wll not nclude an nteracton effect term, that s, the model equaton wll be: Y k = + α + β j + ε k (addtve model) where: s the overall mean, α are effects of battery, β j are effects of devce and ε are expermental errors.
12 f) (5 marks) Create the ANOVA table that would be used n part (e), ncludng Pvalue, and test the man effects. Are the results consstent wth the orgnal model used n the study? Snce we omt the nteracton term from the model, both the degrees of freedom and sums of squares of the nteracton term n the ANOVA table go to the error. The ANOVA table s then: Source DF SS MS Frato Pvalue Factor A Factor B < P <.5 <. Error Total 3.85 END!
STATISTICS 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 informationTopic 23  Randomized Complete Block Designs (RCBD)
Topc 3 ANOVA (III) 31 Topc 3  Randomzed Complete Block Desgns (RCBD) Defn: A Randomzed Complete Block Desgn s a varant of the completely randomzed desgn (CRD) that we recently learned. In ths desgn,
More informationF statistic = s2 1 s 2 ( F for Fisher )
Stat 4 ANOVA Analyss of Varance /6/04 Comparng Two varances: F dstrbuton Typcal Data Sets One way analyss of varance : example Notaton for one way ANOVA Comparng Two varances: F dstrbuton We saw that the
More informationTwofactor model. Statistical Models. Least Squares estimation in LM twofactor model. Rats
tatstcal Models Lecture nalyss of Varance wofactor model Overall mean Man effect of factor at level Man effect of factor at level Y µ + α + β + γ + ε Eε f (, ( l, Cov( ε, ε ) lmr f (, nteracton effect
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 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 yvalue on the ftted lne, y ˆ ˆ = α + x. The are called ftted values.
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 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 informationPsychology 282 Lecture #24 Outline Regression Diagnostics: Outliers
Psychology 282 Lecture #24 Outlne Regresson Dagnostcs: Outlers In an earler lecture we studed the statstcal assumptons underlyng the regresson model, ncludng the followng ponts: Formal statement of assumptons.
More informationTesting for seasonal unit roots in heterogeneous panels
Testng for seasonal unt roots n heterogeneous panels Jesus Otero * Facultad de Economía Unversdad del Rosaro, Colomba Jeremy Smth Department of Economcs Unversty of arwck Monca Gulett Aston Busness School
More informationIntroduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:
CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and
More informationChapter 8 Indicator Variables
Chapter 8 Indcator Varables In general, e explanatory varables n any regresson analyss are assumed to be quanttatve n nature. For example, e varables lke temperature, dstance, age etc. are quanttatve n
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 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 PrentceHall, Inc. Chap. 131 Chapter Topcs Types of Regresson Models Determnng the Smple Lnear
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 informationAnalytical Chemistry Calibration Curve Handout
I. Quckand 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 informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the faketest data; fxed
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 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 informationexperimenteel en correlationeel onderzoek
expermenteel en correlatoneel onderzoek lecture 6: oneway analyss of varance Leary. Introducton to Behavoral Research Methods. pages 246 271 (chapters 10 and 11): conceptual statstcs Moore, McCabe, and
More informationsince [1( 0+ 1x1i+ 2x2 i)] [ 0+ 1x1i+ assumed to be a reasonable approximation
Econ 388 R. Butler 204 revsons Lecture 4 Dummy Dependent Varables I. Lnear Probablty Model: the Regresson model wth a dummy varables as the dependent varable assumpton, mplcaton regular multple regresson
More informationChapter 12 Analysis of Covariance
Chapter Analyss of Covarance Any scentfc experment s performed to know somethng that s unknown about a group of treatments and to test certan hypothess about the correspondng treatment effect When varablty
More informationNow we relax this assumption and allow that the error variance depends on the independent variables, i.e., heteroskedasticity
ECON 48 / WH Hong Heteroskedastcty. Consequences of Heteroskedastcty for OLS Assumpton MLR. 5: Homoskedastcty var ( u x ) = σ Now we relax ths assumpton and allow that the error varance depends on the
More informationEXAMINATION. N0028N Econometrics. Luleå University of Technology. Date: (A1016) Time: Aid: Calculator and dictionary
EXAMINATION Luleå Unversty of Technology N008N Econometrcs Date: 0110516 (A1016) Tme: 09.0013.00 Ad: Calculator and dctonary Teacher on duty (complete telephone number) Robert Lundmark (0701735788)
More informationProblem Set 9 Solutions
Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem
More informationASLevel Maths: Statistics 1 for Edexcel
1 of 6 ASLevel Maths: Statstcs 1 for Edecel S1. Calculatng means and standard devatons Ths con ndcates the slde contans actvtes created n Flash. These actvtes are not edtable. For more detaled nstructons,
More informationAssignment 5. Simulation for Logistics. Monti, N.E. Yunita, T.
Assgnment 5 Smulaton for Logstcs Mont, N.E. Yunta, T. November 26, 2007 1. Smulaton Desgn The frst objectve of ths assgnment s to derve a 90% twosded Confdence Interval (CI) for the average watng tme
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 leastsquares equatons Chapter 3.3 Samplng dstrbuton of b 0, b. Contnued n net lecture
More informationStat 642, Lecture notes for 01/27/ d i = 1 t. n i t nj. n j
Stat 642, Lecture notes for 01/27/05 18 Rate Standardzaton Contnued: Note that f T n t where T s the cumulatve followup tme and n s the number of subjects at rsk at the mdpont or nterval, and d s the
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 informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1s tme nterval. The velocty of the partcle
More information17  LINEAR REGRESSION II
Topc 7 Lnear Regresson II 7 Topc 7  LINEAR REGRESSION II Testng and Estmaton Inferences about β Recall that we estmate Yˆ ˆ β + ˆ βx. 0 μ Y X x β0 + βx usng To estmate σ σ squared error Y X x ε s ε we
More information8.6 The Complex Number System
8.6 The Complex Number System Earler n the chapter, we mentoned that we cannot have a negatve under a square root, snce the square of any postve or negatve number s always postve. In ths secton we want
More informationCopyright 2017 by Taylor Enterprises, Inc., All Rights Reserved. Adjusted Control Limits for P Charts. Dr. Wayne A. Taylor
Taylor Enterprses, Inc. Control Lmts for P Charts Copyrght 2017 by Taylor Enterprses, Inc., All Rghts Reserved. Control Lmts for P Charts Dr. Wayne A. Taylor Abstract: P charts are used for count data
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 informationCopyright 2017 by Taylor Enterprises, Inc., All Rights Reserved. Adjusted Control Limits for U Charts. Dr. Wayne A. Taylor
Taylor Enterprses, Inc. Adjusted Control Lmts for U Charts Copyrght 207 by Taylor Enterprses, Inc., All Rghts Reserved. Adjusted Control Lmts for U Charts Dr. Wayne A. Taylor Abstract: U charts are used
More informationF8: Heteroscedasticity
F8: Heteroscedastcty Feng L Department of Statstcs, Stockholm Unversty What s socalled heteroscedastcty In a lnear regresson model, we assume the error term has a normal dstrbuton wth mean zero and varance
More informationChapter 4 Experimental Design and Their Analysis
Chapter 4 Expermental Desgn and her Analyss Desgn of experment means how to desgn an experment n the sense that how the obseratons or measurements should be obtaned to answer a query n a ald, effcent and
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 informationIII. Econometric Methodology Regression Analysis
Page Econ07 Appled Econometrcs Topc : An Overvew of Regresson Analyss (Studenmund, Chapter ) I. The Nature and Scope of Econometrcs. Lot s of defntons of econometrcs. Nobel Prze Commttee Paul Samuelson,
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 information10701/ Machine Learning, Fall 2005 Homework 3
10701/15781 Machne Learnng, Fall 2005 Homework 3 Out: 10/20/05 Due: begnnng of the class 11/01/05 Instructons Contact questons10701@autonlaborg for queston Problem 1 Regresson and Crossvaldaton [40
More informationPARTIALLY BALANCED INCOMPLETE BLOCK DESIGNS
PARTIALLY BALANCED INCOMPLETE BLOCK DESIGNS V.K. Sharma I.A.S.R.I., Lbrary Avenue, New Delh00. Introducton Balanced ncomplete block desgns, though have many optmal propertes, do not ft well to many expermental
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 leastsquares regresson lne (LSRL) to represent a lnear relatonshp etween two quanttatve explanator
More informationMetaAnalysis of Correlated Proportions
NCSS Statstcal Softare Chapter 457 MetaAnalyss of Correlated Proportons Introducton Ths module performs a metaanalyss of a set of correlated, bnaryevent studes. These studes usually come from a desgn
More informationREGRESSION ANALYSIS II MULTICOLLINEARITY
REGRESSION ANALYSIS II MULTICOLLINEARITY QUESTION 1 Departments of Open Unversty of Cyprus A and B consst of na = 35 and nb = 30 students respectvely. The students of department A acheved an average test
More informationBayesian predictive Configural Frequency Analysis
Psychologcal Test and Assessment Modelng, Volume 54, 2012 (3), 285292 Bayesan predctve Confgural Frequency Analyss Eduardo GutérrezPeña 1 Abstract Confgural Frequency Analyss s a method for cellwse
More informationPASS Sample Size Software
Chapter 57 Introducton Ths procedure power analyzes random effects desgns n whch the outcome (response) s contnuous. Thus, as wth the analyss of varance (ANOVA), the procedure s used to test hypotheses
More informationIntroduction to Econometrics (3 rd Updated Edition, Global Edition) Solutions to OddNumbered EndofChapter Exercises: Chapter 13
Introducton to Econometrcs (3 rd Updated Edton, Global Edton by James H. Stock and Mark W. Watson Solutons to OddNumbered EndofChapter Exercses: Chapter 13 (Ths verson August 17, 014 Stock/Watson 
More informationTHE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens
THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of
More informationChapter 8 Multivariate Regression Analysis
Chapter 8 Multvarate Regresson Analyss 8.3 Multple Regresson wth K Independent Varables 8.4 Sgnfcance tests of Parameters Populaton Regresson Model For K ndependent varables, the populaton regresson and
More informationCHAPER 11: HETEROSCEDASTICITY: WHAT HAPPENS WHEN ERROR VARIANCE IS NONCONSTANT?
Basc Econometrcs, Gujarat and Porter CHAPER 11: HETEROSCEDASTICITY: WHAT HAPPENS WHEN ERROR VARIANCE IS NONCONSTANT? 11.1 (a) False. The estmators are unbased but are neffcent. (b) True. See Sec. 11.4
More informationElectrical double layer: revisit based on boundary conditions
Electrcal double layer: revst based on boundary condtons Jong U. Km Department of Electrcal and Computer Engneerng, Texas A&M Unversty College Staton, TX 77843318, USA Abstract The electrcal double layer
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 informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of ExermentsI MODULE II LECTURE  GENERAL LINEAR HYPOTHESIS AND ANALYSIS OF VARIANCE Dr. Shalabh Deartment of Mathematcs and Statstcs Indan Insttute of Technology Kanur 3.
More informationGoodness of fit and Wilks theorem
DRAFT 0.0 Glen Cowan 3 June, 2013 Goodness of ft and Wlks theorem Suppose we model data y wth a lkelhood L(µ) that depends on a set of N parameters µ = (µ 1,...,µ N ). Defne the statstc t µ ln L(µ) L(ˆµ),
More informationAssessing Studies Based on Multiple Regression
Assessng Studes Based on Multple Regresson (SW Chapter 9) Outlne 1. Internal and External Valdty 2. Threats to Internal Valdty a. Omtted varable bas b. Functonal form msspecfcaton c. Errorsnvarables
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 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 informationJAB Chain. Longtail claims development. ASTIN  September 2005 B.Verdier A. Klinger
JAB Chan Longtal clams development ASTIN  September 2005 B.Verder A. Klnger Outlne Chan Ladder : comments A frst soluton: Munch Chan Ladder JAB Chan Chan Ladder: Comments Black lne: average pad to ncurred
More informationEcon Statistical Properties of the OLS estimator. Sanjaya DeSilva
Econ 39  Statstcal Propertes of the OLS estmator Sanjaya DeSlva September, 008 1 Overvew Recall that the true regresson model s Y = β 0 + β 1 X + u (1) Applyng the OLS method to a sample of data, we estmate
More informationA LINEAR PROGRAM TO COMPARE MULTIPLE GROSS CREDIT LOSS FORECASTS. Dr. Derald E. Wentzien, Wesley College, (302) ,
A LINEAR PROGRAM TO COMPARE MULTIPLE GROSS CREDIT LOSS FORECASTS Dr. Derald E. Wentzen, Wesley College, (302) 7362574, wentzde@wesley.edu ABSTRACT A lnear programmng model s developed and used to compare
More informationI have not received unauthorized aid in the completion of this exam.
ME 270 Sprng 2013 Fnal Examnaton Please read and respond to the followng statement, I have not receved unauthorzed ad n the completon of ths exam. Agree Dsagree Sgnature INSTRUCTIONS Begn each problem
More informationUnit 5: Quadratic Equations & Functions
Date Perod Unt 5: Quadratc Equatons & Functons DAY TOPIC 1 Modelng Data wth Quadratc Functons Factorng Quadratc Epressons 3 Solvng Quadratc Equatons 4 Comple Numbers Smplfcaton, Addton/Subtracton & Multplcaton
More informationEntropy generation in a chemical reaction
Entropy generaton n a chemcal reacton E Mranda Área de Cencas Exactas COICET CCT Mendoza 5500 Mendoza, rgentna and Departamento de Físca Unversdad aconal de San Lus 5700 San Lus, rgentna bstract: Entropy
More informationAntivan der Waerden numbers of 3term arithmetic progressions.
Antvan der Waerden numbers of 3term arthmetc progressons. Zhanar Berkkyzy, Alex Schulte, and Mchael Young Aprl 24, 2016 Abstract The antvan der Waerden number, denoted by aw([n], k), s the smallest
More informationTimeVarying Systems and Computations Lecture 6
TmeVaryng Systems and Computatons Lecture 6 Klaus Depold 14. Januar 2014 The Kalman Flter The Kalman estmaton flter attempts to estmate the actual state of an unknown dscrete dynamcal system, gven nosy
More informationDr. Ing. J. H. (Jo) Walling Consultant Cables Standards Machinery
The common mode crcut resstance unbalance (CMCU) calculaton based on mn. / max. conductor resstance values and par to par resstance unbalance measurements ncludng loop resstance evsed and extended verson
More informationThe topics in this section concern with the second course objective. Correlation is a linear relation between two random variables.
4.1 Correlaton The topcs n ths secton concern wth the second course objectve. Correlaton s a lnear relaton between two random varables. Note that the term relaton used n ths secton means connecton or relatonshp
More informationComparison of the Population Variance Estimators. of 2Parameter Exponential Distribution Based on. Multiple Criteria Decision Making Method
Appled Mathematcal Scences, Vol. 7, 0, no. 47, 070 HIARI Ltd, www.mhkar.com Comparson of the Populaton Varance Estmators of Parameter Exponental Dstrbuton Based on Multple Crtera Decson Makng Method
More informationFor example, if the drawing pin was tossed 200 times and it landed point up on 140 of these trials,
Probablty In ths actvty you wll use some real data to estmate the probablty of an event happenng. You wll also use a varety of methods to work out theoretcal probabltes. heoretcal and expermental probabltes
More informationTests of Single Linear Coefficient Restrictions: ttests and Ftests. 1. Basic Rules. 2. Testing Single Linear Coefficient Restrictions
ECONOMICS 35*  NOTE ECON 35*  NOTE Tests of Sngle Lnear Coeffcent Restrctons: ttests and tests Basc Rules Tests of a sngle lnear coeffcent restrcton can be performed usng ether a twotaled ttest
More informationECE559VV Project Report
ECE559VV Project Report (Supplementary Notes Loc Xuan Bu I. MAX SUMRATE SCHEDULING: THE UPLINK CASE We have seen (n the presentaton that, for downlnk (broadcast channels, the strategy maxmzng the sumrate
More informationFE REVIEW OPERATIONAL AMPLIFIERS (OPAMPS)
FE EIEW OPEATIONAL AMPLIFIES (OPAMPS) 1 The Opamp An opamp has two nputs and one output. Note the opamp below. The termnal labeled wth the () sgn s the nvertng nput and the nput labeled wth the () sgn
More informationPopulation element: 1 2 N. 1.1 Sampling with Replacement: HansenHurwitz Estimator(HH)
Chapter 1 Samplng wth Unequal Probabltes Notaton: Populaton element: 1 2 N varable of nterest Y : y1 y2 y N Let s be a sample of elements drawn by a gven samplng method. In other words, s s a subset of
More informationDEMO #8  GAUSSIAN ELIMINATION USING MATHEMATICA. 1. Matrices in Mathematica
demo8.nb 1 DEMO #8  GAUSSIAN ELIMINATION USING MATHEMATICA Obectves:  defne matrces n Mathematca  format the output of matrces  appl lnear algebra to solve a real problem  Use Mathematca to perform
More informationTHE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructions
THE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructons by George Hardgrove Chemstry Department St. Olaf College Northfeld, MN 55057 hardgrov@lars.acc.stolaf.edu Copyrght George
More informationNonMixture Cure Model for Interval Censored Data: Simulation Study ABSTRACT
Malaysan Journal of Mathematcal Scences 8(S): 3744 (2014) Specal Issue: Internatonal Conference on Mathematcal Scences and Statstcs 2013 (ICMSS2013) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal
More informationSinger & Willett, 2003 October 13, 2003
Snger & Wllett, October, Dong Data Analyss n n the the Multlevel Model for for Change Judy Snger & John Wllett Harvard Unversty Graduate School of Educaton What What we we wll wll cover? cover? Composte
More informationAssignment 4 Solutions
Assgnment 4 Solutons Tmothy Vs February 3, 2006 332 We have n 25, 0.04, P MT 100, and we need to fnd F V. F V P MT (1 + )n 1 100 (1 + 0.04)2 5 1 0.04 4164.59 336 We have F V 8000, n 30, 0.03, and we
More informationLossy Compression. Compromise accuracy of reconstruction for increased compression.
Lossy Compresson Compromse accuracy of reconstructon for ncreased compresson. The reconstructon s usually vsbly ndstngushable from the orgnal mage. Typcally, one can get up to 0:1 compresson wth almost
More information16 Reflection and transmission, TE mode
16 Reflecton transmsson TE mode Last lecture we learned how to represent planetem waves propagatng n a drecton ˆ n terms of feld phasors such that η = Ẽ = E o e j r H = ˆ Ẽ η µ ɛ = ˆ = ω µɛ E o =0. Such
More informationNumerical Transient Heat Conduction Experiment
Numercal ransent Heat Conducton Experment OBJECIVE 1. o demonstrate the basc prncples of conducton heat transfer.. o show how the thermal conductvty of a sold can be measured. 3. o demonstrate the use
More informationChapter 4: Regression With One Regressor
Chapter 4: Regresson Wth One Regressor Copyrght 2011 Pearson AddsonWesley. All rghts reserved. 11 Outlne 1. Fttng a lne to data 2. The ordnary least squares (OLS) lne/regresson 3. Measures of ft 4. Populaton
More informationPolynomials. 1 More properties of polynomials
Polynomals 1 More propertes of polynomals Recall that, for R a commutatve rng wth unty (as wth all rngs n ths course unless otherwse noted), we defne R[x] to be the set of expressons n =0 a x, where a
More informationSupplement to Clustering with Statistical Error Control
Supplement to Clusterng wth Statstcal Error Control Mchael Vogt Unversty of Bonn Matthas Schmd Unversty of Bonn In ths supplement, we provde the proofs that are omtted n the paper. In partcular, we derve
More informationRegression. The Simple Linear Regression Model
Regresson Smple Lnear Regresson Model Least Squares Method Coeffcent of Determnaton Model Assumptons Testng for Sgnfcance Usng the Estmated Regresson Equaton for Estmaton and Predcton Resdual Analss: Valdatng
More informationContinuous vs. Discrete Goods
CE 651 Transportaton Economcs Charsma Choudhury Lecture 34 Analyss of Demand Contnuous vs. Dscrete Goods Contnuous Goods Dscrete Goods x auto 1 Indfference u curves 3 u u 1 x 1 0 1 bus Outlne Data Modelng
More informationAndreas C. Drichoutis Agriculural University of Athens. Abstract
Heteroskedastcty, the sngle crossng property and ordered response models Andreas C. Drchouts Agrculural Unversty of Athens Panagots Lazards Agrculural Unversty of Athens Rodolfo M. Nayga, Jr. Texas AMUnversty
More informationNormally, in one phase reservoir simulation we would deal with one of the following fluid systems:
TPG4160 Reservor Smulaton 2017 page 1 of 9 ONEDIMENSIONAL, ONEPHASE RESERVOIR SIMULATION Flud systems The term sngle phase apples to any system wth only one phase present n the reservor In some cases
More informationInductor = (coil of wire)
A student n 1120 emaled me to ask how much extra he should expect to pay on hs electrc bll when he strngs up a standard 1strand box of ccle holday lghts outsde hs house. (total, cumulatve cost)? Try to
More informationSIMPLE REACTION TIME AS A FUNCTION OF TIME UNCERTAINTY 1
Journal of Expermental Vol. 5, No. 3, 1957 Psychology SIMPLE REACTION TIME AS A FUNCTION OF TIME UNCERTAINTY 1 EDMUND T. KLEMMER Operatonal Applcatons Laboratory, Ar Force Cambrdge Research Center An earler
More informationUNIVERSITY OF TORONTO. Faculty of Arts and Science JUNE EXAMINATIONS STA 302 H1F / STA 1001 H1F Duration  3 hours Aids Allowed: Calculator
UNIVERSITY OF TORONTO Faculty of Arts and Scence JUNE EXAMINATIONS 008 STA 30 HF / STA 00 HF Duraton  3 hours Ads Allowed: Calculator LAST NAME: FIRST NAME: STUDENT NUMBER: Enrolled n (Crcle one): STA30
More informationWeighted Voting Systems
Weghted Votng Systems Elssa Brown Chrssy Donovan Charles Noneman June 5, 2009 Votng systems can be deceptve. For nstance, a votng system mght consst of four people, n whch three of the people have 2 votes,
More informationNoninteracting Spin1/2 Particles in Noncommuting External Magnetic Fields
EJTP 6, No. 0 009) 43 56 Electronc Journal of Theoretcal Physcs Nonnteractng Spn1/ Partcles n Noncommutng External Magnetc Felds Kunle Adegoke Physcs Department, Obafem Awolowo Unversty, IleIfe, Ngera
More informationMin Cut, Fast Cut, Polynomial Identities
Randomzed Algorthms, Summer 016 Mn Cut, Fast Cut, Polynomal Identtes Instructor: Thomas Kesselhem and Kurt Mehlhorn 1 Mn Cuts n Graphs Lecture (5 pages) Throughout ths secton, G = (V, E) s a multgraph.
More informationBit Juggling. Representing Information. representations.  Some other bits.  Representing information using bits  Number. Chapter
Representng Informaton 1 1 1 1 Bt Jugglng  Representng nformaton usng bts  Number representatons  Some other bts Chapter 3.13.3 REMINDER: Problem Set #1 s now posted and s due next Wednesday L3 Encodng
More informationLecture 19. Endogenous Regressors and Instrumental Variables
Lecture 19. Endogenous Regressors and Instrumental Varables In the prevous lecture we consder a regresson model (I omt the subscrpts (1) Y β + D + u = 1 β The problem s that the dummy varable D s endogenous,.e.
More informationLab 4: Twolevel Random Intercept Model
BIO 656 Lab4 009 Lab 4: Twolevel Random Intercept Model Data: Peak expratory flow rate (pefr) measured twce, usng two dfferent nstruments, for 17 subjects. (from Chapter 1 of Multlevel and Longtudnal
More information