Simple Linear Regression and Correlation. Applied Statistics and Probability for Engineers. Chapter 11 Simple Linear Regression and Correlation
|
|
- Ashlyn Paul
- 6 years ago
- Views:
Transcription
1 4//6 Appled Statstcs ad Probablty for Egeers Sth Edto Douglas C. Motgomery George C. Ruger Chapter Smple Lear Regresso ad Correlato CHAPTER OUTLINE Smple Lear Regresso ad Correlato - Emprcal Models -8 Correlato - Smple Lear Regresso -9 Regresso o Trasformed Varables -3 Propertes of the Least Squares - Logstc Regresso Estmators -4 Hypothess Test Smple Lear Regresso -4. Use of t-tests -4. Aalyss of varace approach to test sgfcace of regresso -5 Cofdece Itervals -5. Cofdece tervals o the slope ad tercept -5. Cofdece terval o the mea respose -6 Predcto of New Observatos -7 Adequacy of the Regresso Model -7. Resdual aalyss -7. Coeffcet of determato (R ) Chapter Ttle ad Outle
2 4//6 Learg Objectves for Chapter After careful study of ths chapter, you should be able to do the followg:. Use smple lear regresso for buldg emprcal models to egeerg ad scetfc data.. Uderstad how the method of least squares s used to estmate the parameters a lear regresso model. 3. Aalyze resduals to determe f the regresso model s a adequate ft to the data or to see f ay uderlyg assumptos are volated. 4. Test the statstcal hypotheses ad costruct cofdece tervals o the regresso model parameters. 5. Use the regresso model to make a predcto of a future observato ad costruct a approprate predcto terval o the future observato. 6. Apply the correlato model. 7. Use smple trasformatos to acheve a lear regresso model. Chapter Learg Objectves 3 -: Emprcal Models May problems egeerg ad scece volve eplorg the relatoshps betwee two or more varables. Regresso aalyss s a statstcal techque that s very useful for these types of problems. For eample, a chemcal process, suppose that the yeld of the product s related to the processoperatg temperature. Regresso aalyss ca be used to buld a model to predct yeld at a gve temperature level. Sec - Emprcal Models 4
3 4//6 -: Smple Lear Regresso The smple lear regresso cosders a sgle regressor or predctor ad a depedet or respose varable Y. The epected value of Y at each level of s a radom varable: E(Y ) = b + b We assume that each observato, Y, ca be descrbed by the model Y = b + b + Sec - Smple Lear Regresso 5 -: Smple Lear Regresso Least Squares Estmates The least-squares estmates of the tercept ad slope the smple lear regresso model are bˆ y bˆ (-) bˆ y y (-) where y ( / ) y ad ( / ). Sec - Smple Lear Regresso 6 3
4 4//6 4 -: Smple Lear Regresso 7 The ftted or estmated regresso le s therefore (-3) Note that each par of observatos satsfes the relatoshp where s called the resdual. The resdual descrbes the error the ft of the model to the th observato y. y b b ˆ ˆ ˆ e y,,,, ˆ ˆ b b y y e ˆ Sec - Smple Lear Regresso -: Smple Lear Regresso Notato 8 S y y y S y Sec - Smple Lear Regresso
5 4//6 EXAMPLE - Oyge Purty We wll ft a smple lear regresso model to the oyge purty data Table -. The followg quattes may be computed: 3.9 y, y 9.65 y 7, y, (3.9) S ad y S y y (3.9)(,843.), Sec - Smple Lear Regresso 9 EXAMPLE - Oyge Purty - cotued Therefore, the least squares estmates of the slope ad tercept are ad bˆ y bˆ S bˆ S y ( ) The ftted smple lear regresso model (wth the coeffcets reported to three decmal places) s yˆ Sec - Smple Lear Regresso 5
6 4//6 -: Smple Lear Regresso Estmatg The error sum of squares s SS E e y yˆ It ca be show that the epected value of the error sum of squares s E(SS E ) = ( ). Sec - Smple Lear Regresso -: Smple Lear Regresso Estmatg A ubased estmator of s ˆ SS E (-4) where SS E ca be easly computed usg SSE SST ˆb Sy (-5) Sec - Smple Lear Regresso 6
7 4//6-3: Propertes of the Least Squares Estmators Slope Propertes E( bˆ) b Itercept Propertes V (ˆ) b S E(ˆ b ) b ad V (ˆ b ) S Sec -3 Propertes of the Least Squares Estmators 3-4: Hypothess Tests Smple Lear Regresso -4. Use of t-tests Suppose we wsh to test H : b = b, H : b b, A approprate test statstc would be T bˆ b ˆ /, S (-6) Sec -4 Hypothess Tests Smple Lear Regresso 4 7
8 4//6-4: Hypothess Tests Smple Lear Regresso -4. Use of t-tests The test statstc could also be wrtte as: T bˆ ˆ b se (ˆ b ) We would reject the ull hypothess f, t > t a/, - Sec -4 Hypothess Tests Smple Lear Regresso 5-4: Hypothess Tests Smple Lear Regresso -4. Use of t-tests Suppose we wsh to test H : b = b, H : b b, A approprate test statstc would be T ˆ bˆ b, S bˆ b se(ˆ b ), (-7) Sec -4 Hypothess Tests Smple Lear Regresso 6 8
9 4//6-4: Hypothess Tests Smple Lear Regresso -4. Use of t-tests We would reject the ull hypothess f t > t a/, - Sec -4 Hypothess Tests Smple Lear Regresso 7-4: Hypothess Tests Smple Lear Regresso -4. Use of t-tests A mportat specal case of the hypotheses of Equato -8 s H : b = H : b These hypotheses relate to the sgfcace of regresso. Falure to reject H s equvalet to cocludg that there s o lear relatoshp betwee ad Y. Sec -4 Hypothess Tests Smple Lear Regresso 8 9
10 4//6-4: Hypothess Tests Smple Lear Regresso EXAMPLE - Oyge Purty Tests of Coeffcets We wll test for sgfcace of regresso usg the model for the oyge purty data from Eample -. The hypotheses are H : b = H : b ad we wll use a =.. From Eample - ad Table - we have b ˆ 4.947, S.6888, ˆ.8 so the t-statstc Equato -6 becomes bˆ ˆ b t.35 ˆ / S se(ˆ b).8/.6888 Practcal Iterpretato: Sce the referece value of t s t.5,8 =.88, the value of the test statstc s very far to the crtcal rego, mplyg that H : b = should be rejected. There s strog evdece to support ths clam. The P-value for ths test s ~ 9 P.3. Ths was obtaed maually wth a calculator. Table - presets the Mtab output for ths problem. Notce that the t-statstc value for the slope s computed as.35 ad that the reported P-value s P =.. Mtab also reports the t-statstc for testg the hypothess H : b =. Ths statstc s computed from Equato -7, wth b, =, as t = Clearly, the, the hypothess that the tercept s zero s rejected. Sec -4 Hypothess Tests Smple Lear Regresso 9-4: Hypothess Tests Smple Lear Regresso -4. Aalyss of Varace Approach to Test Sgfcace of Regresso The aalyss of varace detty s y y yˆ y y yˆ Symbolcally, (-8) SS T = SS R + SS E (-9) Sec -4 Hypothess Tests Smple Lear Regresso
11 4//6-4: Hypothess Tests Smple Lear Regresso -4. Aalyss of Varace Approach to Test Sgfcace of Regresso If the ull hypothess, H : b = s true, the statstc F SS SS / E R / MS R MSE (-) follows the F,- dstrbuto ad we would reject f f > f a,,-. Sec -4 Hypothess Tests Smple Lear Regresso -4: Hypothess Tests Smple Lear Regresso -4. Aalyss of Varace Approach to Test Sgfcace of Regresso The quattes, MS R ad MS E are called mea squares. Aalyss of varace table: Source of Sum of Squares Degrees of Mea Square F Varato Freedom Regresso SSR ˆb S y MS R MS R /MS E Error SSE SST ˆb Sy - MS E Total SS T - Note that MS E = ˆ Sec -4 Hypothess Tests Smple Lear Regresso
12 4//6-4: Hypothess Tests Smple Lear Regresso EXAMPLE -3 Oyge Purty ANOVA We wll use the aalyss of varace approach to test for sgfcace of regresso usg the oyge purty data model from Eample -. Recall that SS ˆ T 73.38, b 4.947, S y =.7744, ad =. The regresso sum of squares s SS R b S ˆ y ad the error sum of squares s (4.947) SS E = SS T - SS R = =.5 The aalyss of varace for testg H : b = s summarzed the Mtab output Table -. The test statstc s f = MS R /MS E = 5.3/.8 = 8.86, for whch we fd that the P-value s ~ 9 P.3, so we coclude that b s ot zero. There are frequetly mor dffereces termology amog computer packages. For eample, sometmes the regresso sum of squares s called the model sum of squares, ad the error sum of squares s called the resdual sum of squares. Sec -4 Hypothess Tests Smple Lear Regresso 3-5: Cofdece Itervals -5. Cofdece Itervals o the Slope ad Itercept Defto Uder the assumpto that the observato are ormally ad depedetly dstrbuted, a ( - a)% cofdece terval o the slope b smple lear regresso s ˆ ˆ t ˆ ˆ b a/, b b ta/, (-) S S Smlarly, a ( - a)% cofdece terval o the tercept b s bˆ t a/, ˆ S b bˆ t a/, ˆ S (-) Sec -5 Cofdece Itervals 4
13 4//6-5: Cofdece Itervals EXAMPLE -4 Oyge Purty Cofdece Iterval o the Slope We wll fd a 95% cofdece terval o the slope of the regresso le usg the data Eample -. Recall that bˆ 4.947, S.6888, ad ˆ. 8 (see Table -). The, from Equato - we fd Or Ths smplfes to bˆ t ˆ b bˆ t.5,8.5,8 S.8 b 7.73 Practcal Iterpretato: Ths CI does ot clude zero, so there s strog evdece (at a =.5) that the slope s ot zero. The CI s reasoably arrow (.766) because the error varace s farly small. ˆ S b Sec -5 Cofdece Itervals 5-5: Cofdece Itervals -5. Cofdece Iterval o the Mea Respose Defto ˆ Y bˆ bˆ A ( - a)% cofdece terval about the mea respose at the value of, say Y, s gve by ˆ Y ta/, ˆ Y S ˆ Y ta/, ˆ S (-3) where ˆ Y bˆ bˆ s computed from the ftted regresso model. Sec -5 Cofdece Itervals 6 3
14 4//6-5: Cofdece Itervals Eample -5 Oyge Purty Cofdece Iterval o the Mea Respose We wll costruct a 95% cofdece terval about the mea respose for the data Eample -. The ftted model s ˆ Y , ad the 95% cofdece terval o Y s foud from Equato -3 as (.96) ˆ..8 Y.6888 Suppose that we are terested predctg mea oyge purty whe = %. The ˆ Y (.) ad the 95% cofdece terval s or Therefore, the 95% CI o Y. s Y Ths s a reasoable arrow CI. (..96).6888 Sec -5 Cofdece Itervals 7-6: Predcto of New Observatos Predcto Iterval A ( - a) % predcto terval o a future observato Y at the value s gve by yˆ t a/, ˆ Y yˆ S t a/, ˆ S (-4) The value ŷ s computed from the regresso model yˆ bˆ ˆ b. Sec -6 Predcto of New Observatos 8 4
15 4//6-6: Predcto of New Observatos EXAMPLE -6 Oyge Purty Predcto Iterval To llustrate the costructo of a predcto terval, suppose we use the data Eample - ad fd a 95% predcto terval o the et observato of oyge purty =.%. Usg Equato -4 ad recallg from Eample -5 that yˆ 89.3, we fd that the predcto terval s Y whch smplfes to y 9.63 Ths s a reasoably arrow predcto terval. Sec -6 Predcto of New Observatos 9-7: Adequacy of the Regresso Model Fttg a regresso model requres several assumptos.. Errors are ucorrelated radom varables wth mea zero;. Errors have costat varace; ad, 3. Errors be ormally dstrbuted. The aalyst should always cosder the valdty of these assumptos to be doubtful ad coduct aalyses to eame the adequacy of the model Sec -7 Adequacy of the Regresso Model 3 5
16 4//6-7: Adequacy of the Regresso Model -7. Resdual Aalyss The resduals from a regresso model are e = y - ŷ, where y s a actual observato ad ŷ s the correspodg ftted value from the regresso model. Aalyss of the resduals s frequetly helpful checkg the assumpto that the errors are appromately ormally dstrbuted wth costat varace, ad determg whether addtoal terms the model would be useful. Sec -7 Adequacy of the Regresso Model 3-7: Adequacy of the Regresso Model EXAMPLE -7 Oyge Purty Resduals The regresso model for the oyge purty data Eample - s yˆ Table -4 presets the observed ad predcted values of y at each value of from ths data set, alog wth the correspodg resdual. These values were computed usg Mtab ad show the umber of decmal places typcal of computer output. A ormal probablty plot of the resduals s show Fg. -. Sce the resduals fall appromately alog a straght le the fgure, we coclude that there s o severe departure from ormalty. The resduals are also plotted agast the predcted value Fg. - ad agast the hydrocarbo levels Fg. -. These plots do ot dcate ay serous model adequaces. ŷ Sec -7 Adequacy of the Regresso Model 3 6
17 4//6-7: Adequacy of the Regresso Model Eample -7 Sec -7 Adequacy of the Regresso Model 33-7: Adequacy of the Regresso Model Eample -7 Fgure - Normal probablty plot of resduals, Eample -7. Sec -7 Adequacy of the Regresso Model 34 7
18 4//6-7: Adequacy of the Regresso Model Eample -7 Fgure - Plot of resduals versus predcted oyge purty, ŷ, Eample -7. Sec -7 Adequacy of the Regresso Model 35-7: Adequacy of the Regresso Model -7. Coeffcet of Determato (R ) The quatty R SS R SS T SS SS E T s called the coeffcet of determato ad s ofte used to judge the adequacy of a regresso model. R ; We ofte refer (loosely) to R as the amout of varablty the data eplaed or accouted for by the regresso model. Sec -7 Adequacy of the Regresso Model 36 8
19 4//6-7: Adequacy of the Regresso Model -7. Coeffcet of Determato (R ) For the oyge purty regresso model, R = SS R /SS T = 5.3/73.38 =.877 Thus, the model accouts for 87.7% of the varablty the data. Sec -7 Adequacy of the Regresso Model 37-8: Correlato We assume that the jot dstrbuto of X ad Y s the bvarate ormal dstrbuto preseted Chapter 5, ad Y ad Y are the mea ad varace of Y, X ad X are the mea ad varace X, ad r s the correlato coeffcet betwee Y ad X. Recall that the correlato coeffcet s defed as where XY s the covarace betwee Y ad X. The codtoal dstrbuto of Y for a gve value of X = s where fy y r XY X Y ep Y y b b Y (-5) (-6) b b Y X Y r X r Y X (-7) (-8) Sec -8 Correlato 38 9
20 4//6-8: Correlato It s possble to draw fereces about the correlato coeffcet r ths model. The estmator of r s the sample correlato coeffcet Note that R X X Y Y Y X X / S S SS / XX XY T (-9) b ˆ SS S T XX / R (-) We may also wrte: R bˆ S S XX YY bˆ S SS X Y T SS SS R T Sec -8 Correlato 39-8: Correlato It s ofte useful to test the hypotheses H : r = H : r The approprate test statstc for these hypotheses s T R (-) R Reject H f t > t a/,-. Sec -8 Correlato 4
21 4//6-8: Correlato The test procedure for the hypothess H : r = H : r where r s somewhat more complcated. I ths case, the approprate test statstc s Z = (arctah R - arctah r )( - 3) / (-) Reject H f z > z a/. Sec -8 Correlato 4-8: Correlato The appromate (- a)% cofdece terval s / / tah za za arctah r r tah arctah r 3 3 (-3) Sec -8 Correlato 4
22 4//6-8: Correlato EXAMPLE -8 Wre Bod Pull Stregth I Chapter (Secto -3) a applcato of regresso aalyss s descrbed whch a egeer at a semcoductor assembly plat s vestgatg the relatoshp betwee pull stregth of a wre bod ad two factors: wre legth ad de heght. I ths eample, we wll cosder oly oe of the factors, the wre legth. A radom sample of 5 uts s selected ad tested, ad the wre bod pull stregth ad wre legth arc observed for each ut. The data are show Table -. We assume that pull stregth ad wre legth are jotly ormally dstrbuted. Fgure -3 shows a scatter dagram of wre bod stregth versus wre legth. We have used the Mtab opto of dsplayg bo plots of each dvdual varable o the scatter dagram. There s evdece of a lear relatoshp betwee the two varables. The Mtab output for fttg a smple lear regresso model to the data s show below. Sec -8 Correlato 43-8: Correlato Fgure -3 Scatter plot of wre bod stregth versus wre legth, Eample -8. Sec -8 Correlato 44
23 4//6-8: Correlato Mtab Output for Eample -8 Regresso Aalyss: Stregth versus Legth The regresso equato s Stregth = Legth Predctor Coef SE Coef T P Costat Legth S = 3.93 R-Sq = 96.4% R-Sq(adj) = 96.% PRESS = 7.44 R-Sq(pred) = 95.54% Aalyss of Varace Source DF SS MS F P Regresso Resdual Error Total Sec -8 Correlato 45 Eample -8 (cotued) -8: Correlato Now S = ad S y = 7.73, ad the sample correlato coeffcet s r 7.73 / S SS S y T /.988 Note that r = (.988) =.964 (whch s reported the Mtab output), or that appromately 96.4% of the varablty pull stregth s eplaed by the lear relatoshp to wre legth. Sec -8 Correlato 46 3
24 4//6 Eample -8 (cotued) -8: Correlato Now suppose that we wsh to test the hypotheses H : r = H : r wth a =.5. We ca compute the t-statstc of Equato - as t r r Ths statstc s also reported the Mtab output as a test of H : b =. Because t.5,3 =.69, we reject H ad coclude that the correlato coeffcet r. Sec -8 Correlato 47 Eample -8 (cotued) -8: Correlato Fally, we may costruct a appromate 95% cofdece terval o r from Equato -3. Sce arctah r = arctah.988 =.345, Equato -3 becomes tah.345 r tah.345 whch reduces to.9585 r.99 Sec -8 Correlato 48 4
25 4//6-9: Trasformato ad Logstc Regresso We occasoally fd that the straght-le regresso model Y = b + b + approprate because the true regresso fucto s olear. Sometmes olearty s vsually determed from the scatter dagram, ad sometmes, because of pror eperece or uderlyg theory, we kow advace that the model s olear. Occasoally, a scatter dagram wll ehbt a apparet olear relatoshp betwee Y ad. I some of these stuatos, a olear fucto ca be epressed as a straght le by usg a sutable trasformato. Such olear models are called trscally lear. Sec -9 Trasformato ad Logstc Regresso 49-9: Trasformato ad Logstc Regresso EXAMPLE -9 Wdmll Power A research egeer s vestgatg the use of a wdmll to geerate electrcty ad has collected data o the DC output from ths wdmll ad the correspodg wd velocty. The data are plotted Fgure -4 ad lsted Table -5 Table -5 Observed Values ad Regressor Varable for Eample -9. Observato Wd Velocty (mph), DC Output, y Number, Sec -9 Trasformato ad Logstc Regresso 5 5
26 4//6-9: Trasformato ad Logstc Regresso Eample -9 (Cotued) Fgure -4 Plot of DC output y versus wd velocty for the wdmll data. Fgure -5 Plot of resduals e versus ftted values y for the wdmll data. ˆ Sec -9 Trasformato ad Logstc Regresso 5-9: Trasformato ad Logstc Regresso Eample -9 (Cotued) Fgure -6 Plot of DC output versus = / for the wdmll data. Sec -9 Trasformato ad Logstc Regresso 5 6
27 4//6-9: Trasformato ad Logstc Regresso Eample -9 (Cotued) Fgure -7 Plot of resduals versus ftted values y for the trasformed model for the wdmll data. ˆ Fgure -8 Normal probablty plot of the resduals for the trasformed model for the wdmll data. A plot of the resduals from the trasformed model versus yˆ s show Fgure -7. Ths plot does ot reveal ay serous problem wth equalty of varace. The ormal probablty plot, show Fgure -8, gves a mld dcato that the errors come from a dstrbuto wth heaver tals tha the ormal (otce the slght upward ad dowward curve at the etremes). Ths ormal probablty plot has the z-score value plotted o the horzotal as. Sce there s o strog sgal of model adequacy, we coclude that the trasformed model s satsfactory. Sec -9 Trasformato ad Logstc Regresso 53 Importat Terms & Cocepts of Chapter Aalyss of varace test regresso Cofdece terval o mea respose Correlato coeffcet Emprcal model Cofdece tervals o model parameters Itrscally lear model Least squares estmato of regresso model parameters Logstcs regresso Model adequacy checkg Odds rato Predcto terval o a future observato Regresso aalyss Resdual plots Resduals Scatter dagram Smple lear regresso model stadard error Statstcal test o model parameters Trasformatos Chapter Summary 54 7
Probability and. Lecture 13: and Correlation
933 Probablty ad Statstcs for Software ad Kowledge Egeers Lecture 3: Smple Lear Regresso ad Correlato Mocha Soptkamo, Ph.D. Outle The Smple Lear Regresso Model (.) Fttg the Regresso Le (.) The Aalyss of
More informationSimple 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 informationObjectives 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: 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 informationChapter 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 informationStatistics 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 informationENGI 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 informationSimple 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 informationSTA302/1001-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 informationMultiple 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 informationApplied Statistics and Probability for Engineers, 5 th edition February 23, b) y ˆ = (85) =
Appled Statstcs ad Probablty for Egeers, 5 th edto February 3, y.8.7.6.5.4.3.. -5 5 5 x b) y ˆ.3999 +.46(85).6836 c) y ˆ.3999 +.46(9).744 d) ˆ.46-3 a) Regresso Aalyss: Ratg Pots versus Meters per Att The
More informationRegresso 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 informationb. 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 informationLinear Regression with One Regressor
Lear Regresso wth Oe Regressor AIM QA.7. Expla how regresso aalyss ecoometrcs measures the relatoshp betwee depedet ad depedet varables. A regresso aalyss has the goal of measurg how chages oe varable,
More information12.2 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 informationChapter 14 Logistic Regression Models
Chapter 4 Logstc Regresso Models I the lear regresso model X β + ε, there are two types of varables explaatory varables X, X,, X k ad study varable y These varables ca be measured o a cotuous scale as
More informationresidual. (Note that usually in descriptions of regression analysis, upper-case
Regresso Aalyss Regresso aalyss fts or derves a model that descres the varato of a respose (or depedet ) varale as a fucto of oe or more predctor (or depedet ) varales. The geeral regresso model s oe of
More informationOrdinary 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 informationStatistics. 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 informationCorrelation 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 informationChapter 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 informationUNIVERSITY 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 informationSummary 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 informationSimple 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 informationFunctions of Random Variables
Fuctos of Radom Varables Chapter Fve Fuctos of Radom Varables 5. Itroducto A geeral egeerg aalyss model s show Fg. 5.. The model output (respose) cotas the performaces of a system or product, such as weght,
More informationLecture 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 informationChapter Two. An Introduction to Regression ( )
ubject: A Itroducto to Regresso Frst tage Chapter Two A Itroducto to Regresso (018-019) 1 pg. ubject: A Itroducto to Regresso Frst tage A Itroducto to Regresso Regresso aalss s a statstcal tool for the
More informationChapter 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 informationMultiple 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 informationExample: 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 informationReaction 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 informationLecture 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 informationESS 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 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 informationSimple Linear Regression and Correlation.
Smple Lear Regresso ad Correlato. Correspods to Chapter 0 Tamhae ad Dulop Sldes prepared b Elzabeth Newto (MIT) wth some sldes b Jacquele Telford (Johs Hopks Uverst) Smple lear regresso aalss estmates
More informationMidterm Exam 1, section 1 (Solution) Thursday, February hour, 15 minutes
coometrcs, CON Sa Fracsco State Uversty Mchael Bar Sprg 5 Mdterm am, secto Soluto Thursday, February 6 hour, 5 mutes Name: Istructos. Ths s closed book, closed otes eam.. No calculators of ay kd are allowed..
More informationChapter 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 informationEconometric 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 informationChapter 2 Supplemental Text Material
-. Models for the Data ad the t-test Chapter upplemetal Text Materal The model preseted the text, equato (-3) s more properl called a meas model. ce the mea s a locato parameter, ths tpe of model s also
More informationLecture 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 informationHandout #8. X\Y f(x) 0 1/16 1/ / /16 3/ / /16 3/16 0 3/ /16 1/16 1/8 g(y) 1/16 1/4 3/8 1/4 1/16 1
Hadout #8 Ttle: Foudatos of Ecoometrcs Course: Eco 367 Fall/05 Istructor: Dr. I-Mg Chu Lear Regresso Model So far we have focused mostly o the study of a sgle radom varable, ts correspodg theoretcal dstrbuto,
More informationAnalysis of Variance with Weibull Data
Aalyss of Varace wth Webull Data Lahaa Watthaacheewaul Abstract I statstcal data aalyss by aalyss of varace, the usual basc assumptos are that the model s addtve ad the errors are radomly, depedetly, ad
More information( ) = ( ) ( ) 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 informationThe equation is sometimes presented in form Y = a + b x. This is reasonable, but it s not the notation we use.
INTRODUCTORY NOTE ON LINEAR REGREION We have data of the form (x y ) (x y ) (x y ) These wll most ofte be preseted to us as two colum of a spreadsheet As the topc develops we wll see both upper case ad
More informationLecture 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 informationUNIVERSITY 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 informationStatistics: 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 informationChapter 2 Simple Linear Regression
Chapter Smple Lear Regresso. Itroducto ad Least Squares Estmates Regresso aalyss s a method for vestgatg the fuctoal relatoshp amog varables. I ths chapter we cosder problems volvg modelg the relatoshp
More informationHomework Solution (#5)
Homework Soluto (# Chapter : #6,, 8(b, 3, 4, 44, 49, 3, 9 ad 7 Chapter. Smple Lear Regresso ad Correlato.6 (6 th edto 7, old edto Page 9 Rafall volume ( vs Ruoff volume ( : 9 8 7 6 4 3 : a. Yes, the scatter-plot
More informationChapter 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 informationENGI 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 informationECON 482 / WH Hong The Simple Regression Model 1. Definition of the Simple Regression Model
ECON 48 / WH Hog The Smple Regresso Model. Defto of the Smple Regresso Model Smple Regresso Model Expla varable y terms of varable x y = β + β x+ u y : depedet varable, explaed varable, respose varable,
More informationLinear Regression Siana Halim
Lear Regresso Saa Halm Draper,N.R; Smth, H.; Appled Regresso Aalyss,3rd Edto, Joh Wley & Sos, Ic. 998 Motgomery, D.C; Peck, E.A; Itroducto to Lear Regresso Aalyss, d Edto, 99 Outle Itroducto Fttg a Straght
More informationSTATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. x, where. = y - ˆ " 1
STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS Recall Assumpto E(Y x) η 0 + η x (lear codtoal mea fucto) Data (x, y ), (x 2, y 2 ),, (x, y ) Least squares estmator ˆ E (Y x) ˆ " 0 + ˆ " x, where ˆ
More informationSTA 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 informationMultivariate 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 information4. Standard Regression Model and Spatial Dependence Tests
4. Stadard Regresso Model ad Spatal Depedece Tests Stadard regresso aalss fals the presece of spatal effects. I case of spatal depedeces ad/or spatal heterogeet a stadard regresso model wll be msspecfed.
More informationSpecial 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 informationModule 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 informationSTA 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 informationREVIEW 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 informationMultiple 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 informationA New Family of Transformations for Lifetime Data
Proceedgs of the World Cogress o Egeerg 4 Vol I, WCE 4, July - 4, 4, Lodo, U.K. A New Famly of Trasformatos for Lfetme Data Lakhaa Watthaacheewakul Abstract A famly of trasformatos s the oe of several
More informationECONOMETRIC THEORY. MODULE VIII Lecture - 26 Heteroskedasticity
ECONOMETRIC THEORY MODULE VIII Lecture - 6 Heteroskedastcty Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur . Breusch Paga test Ths test ca be appled whe the replcated data
More information{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution:
Chapter 4 Exercses Samplg Theory Exercse (Smple radom samplg: Let there be two correlated radom varables X ad A sample of sze s draw from a populato by smple radom samplg wthout replacemet The observed
More informationLecture 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 informationLINEAR REGRESSION ANALYSIS
LINEAR REGRESSION ANALYSIS MODULE V Lecture - Correctg Model Iadequaces Through Trasformato ad Weghtg Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur Aalytcal methods for
More informationChapter 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 informationX X X E[ ] E X E X. is the ()m n where the ( i,)th. j element is the mean of the ( i,)th., then
Secto 5 Vectors of Radom Varables Whe workg wth several radom varables,,..., to arrage them vector form x, t s ofte coveet We ca the make use of matrx algebra to help us orgaze ad mapulate large umbers
More informationbest 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 informationLecture 3 Probability review (cont d)
STATS 00: Itroducto to Statstcal Iferece Autum 06 Lecture 3 Probablty revew (cot d) 3. Jot dstrbutos If radom varables X,..., X k are depedet, the ther dstrbuto may be specfed by specfyg the dvdual dstrbuto
More informationChapter 4 Multiple Random Variables
Revew for the prevous lecture: Theorems ad Examples: How to obta the pmf (pdf) of U = g (, Y) ad V = g (, Y) Chapter 4 Multple Radom Varables Chapter 44 Herarchcal Models ad Mxture Dstrbutos Examples:
More informationExample. Row Hydrogen Carbon
SMAM 39 Least Squares Example. Heatg ad combusto aalyses were performed order to study the composto of moo rocks collected by Apollo 4 ad 5 crews. Recorded c ad c of the Mtab output are the determatos
More informationPrevious lecture. Lecture 8. Learning outcomes of this lecture. Today. Statistical test and Scales of measurement. Correlation
Lecture 8 Emprcal Research Methods I434 Quattatve Data aalss II Relatos Prevous lecture Idea behd hpothess testg Is the dfferece betwee two samples a reflecto of the dfferece of two dfferet populatos or
More informationMean 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 informationEconometrics. 3) Statistical properties of the OLS estimator
30C0000 Ecoometrcs 3) Statstcal propertes of the OLS estmator Tmo Kuosmae Professor, Ph.D. http://omepre.et/dex.php/tmokuosmae Today s topcs Whch assumptos are eeded for OLS to work? Statstcal propertes
More informationChapter 3 Sampling For Proportions and Percentages
Chapter 3 Samplg For Proportos ad Percetages I may stuatos, the characterstc uder study o whch the observatos are collected are qualtatve ature For example, the resposes of customers may marketg surveys
More information22 Nonparametric Methods.
22 oparametrc Methods. I parametrc models oe assumes apror that the dstrbutos have a specfc form wth oe or more ukow parameters ad oe tres to fd the best or atleast reasoably effcet procedures that aswer
More informationMidterm Exam 1, section 2 (Solution) Thursday, February hour, 15 minutes
coometrcs, CON Sa Fracsco State Uverst Mchael Bar Sprg 5 Mdterm xam, secto Soluto Thursda, Februar 6 hour, 5 mutes Name: Istructos. Ths s closed book, closed otes exam.. No calculators of a kd are allowed..
More informationUniversity of Belgrade. Faculty of Mathematics. Master thesis Regression and Correlation
Uversty of Belgrade Vrtual Lbrary of Faculty of Mathematcs - Uversty of Belgrade Faculty of Mathematcs Master thess Regresso ad Correlato The caddate Supervsor Karma Ibrahm Soufya Vesa Jevremovć Jue 014
More informationSimple Linear Regression Analysis
LINEAR REGREION ANALYSIS MODULE II Lecture - 5 Smple Lear Regreo Aaly Dr Shalabh Departmet of Mathematc Stattc Ida Ittute of Techology Kapur Jot cofdece rego for A jot cofdece rego for ca alo be foud Such
More informationTESTS BASED ON MAXIMUM LIKELIHOOD
ESE 5 Toy E. Smth. The Basc Example. TESTS BASED ON MAXIMUM LIKELIHOOD To llustrate the propertes of maxmum lkelhood estmates ad tests, we cosder the smplest possble case of estmatg the mea of the ormal
More informationLecture 1: Introduction to Regression
Lecture : Itroducto to Regresso A Eample: Eplag State Homcde Rates What kds of varables mght we use to epla/predct state homcde rates? Let s cosder just oe predctor for ow: povert Igore omtted varables,
More informationLecture 1: Introduction to Regression
Lecture : Itroducto to Regresso A Eample: Eplag State Homcde Rates What kds of varables mght we use to epla/predct state homcde rates? Let s cosder just oe predctor for ow: povert Igore omtted varables,
More informationFundamentals of Regression Analysis
Fdametals of Regresso Aalyss Regresso aalyss s cocered wth the stdy of the depedece of oe varable, the depedet varable, o oe or more other varables, the explaatory varables, wth a vew of estmatg ad/or
More informationSimulation 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 informationLecture 2: Linear Least Squares Regression
Lecture : Lear Least Squares Regresso Dave Armstrog UW Mlwaukee February 8, 016 Is the Relatoshp Lear? lbrary(car) data(davs) d 150) Davs$weght[d]
More informationChapter 5 Transformation and Weighting to Correct Model Inadequacies
Chapter 5 Trasformato ad Weghtg to Correct Model Iadequaces The graphcal methods help detectg the volato of basc assumptos regresso aalss. Now we cosder the methods ad procedures for buldg the models through
More informationSTK3100 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 informationIntroduction to F-testing in linear regression models
ECON 43 Harald Goldste, revsed Nov. 4 Itroducto to F-testg lear regso s (Lecture ote to lecture Frday 4..4) Itroducto A F-test usually s a test where several parameters are volved at oce the ull hypothess
More informationConfidence Intervals for Double Exponential Distribution: A Simulation Approach
World Academy of Scece, Egeerg ad Techology Iteratoal Joural of Physcal ad Mathematcal Sceces Vol:6, No:, 0 Cofdece Itervals for Double Expoetal Dstrbuto: A Smulato Approach M. Alrasheed * Iteratoal Scece
More informationLine 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 informationLecture 2: The Simple Regression Model
Lectre Notes o Advaced coometrcs Lectre : The Smple Regresso Model Takash Yamao Fall Semester 5 I ths lectre we revew the smple bvarate lear regresso model. We focs o statstcal assmptos to obta based estmators.
More informationLogistic regression (continued)
STAT562 page 138 Logstc regresso (cotued) Suppose we ow cosder more complex models to descrbe the relatoshp betwee a categorcal respose varable (Y) that takes o two (2) possble outcomes ad a set of p explaatory
More informationTHE 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 informationMaximum 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 informationMS exam problems Fall 2012
MS exam problems Fall 01 (From: Rya Mart) 1. (Stat 401) Cosder the followg game wth a box that cotas te balls two red, three blue, ad fve gree. A player selects two balls from the box at radom, wthout
More informationLecture Note to Rice Chapter 8
ECON 430 HG revsed Nov 06 Lecture Note to Rce Chapter 8 Radom matrces Let Y, =,,, m, =,,, be radom varables (r.v. s). The matrx Y Y Y Y Y Y Y Y Y Y = m m m s called a radom matrx ( wth a ot m-dmesoal dstrbuto,
More informationTHE ROYAL STATISTICAL SOCIETY HIGHER CERTIFICATE
THE ROYAL STATISTICAL SOCIETY 00 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE PAPER I STATISTICAL THEORY The Socety provdes these solutos to assst caddates preparg for the examatos future years ad for the
More informationBayes 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