Handout #4. Statistical Inference. Probability Theory. Data Generating Process (i.e., Probability distribution) Observed Data (i.e.

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1 Hadout #4 Ttle: FAE Coure: Eco 368/01 Sprg/015 Itructor: Dr. I-Mg Chu Th hadout ummarze chapter 3~4 from the referece PE. Relevat readg (detaled oe) ca be foud chapter 6, 13, 14, 19, 3, ad 5 from MPS. Chapter 6 ad 7 from BR alo provde formato about what covered th hadout. Stattcal Iferece Probablty Theory Data Geeratg Proce (.e., Probablty dtrbuto) Oberved Data (.e., Sample) Stattcal Iferece Populato v. Sample Who to blame for govermet hutdow? Accordg to a ABC New/Wahgto Pot poll, 63% of Amerca dapprove the way GOP hadlg the budget mpae 1. What the average aual houehold come the U.S. 014? What the curret 30 year fxed mortgage rate? What the crme rate Camde Couty 014? We re lvg a world wth all kd of dfferet fgure ad they provde formato to help u make better deco. How do we get thee fgure? Are they accurate? How do we tur data to ueful formato? Th hadout focue o tattcal ferece. Gve that the ze of the populato we wat to tudy ofte huge, t mpractcal to tudy the populato feature (.e. mea, varace, proporto, quatle, etc) by coductg ceue. Itead, a mall porto of the populato elected for tudyg the charactertc of the populato. For example, the Ceu Bureau o behalf of Bureau of Labor coduct urvey o 60,000 houehold mothly order to fd the job market codto ad etmate the uemploymet rate. However, th method may troduce ucertaty to our aaly. Th ucertaty termed amplg varato. For example, f aother 60,000 houehold are choe ad urveyed, the etmate of uemploymet rate ca be dfferet. How we hadle th ucertaty to make ure the formato obtaed trutworthy, term of probabltc tatemet, wll be explaed th hadout

2 Some mportat uage of Normal,, t ad F dtrbuto tattcal ferece Suppoe a ample of X (=1 ) are draw from a N(, ) RV, the wthout proof: Z = X μ σ N(0, 1) for all (1) ( 1) σ -1 () X μ / t -1 (3) m / m / F m, (4) Where X = = X 1 (X 1 X) 1 Theorem: Suppoe a radom ample of ze draw from a populato wth mea ad varace, the the ample average X wll have the followg property: E( X) = & V( X) = σ, Where X = X 1 (Ca you prove both properte?) σ Suppoe we alo kow the populato ha a ormal dtrbuto, the X~N(, ). Theorem: (Weak) Law of Large Number Let X 1, X,, X be a d equece of radom varable ad E(X ) =, let S = X. S, a Alteratvely, the above tatemet ca be wrtte a P( S ) 1, a, > 0. (e.g. Bomal RV ug R) 1

3 Smulato: Ofte we do t kow the feature about a populato ad t make the ferece job challegg. However, we ca ue mulated data to tudy the lkage betwee populato feature ad the correpodg ample charactertc. e.g. geerate a radom ample from a kow radom varable ad tudy t properte. The Cetral Lmt Theorem If large ample ( practce, the ze of each ample o le tha 30) are radomly elected from a populato wth mea ad varace, the the ample average X wll have the followg property regardle of the hape of the dtrbuto of the paret populato. σ X~N(, ) Illutrate Cetral Lmt Theorem ug Smulato Bomal Dtrbuto Htogram of xbar_3 dety Dety x xbar_3 Htogram of xbar_10 Htogram of xbar_30 Dety Dety xbar_ xbar_30 Explaato: X a radom varable wth Bomal dtrbuto (.e., X bom(10, 0.)). The upper left pael how what t (theoretcal) probablty dtrbuto look lke; t aymmetrc ad kewed rght. We chooe repeated ample (.e., 500 ample) wth ample ze 3, 10 ad 30, repectvely. The ret of three htogram how the dtrbuto of the ample average (whe =3, 10 ad 30), what do you oberve? 3

4 The R code ued to preet CLT ca be foud the lat part of 368_HD04.txt fle. You ca reave th part a a R fle (e.g. the fle exteo.r; e.g., CLT_Bomal.R) ad execute t R a follow: Start R ad chooe Source R code from the pull dow meu uder Fle the coole, the ame dagram a how above wll appear o the cree. You ca chooe Ope crpt to read or modfy the code f t eeded. The detal of the code wll be gradually covered a we proceed. Etmato of Populato Charactertc A. Pot etmate (populato mea, varace, proporto, ad covarace) X 1 Sample Mea: X = E( X ) = X Sample Varace: X = (X 1 X) 1 E( X ) = X Sample Proporto: pˆ = X 1 Sample Proporto Varace: pˆ = (where X = 0 or 1) E( pˆ ) = p ˆ( 1 pˆ ) All of thee formula are termed etmator. Sample Covarace: X,Y = m 1j 1 (X X) * (Y Y) 1 j Sample Co var ace(x, Y) Sample Correlato: r = X Y Notce: (A) There are three method for obtag etmator; () Leat Square, () Maxmum Lkelhood ad () Method of Momet. (B) If the expectato of the etmator equal the true parameter, we ay the etmator ubaed. For example, E( X) = X. (C) The other two ce properte for a etmator addto to ubaede are mmum varace ad effcecy (compare the properte metoed pot (B) ad (C) to the theorem troduced page ad 3) 4

5 B. Cofdece Iterval (CI) Stadardzato of X Z = X μ σ/ ~ N(0, 1) I practce uually ukow, therefore, the ample tadard devato ued to replace. However, from equato (3) o page we lear that the dtrbuto o loger tadard ormal, we hould wrte X / ~ t -1 Ue the t dtrbuto table or from R we ca fd that Pr(-H < t -1 < H 3 ) = 0.95 X Pr(-H < / < H) = 0.95 Pr(-H* < X - < H* ) = 0.95 Pr(H* > - X > -H* 4 ) = 0.95 Pr( X + H* > > X - H* ) = 0.95 We ca ay that there a 95% chace that the mea wll le the rage of X H*. We ca replace H by t 1- /, -1, where referred to a rejecto rego ued hypothe. e.g. Let aume that X = 68 (tudet weght), = 10 ad = 36 from a example of tudet weght collected * % chace that mea wll fall the rage (64.6, 71.38). e.g. I a radom ample of 64 law frm, legal charge per hour are foud to have a mea of $40 wth a tadard devato 5. Obta a 99% CI for the average legal charge per hour the law profeo a a whole. If varace kow, we ca ue the tadard ormal dtrbuto (Z). 3 H vare wth the ze of cofdece terval a well a the dtrbuto ued (Z or t). 4 a < X < b -a > -X > -b 5

6 C. Hypothe Tetg Def: A tattcal hypothe a cojecture about the amplg dtrbuto of a radom varable. Whe a hypothe completely pecfe the dtrbuto, t called a mple hypothe; f t ot, t referred to a a compote hypothe. Neyma ad Pearo troduce ther hypothe tetg approach by pecfyg the ull hypothe ad alteratve hypothe. The former deoted by H 0 ad the latter H 1 or H A. The ull tatemet a tatu quo ad our goal how that whether the evdece from the data trog eough to reject the ull hypothe. e.g. H 0 : = 4,000 H 1 : = 45,000 H 0 : 4,000 H 1 : 4,000 Type I ad Type II Error H 0 true Accept H 0 Reject H 0 Type I ( ) H 0 fale Type II ( ) Power = 1 - Power: the ablty to detect a fale ull hypothe. I practce we lke our tet to have hgh power gve the reject rego. Hypothe Tetg probablty Note: = 0.05 the rejecto rego. x 6

7 Oe-tal Tet Let ue the prevou umercal example from page 5. There we aume that X = 68, = 10 ad = 36. Th example about polcy effectvee. The tax leved o weet drk order to help chldre reduce ther weght (chldhood obety a popular reearch topc recet year). We are tereted examg whether the average weght actually decreaed the weet drk tax? H 0 : = 70 Null Hypothe H A : < 70 Alteratve Hypothe If the ull hypothe true, the TS (tet tattc) ha a t dtrbuto ad ca be repreeted a follow: X X TS = ~ t-1 = = -1. We ca t ot reject that = 70. / / 10/ 36 Cocluo: the ew tax o weet food may ot be effectve to reduce the tudet weght. Let chooe a level of gfcace that equal 5% 5 ; the area uder the curve ad o the left of the value Gve the ull true, the chace that TS would fall the level of gfcace oly 5%. Meag, t ulkely to happe. Therefore, the reaoable judgmet to reject the ull hypothe ad accept alteratve hypothe. Note that there a 5% chace that the ull ca be true. If th the cae, we may make the type I error 6. f(t) f(x) % Reject H t x 5 It ca alo be 1% or 10% deped o reearcher choce. 6 The Type II error refer to the tuato where we accept a fale ull hypothe. To reduce Type I error would reult a creae of Type II error. I practce we ether reject the ull hypothe or reerve the judgmet about t. By dog th we ever accept ull hypothe ad therefore we caot make Type II error. 7

8 Two-tal Tet e.g. The mea lfetme of a radom ample of 80 lght bulb produced by a factory foud to be 1460 hour, wth a tadard devato = 110 hour. If the mea lfetme of all the lght bulb produced by the factory, tet the hypothe = 1500 agat the alteratve hypothe that 1500, ug the level of gfcace H 0 : = 1500 Null Hypothe H A : 1500 Alteratve Hypothe f(t) f(x) %.5% Reject H 0 x Reject H t 0 TS = X μ / = / 80 = -3.5 We reject that = ***The other method to coduct hypothe tetg to ue P-value. P-value defed a the lowet gfcace level at whch the ull hypothe ca be rejected. Let ue the oetal tet from page 7. Sce the TS -1., we ca fd the P-value or 11.9%. Graphcally, t a area o the left of TS (-1.) uder the t curve. 8