ELEG 3143 Probability & Stochastic Process Ch. 5 Elements of Statistics
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1 Deprtet of Electricl Egieerig Uiversity of Arkss ELEG 3143 Probbility & Stochstic Process Ch. 5 Eleets of Sttistics Dr. Jigxi Wu wuj@urk.edu
2 OUTLINE Itroductio: wht is sttistics? Sple e d sple vrice Cofidece itervls Hypothesis testig
3 INTRODUCTION 3 Sttistics A bridge betwee the probbility theory d the rel world The sciece of gtherig d lyzig dt, d with the drwig of coclusios or ifereces fro the dt. Exple 1. collect dt bout the life sp of ll feles i US. lyze the dt to fid out:» Wht is the verge (e) fele life sp i US?» Wht is the vrice of the fele life sp?» Wht is the distributio of the fele life sp? Exple 1. cosider couictio syste trsittig -1 d 1. received sigl is distorted by oise (e.g., Tx 1, Rx 0.) 3. Bsed o the oisy received ifortio, fid out wht ws trsitted (1 or -1)
4 INTRODUCTION 4 Clssifictios of sttistics Splig theory: How to select sples fro soe collectio of dt tht is too lrge to be exied copletely. Estitio theory: Mke soe estitio or predictio bsed o the dt tht re vilble (e.g. estite the verge life sp) Hypothesis testig Attepts to decide which of two or ore hypotheses bout the dt re true (e.g. fid out whether 1 or -1 re trsitted i couictio syste)
5 OUTLINE 5 Itroductio: wht is sttistics? Sple e d sple vrice Cofidece itervl Hypothesis testig
6 SAMPLE MEAN AND VARIANCE 6 Defiitio: popultio the collectio of ALL objects or eleets uder study. E.g. esure the fele life sp i US eber: the life sp of oe fele i US (c be odeled s RV: ) Popultio: the life sps of ALL the feles i US Most of the tie, it is extreely difficult d expesive, if ot ipossible, to get the dt for the etire popultio. We tke liited uber of sples to represet the popultio Defiitio: rdo sple (or, sple) A rdo sple is prt of the popultio tht hs bee selected t rdo. E.g. Cosider popultio with N ebers. We c rdoly pick << N ebers. The ebers for sple of the popultio. All ebers of the popultio re eqully likely beig picked. The picks re idepedet of ech others.
7 SAMPLE MEAN AND VARIANCE 7 Rdo sple Cosider rdo sple with ebers Ech eber is rdo vrible The rdo vribles re utully idepedet The rdo vribles re ideticlly distributed With the rdo sple, we c ifer (estite) the properties of the popultio with N >> ebers E.g. e, vrice, distributio, etc. Ituitively, the lrger the vlue of, the better the estitio. We will prove this ituitio.
8 SAMPLE MEAN AND VARIANCE 8 Sple e The sple e of is 1 It is estitio of the ctul e of the popultio i 1 is rdo vrible, becuse it is fuctio of RVs i E ( ) The first oet of the sple e E [ ] Defiitio: ubised estitor The e of the estitio is the se s its true vlue The sple e is ubised estitio of the e E [ ]
9 SAMPLE MEAN AND VARIANCE The d cetrl oet (vrice) of the sple e Recll: vrice esures the devitio fro the e The sller the vrice, the less the rdoess If vrice is 0, the the RV is costt The lrger the sple size, the ore ccurte the estitio. 9 ] [ ] [ ) ( E E E Vr Vr ) (
10 SAMPLE MEAN AND VARIANCE 10 Exple A popultio of 10 resistors is to be tested. The true stdrd devitio is 5 Oh, d the true e is 100 Oh. How lrge ust be the sple size, if we wt to obti sple e, whose stdrd devitio is % of the true popultio e? If the sple size is 8, wht is the stdrd devitio of the sple e?
11 SAMPLE MEAN AND VARIANCE 11 Exple The sple of rdo tie fuctio follows pdf give s follows: f ( x ) 1 ( x 3) exp The fuctio is spled so s to obti idepedet sple vlues. How y sple vlues re required to obti sple e, whose stdrd devitio is 1% fro the true e?
12 SAMPLE MEAN AND VARIANCE Sple vrice The sple vrice of is defied s The is used to ke ubised estitor. is rdo vrible. First oet of the sple vrice The sple vrice is ubised estitio of 1 i i S 1 ) ( 1 1 ~ 1 1 ~ S ~ S ~ S ] ~ [ S E
13 SAMPLE MEAN AND VARIANCE 13 Exple Cosider 5 rdo ubers: 0.3, 0., 0.8, 0.7, Fid the sple e d sple vrice. If the 5 rdo ubers re rdoly picked fro popultio of rdo ubers tht re uiforly distributed i [0, 1], fid the vrice of the sple e.
14 SAMPLE MEAN AND VARIANCE 14 Exple Write Mtlb progr to geerte 100 idepedet rdo ubers. The rdo uber re sples of uiforly distributed rdo vrible i (0, 10). (1) Write fuctios to fid the sple e d sple vrice. () Wht is the vrice of the sple e? % % i. cler ll; rdo_sple = 10*rd (1, 100); % geerte 100 rdo ubers sple_e = fid_sple_e(rdo_sple); sple_vr = fid_sple_vr(rdo_sple); % % fid_sple_e. fuctio output = fid_sple_e(iput) _sple = legth(iput); output = su(iput)/_sple;
15 SAMPLE MEAN AND VARIANCE 15 Exple (Cot d) % % fid_sple_vr. fuctio output = fid_sple_vr(iput) % fid out how y ebers re i the sple _sple = legth(iput); % clculte the sple e sple_e = fid_sple_e(iput); % clculte the sple vrice output = su( (iput-sple_e).^ )/(_sple-1);
16 OUTLINE 16 Itroductio: wht is sttistics? Sple e d sple vrice Cofidece itervl Hypothesis testig
17 CONFIDENCE INTERVAL 17 Cetrl Liit Theore Let be rdo sple of size. They re idepedet d ideticlly distributed with e d vrice. Whe, the sple e, distributio with e d vrice. 1 i 1 i, coverges i distributio to Gussi Whe is lrge ( > 30), follows pproxitely to Gussi distributio, regrdless of the distributio of i
18 CONFIDENCE INTERVAL 18 Cofidece itervl Exple: estite the e,, of popultio of dt. The sple e, 1, is RV thus it could be quite differet fro the true e. i 1 i Specify itervl tht is highly likely to coti the true vlue of E.g. It is 99% likely tht the true vlue of is i the itervl [, ] Pr [, ] is clled the 99% cofidece itervl of the e
19 19 Cofidece Itervl Cofidece itervl v.s. sple e Sple e ttepts to use sigle uber, represet the sple e The sple e is RV, thus it could be quite differet fro the true e. How uch c we trust this sigle uber estitio? Cofidece itervl, isted, ttepts to specify itervl tht is highly likely to coti the true vlue of the estitio. 1 i 1 i, to The q 100 % cofidece itervl for the estitio of the preter is defied s [, ], such tht Pr q
20 CONFIDENCE INTERVAL Cofidece Itervl (Cot d) Bsed o the cetrl liit theore, is Gussi distributed with e d vrice 0 Pr Pr dx x f ) ( z Q dz e dx x f 1 1 ) ( Q 1 Pr Q 1 Pr
21 CONFIDENCE INTERVAL Exple A very lrge popultio of resistor vlues hs true e of 100 Oh d stdrd devitio of 5 Oh. Fid the 95% cofidece itervl (cofidece liits) o the sple e if the sple size is 100. Q(1.96) = Q 1 Pr
22 OUTLINE Itroductio: wht is sttistics? Sple e d sple vrice Cofidece itervl Hypothesis testig
23 HYPOTHESIS TESTING 3 Hypothesis testig Testig ssertio bout popultio bsed o rdo sple. Exple: Hypothesis: give coi is fir Test: flip the coi 100 ties, cout the uber of heds Exple: If the coi is fir, we expect pproxitely 50 heds. E.g. if the uber of heds is i [47, 53], the hypothesis is true. The hypothesis is flse otherwise. The itervl [47, 53] is chose rbitrrily. How to systeticlly choose the itervl? Hypothesis: the light bulb fro certi ufcture c lst 1000 hrs. Test: tke 50 light bulbs, esure their life, d fid the sple e If the sple e is greter th t hrs, the hypothesis is true. How do we deterie the vlue t? 900 hrs? 950 hrs? 999 hrs?
24 HYPOTHESIS TESTING 4 Hypothesis testig: Null hypothesis The hypothesis to be tested Altertive hypothesis The copleet (opposite) of the ull hypothesis Exple: Test the hypothesis tht give coi is fir. Test the hypothesis tht lightbulb c lst 1,000 hours Test the hypothesis tht certi ptiet does ot hve ccer
25 HYPOTHESIS TESTING 5 Errors Type I error: Reject whe is true H H 0 0 Flse positive, flse lr Type II error: Accept whe is flse Flse egtive H H 0 0
26 HYPOTHESIS TESTING 6 Hypothesis testig: sigificce testig Test hypothesis bout preter p of rdo vrible Exple: test whether coi is fir : Beroulli RV with preter p: P(=1)=p, P(=0)=1-p : p = 0.5 (the coi is fir). H 0 H 0 Objective: ccept or reject the hypothesis bsed o rdo sple
27 HYPOTHESIS TESTING 7 Exple A certi coi is clied to be fir with 95% cofidece itervl. To test the hypothesis, we flip the coi 100 ties, d fid the sple e,. If, c we ccept the cli? 0.43 Sol: Hypothesis: this is fir coi p 0.5, fid out if 0.43 p p 0. 5 is i the 95% cofidece itervl Pr 1 Q
28 HYPOTHESIS TESTING 8 Exple: A resistor ufcture is testig the qulity of btch of resistors with oil vlue 1K Oh, with 95% cofidece level. A sple of 100 resistors re tested, d the sple e is 1040 Oh, d the sple stdrd devitio is 100 Oh. Do the resistors pss the qulity check?
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