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# Elementary Statistics STA 013

UCD

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This 22 page Class Notes was uploaded by Carmen Mayer on Tuesday September 8, 2015. The Class Notes belongs to STA 013 at University of California - Davis taught by Jie Peng in Fall. Since its upload, it has received 33 views. For similar materials see /class/191911/sta-013-university-of-california-davis in Statistics at University of California - Davis.

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Date Created: 09/08/15

STATISTICS 13 Lec rur39e 10 Jan 26 2007 v Review gt Linear regression gt Formula gt In rerpre ro rion of regression resul r gt Regression effect Pr39obabiliTy An In rnoduc rion gt Pr obabili ry is used as a maThemaTical Tool To under39s randdescr39ibe random phenomena chance variaTion and uncer rain ry gt Example emission Tes rs on mo ror39 vehicles Four39 r epea red measuremenTs on The same car39 HCgmmile138183 322 325 COgmmile118 149 232 286 There is subsTanTial var39ia rion How To make a precision measure gt ApplicaTions geneTics kineTics quali ry con rr39ol finance e rc Basic Concest ExperimenTs gt Probabili ry is used as a model rool for si rua rions for which ou rcomes occur randomly such si rua rions are called experimen rs and The se r of all possible ou rcomes is called The sample space gt Examples Toss a coin Measure The blood pressure Draw a poker hand from a carddeck Throw rwo dice Basic Concest Simple EvenTs gt A simple even r is The ou rcome of a single repe ri rion of a random experimen r gt Example Experimen r ross a coin once Simple even rs Head H Tail T gt One and only one simple even r can occur when an experimen r is performed once gt One can assign a probabili ry To each simple even r in The sample space Basic Concest Sample Space gt The se r of all possible simple even rs of an experimen r is called The sample space usually deno red by S corresponding ro Tha r experimen r gt Example Experimen r Toss rwo coins once Sample space Basic Concest EvenTs gt An evenT usually denoTed by E F A B eTc is a collecTion of simple evenTs In anoTher39 word a subseT of The sample space gt Any subseT of The sample space including The empTy seT is an evenT gt Example Throw a die 5 123456 A An odd numb r 135 B A number gt L 3456 Basic Concest MuTuolly Exclusive gt Two even rs A and B are mu rually exclusive if whenever one of Them occurs The other conno r occur gt Example Throw a die A observe an odd number B observe a number greater than 2 C observe a 6 D observe a 3 Probabili ry RelaTive Frequency View gtHow of ren does an even rquot occur39 9 Relative frequency fn gtAs sample size 17 becomes large Rela rive frequency gt Probabili ry ProbabiliTy of an EvenT ProperTies gt Pr39obabiliTy of an evenT is a number beTween O and 1 gt Sum of The pr39obabiliTies of all The simple evenTs in 5 sample space equals To 1 gt Pr39obabiliTy of any evenT A denoTed by PA is The sum of The pr39obabiliTies of all The simple evenTs ThaT consTiTuTe A gt PA 0 means A never39 occur39s gt PA 1 means A always occur39s Examples gt Toss a fair39 coin Twice Wha r is The probability of observing a r leas r one head 9 gt In This case all simple even rs are equally likely Fir39s r ross Second ross Ei PEi HH 14 HT 14 TH 14 TT 14 Sample Space wi rh Equally Likely Simple Even rs gtIf The simple even rs are equally likely Then for39 even r A number of simple events in A PA quot A N total number of simple events gtNeed coun ring rules To find 17 and N The MulTiplicaTive Rule gt If an experimenT is performed in Two sTages wiTh m possibilities for The firsT sTage and n possibiliTies for The second sTage Then There are Inn ways To accomplish The experimenT gt This rule is easily exTended To ksTages wiTh The number of ways equal To n1 n2 n3 nk Example Toss Two coins 39 ToTal number of simple evenTs Example Urn problem gt Urn problem An urn conTains Three red balls and Two yellow balls Two balls are randomly chosen wiThouT replacemenT whaT is The probabiliTy ThaT The Two balls are The same color gt Answer There are N simple evenTs in The sample space and They are all equally likely due To random selecTion A evenT ATwo balls are The same color Example Urn problem ConT gt How abouT The Two balls are chosen wiTh replacemenT gt Answer2 There are N simple evenTs in The sample space and They are all equally likely evenT ATwo balls are The same color39 r391r391r391r392r391r3r2r391r392r392 r2r 3r393r391r393r2r393r393 y1y1y1y2y2y1y2y2 so number39 of simple evenTs in A HA ProbabiliTy As ProporTion gt Exam le In a eneTics experimenT The researcher maTed Two Drosophila fruiT flies and 0 served wo TraITs wmg Size and eye color of 300 offspring Wing Size Eye Color Normal MiniaTure Normal 141 Vermillion 3 150 gt One of These offspring is randoml selecTed and observed for The Two TraiTs WhaT is The sample space WhaT probabili y you should assign To each simple evenT in The sample space gt Answer The sample space consisTs of four simple evenTs E1normal wing and normal eye E2normal win and vermillion eye E3miniaTure wing and normal eye E4minIaTure wing and vermI lion eye A reasonable assi nmenT of probabiliTies To simple evenTs is bIJIEroporTion so PE1141300 47 PE23300001 PE36300002 4150300050 NoTe ThaT all The properTies of probabiliTies are saTisfied so This is a legiTimaTe aSSIgnmenT gt WhaT is The probabiliTy ThaT The fly has normal wing size gt Answer CounTing Permu ro rions gtThe number of ways you can choose r39 objec rs in order39 from n dis rinc r objecTs Pr 2 n n r where n nn 1n 221 and 0 1 nn 1n 2n r1 Example Candi daTes Suppose we wanT To rank five candidaTes Adam BeTTe Carl David Elle in preference for a posiTion how many differenT rankings are possible Answer Suppose we can only choose Two of Them as possible candidaTes and Then rank The Two chosen persons Then how many possible rankings are There Answer ABACADAEBABCBDBECACBCDCEDADBDCDEEAEBEC ED Example Cards gt 4 cards on The Table 2 3 4 5 gt Dr39aw car39ds one afTer39 anoTher39 gt Win if The second card has higher denominaTion WhaT is The pr39obabiliTy gt Simple evenTs a Two car39d sequence order39 of drawing is impor39TanT all equally likely gt ToTal number39 of evenTs N gt EvenT A The second card has higher denominaTion CounTing CombinaTions gtThe number of ways you can choose r39 objecTs wi rhou r regarding order from n dis rinc r objec rs rn r Example Candi daTes gt Suppose we wanT To form a Twomember commiTTee from five candidaTes Adam BeTTe Carl David Elle how many differenT commiTTees are possible gt Answer NoTe ThaT unlike The ranking problem here The order of choiceselecTion is noT imporTanT possible commiTTees are whaT39s The difference compared To The ranking problem Example Cards gt5 cards on The Table gtHow many differen r ways To salad 3 cards To form a 3 car39d hand gtOr39der39 of The drawing is no r impor ran r

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