Sttistics of Exm Mth Itrouctory Sttistics Professor B. Ábrego Lecture Sectios.3,.4 Me 7. SD.7 Mi 3 Q Me 7 Q3 8 Mx 0 0 0 0 0 0 70 80 0 Importt Uses of Coitiol Probbility To compre smplig with or without replcemet. To stuy effectiveess of meicl tests Coitiol Probbility Meicl Tests I meicie, screeig tests give quick iictio of whether or ot perso is likely to hve prticulr isese. Becuse screeig tests re itee to be reltively quick oivsive, they re ofte ot s ccurte s other tests tht tke loger or re more ivsive. A two-wy tble is ofte use to show the four possible outcomes of screeig test. To stuy effectiveess of sttisticl testig Negtive Disese Preset Abset c b + b c + + c b + + b + c +
Coitiol Probbility Meicl Tests Exmple of Rre Disese o,000 ptiets Negtive Negtive Disese Preset Abset c b + b c + + c b + + b + c + preictive vlue (PPV) P (isese test positive) + c Negtive preictive vlue (NPV) P (o isese test egtive) b + Sesitivity P (test positive isese preset) + b Specificity P (test egtive o isese) c + Preset Disese Abset 0,,0,4,000 PPV P (isese test positive) 0. NPV P (o isese test egtive) 0.8 4 Sesitivity P (test positive isese preset) 0.0 Specificity P (test egtive o isese).4 0 Exmple P34 (p. 33) Exmple P34 (p. 33) A lbortory techici is beig teste o her bility to etect cotmite bloo smples. Amog 0 smples give to her, 0 re cotmite, ech with bout the sme egree of cotmitio. Suppose the techici mkes the correct ecisio 0% of the time (regrless of cotmitio or ot). Mke tble showig wht you woul expect to hppe. Wht is her flse positive rte? Wht is her flse egtive rte? How woul these rtes chge if she were give 0 smples with 0 cotmite? Cotmite? Detectio of Cotmitio (test) Negtive 8 0 8 7 80 74 0 Detectio of Cotmitio (test) Preictive Vlue 8/ 0.3 Cotmite? 8 8 Negtive 7 74 0 80 0 Negtive preictive Vlue 7/74 0.77 Sesitivity 8/0 0.0 Specificity 7/80 0.
Coitiol Probbility Sttisticl Iferece. Iepeet Evets Sttistici: Suppose you rw 3 workers t rom from the set of hourly workers. This estblishes rom smplig s the moel for the stuy. Lwyer: Oky. Sttistici: It turs out tht there re 0, possible smples of size 3, oly of them give verge ge of 8 or more. Lwyer: So the probbility is /0, or.0. Sttistici: Right. Lwyer: There s oly % chce the compy i t iscrimite % chce tht they i. Sttistici:, tht s ot true. Lwyer: But you si... Sttistici: I si tht if the ge-eutrl moel of rom rws is correct, the there s oly % chce of gettig verge ge of 8 or more. Lwyer: So the chce the compy is guilty must be %. Sttistici: Slow ow. If you strt by ssumig the moel is true, you c compute the chces of vrious results. But you re tryig to strt from the results compute the chce tht the moel is right or wrog. You c t o tht. P ( v. ge 8 rom rws).0 P ( o iscr. v. ge 8)?? Evets A B re iepeet if oly if P (A B) P (A). Equivletly, A B re iepeet if oly if P (B A) P (B). I other wors, kowig tht B hppee oes ot ffect the probbility of A hppeig, coversely kowig tht A hppee oes ot ffect the probbility of B. Exmple: Wter, Geer, Iepeece (p.3) Driks Bottle Wter? Ietifie Tp Wter? 4 3 0 34 70 0 Geer Mle Femle Ietifie Tp Wter? Show tht the evets is mle correctly ietifies tp wter re iepeet tht the evets riks bottle wter correctly ietifies tp wter re ot iepeet. 3 0 4 3 0 Discussio D Suppose you choose stuet t rom from your school. I ech cse, oes kowig tht evet A hppee icrese the probbility of evet B, ecrese the probbility of evet B, or leve the probbility of evet B uchge?. A: The stuet is footbll plyer. B: The stuet weighs less th 0 pous. b. A: The stuet hs log figerils. B: The stuet is femle. c. A: The stuet is freshm. B: The stuet is mle.. A: The stuet is freshm. B: The stuet is seior.
Multiplictio Rule for Iepeet Evets. Evets A B re iepeet if oly if P (A B) P (A). P (B). More geerlly, evets A, A,, A re iepeet if oly if P (A A A) P (A ). P (A )... P (A). Exmples. If you flip coi four times wht is the probbility tht you get hes o ll flips.. About % of youg Americ ults ges to o t hve helth isurce. Suppose you tke rom smple of te Americ ults i this ge group. Wht is the probbility tht t lest oe of them oes t hve helth isurce? 3 Iepeece with Rel Dt Me SD revisite I relity you rrely will ecouter situtio where P (A B) P (A). P (B). But if these two umbers re very close to ech other sttistici c coclue tht A B re iepeet evets. Exmple: As of July, 000, the Los Ageles Dogers h wo totl of 4 gmes lost 37 gmes. The brekow by whether the gme ws plye urig the y or t ight is show. Are the evets wi y gme iepeet? Time of Gme Dy Night 4 Wo the Gme? 7 37 7 78 We hve three wys of specifyig popultio:. List of ll (iiviul) uits. Frequecy Tble (p. 8) 3. Reltive Frequecy or Proportio Tble (p. 3) How c we clculte me SD o ech?
List of ll uits Frequecy Tbles Number 3 4 Pey Pey Pey Pey Vlue x x - popultio me x + + + + + + + + + 4 Pey Vlue x Frequecy f 3 x. f 0 popultio me + + 0 4 x f 7 8 Pey cois Sum cets SD ( x ) + + + + + + + + 3 + 3 0 3.44 Sum f SD ( x ) f + 3 + 3 0 3.44 7 Reltive Frequecy or Proportio Tble Summry of Me/SD Pey Vlue x Proportio of cois P(x) 0. x. P(x) 0. x popultio me x P ( ) 0. +. +.0 4 List of ll uits Frequecy Tble Reltive Frequecy (or Proportio) Tble Sum 0.3 0...0 4.0 SD ( x ) P ( x) (0.) + (0.3) + 3 (0.) 4. + 0.3 + 7. 3.44 x ( x ) x f ( x ) f x P (x ) ( x ) P ( x)