? Economical statistics

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1 Probablty calculato ad statstcs Probablty calculato Mathematcal statstcs Appled statstcs? Ecoomcal statstcs populato statstcs medcal statstcs etc. Example: blood type Dstrbuto A AB B Elemetary evets: A, B, AB, Sample space. Probabltes: P(A), P(B), P(AB), P() Exclusve evets: P(AB) P(A) P(B) P() Is there aythg to do? 45% 4% 35% 3% 5% % 5% % 5% % Dstrbuto of blood types Hugary A B AB How ca we get ths formato? How much s the relablty? P( A) + P( B) + P( AB) + P() A evet for example: A atge s preset: probablty P(A)+P(AB) oe ad oly oe atge s preset:? Theoretcal way (rare) (e.g. throw of dce: probablty of a elemetary evet: /6.) Expermetal way (expermet or tral. tral: measuremet, observato, askg, etc.) probablty P(A)+P(B)

2 ,6,5,4,3,, F N,8,7,6,5,4,3,, F N,6,5,4,3,,,4,35,3,5,,5,,5,35,3,5,,5,,5 F FN N 4F 3FN FN F3N 4N 6F 5FN 4FN 3F3N F4N F5N 6N,6,5,4,3,,,45,4,35,3,5,,5,,5,4,35,3,5,,5,,5 F FN N 4F 3FN FN F3N 4N 6F 5FN 4FN 3F3N F4N F5N 6N Next please! Tral P(F).5 P(M).5 P(F).75 P(M).5 Example: the rato of the male ad female. The sample space has elemets: male, female probabltes: P(M) és P(F). It s true: P(F)+P(M) o. of elemets: ő férf ő férf What s the geder? outcome: M(ale) or F(emale) P(F).5 P(M).5 P(F).75 P(M).5 M F M F o. of elemets: 4 ő férf ő férf o. of elemets: ( tral) freq. ő férf freq. ő férf o. of elemets: 6 ő férf ő férf Prcple of samplg Populato ad sample Cocluso larger o. of elemets smaller dffereces, more relable result. No. of elemets: as large as possble. (Wth the bouds of reaso.) Radom samplg. I medce: If there s o exclusve occaso, the must be radom. Populato (statstcal uverse): A large set or collecto of tems that have a commo observable property or propertes. Ths may cosst of fte or fte o. of tems. Theoretcal uverse s also possble wth potetally observable elemets. Sample: A small porto of the populato selected accordg to a certa rule or rules. ssble cases.

3 Samplg error No-samplg error Org: we deal wth the sample oly (a smaller part of the statstcal uverse). Sample survay error e.g.: respose error, processg error etc. We are ot able to avod but we are able to aalyze ad take to the cosderato usg statstcal methods! Gyecology Next please! A extreme example: No-radom samplg! (prevous example) Estmato Type of the estmato How hgh s the tree? Pot estmato Estmato by oe value. Iterval estmato Estmato by terval (t s sde the rage wth hgh relablty). About 7 m. warrat of capto heght: about 75 cm warrat of capto hght: 7-75 cm Estmato: such kd of procedure, that orders a value to a varable or to a case o the base of complete, emprcal data.

4 Propertes of a good estmato Categorcal quatty tral: select a people ad do a test! Ubased: Effcet: Cosstet: Suffcet: The expected value of the estmato s the requred parameter the case of every possble o. of elemets. The squered error of the estmato from the paramater has mmum. Icreasg the sample sze creases the probablty of the estmator beg close to the populato parameter. Cotas every formato that possble to get from the sample (E.g. a mea ad stadard devato are suffcet t he case of the ormal dstrbuto). outcome: A or B or AB or. Select eough large o. of people! : o. of elemets. Sample: people from the populato. Blood type A B AB frequecy k A k B k AB k Estmato of a probablty The error of the relatve frequecy P(A) probablty of the A blood type. The expected value of the frequecy of A:. P(A). Estmato of. P(A) o the base of the sample: k A Bomal dstrbuto. expected value: p varace: p(-p) (Oop! Probablty calculato?) elemets: k elemets have A blood type, (-k) ot. Pot estmato of P(A) : k A /. Estmato of the sd of the k A value: s k/ s thesdofk/, or stadard error of t. s k ( P( )) P( A) A O.K., but aother sample results other value. How much s the realblty of ths value? Estmato of the sd of the k A / value: P( A) ( P( A) ) P( A) ( P( A ) ) s k / Istead of P(A) use k A /!

5 Cofdece terval Cotuous quatty Usg ths value we are able to determe a terval. (terval estmato) Example: heght Is t correct? Sample space ftely large! k ± s k / 68% cofdece terval, 68% cofdece level belogs to ths. meag: If we repeat the observato may tmes, about the 68% of the cofdece tervals cotas the P(A). The relablty of the terval estmato s about 68%. heght: 7 cm. Fte o. of elemets the sample. Theoretcally there s o two equal elemets. (frequeces: or ) False cocluso, No! Ca t be used. Exact measuremet s mpossble, Ift acurracy were requred. Samplg the case of cotuous quatty μ ad σ Exact statemet: Heght (x): 7.5 x < 7.5 cm Istead of a cocrete value we use a terval so-called class. (We ca use them as the dscrete values) σ characterzes the devato of the data aroud the μ. About 68% of the data are aroud the μ the σ wde terval. ( μ ± σ ) 68% ( μ ± σ ) 95% p probablty, that x s the gve class. ( μ ± )?

6 Samplg dstrbuto Every x elemet the samples dffer from each other. Dstrbuto s used to descrbe. The dstrbuto of x values s same as the dstrbuto of the populato. The expected value of the average ad t s varace Ths s a smple sum. x M ( x) M ( x ) ( μ ) μ x σ D ( x) D ( x ) ( ) σ M ( x ) μ ad D ( x ) σ The expected value of the averages s equal to the m of the populato, varace s tmes smaller. Estmato the case of the cotuous quatty Estmato of the expected value parameters of the dstrbuto: expected value ad theoretcal sd. Deftos: M ( x) p x approxmate p by k /! Pot estmato: average. Ubased: Expected value: M ( x) xf ( x) dx p x Theoretcal sd: D ( x) [ x M ( x) ] f ( x) dx p ( x k x k x x x M (x) μ

7 p ( x μ? Estmato of σ k p ( x ( x k ( x ( x approxmate p by k /! ormally ukow, oly the average s kow. lmt: ( x Prevously was proved: Good estmato? ( x x) > Calculate the average of several samples cotag elemets! crease! (expected value) σ > M ( x x) Ths s a based estmato! ( x x)? The dfferece derves from the dfferece of μ ad average. ( x M [( x ] Corrected emprcal s d crease! ( s ) σ σ M + σ M * s s s ( s ) ( x x) * varace of the average: σ sd of the average: σ But σ s ormally ukow. Stadard error s s a good estmato of σ. s x s Ths s the sd of the average or t s stadard error. σ Ths s the varace of the samples. We use s to symbolze the corrected emprcal sd.

8 Cofdece terval of the expected value If we kow the stadard error we are able to determe the cofdece terval of the expected value. Propertes of the terval estmato 68%? Is t too small? We ca crease, e.g.: ths case about 95% s the cofdece level, but the formato cotet s less. [ x ± ] s x [ ] x ± s x Ths terval cludes μ about wth 68% cofdece. terval realablty formato exact formula: [ x ± ] t p s x where t p : value of the t-dstrbuto wth (-) degree. (cofdece level (-p)) Relato amog parameters Referece or ormal rage sample populato What dose t mea? K WBC male 3,5-5 mmol/l /l female 3,5-5 mmol/l /l average μ HCT 4-54 % 38-5 % sd σ stadard error But, frequetly I do t kow?... Thats thereasotouse statstcs! I the case of quatty havg ormal dstrbuto see the fgure! (Istead of μ ad σ ormally we use ther estmato from a large sample). Ayway we use the terval cotag 95% of the data.

9 Problem expermet A possble cocluso: Δt postve or zero: effectve Δt egatve: effectve. Effectve the medce or ot? Is t true? What s the stuato f we do t use t? What s the role of the radom fluctuatos?