Lect 4: Hypothess Test ad Model Valdato
Goodess of : Cofdece Iterval for μ of 5 X: ormal dstrbuto σ : ow s also ormal f () (-α) C.I. C.I for X (- ) P{ w w X w X P{ } / / / w w P{ z } / / P{ z z z } w z / X w} X w z C.I for X (- ) : fd w f() C.I = w [ X-w, X w] [X-z, X z α α ] X X w w f (z) ormalze - - [Note] () (-) w : probablty that X CI s small. -z α () X s ot ormal but 30, [ X-z, X z ] α α s stll worg. () X s ormal but σ s uow use t-dstrbuto wth s stead of σ C.I for X (- ) : [ X-t, X t α α ] z α z EE6: Lect. 4 Hypothess Test ad Model Valdato 04 Tag Go Km
3 of 5 04 Tag Go Km EE6: Lect. 4 Hypothess Test ad Model Valdato C.I for σ (- ) () < 30 (small) -α = P {a σ b } fd a ad b Use ormalzed dstrbuto of. } / ) ( / ) ( { } / ) ( / { } / ) ( / { } / / { ) (, P P P P b a Goodess of : Cofdece Iterval for σ C.I for σ (- ): ) (, ) ( ) (, / ) (, / / / ) ( f (-α) C.I. / / CI for (- ):, a a z z () 30 (large) ow s / z X X
Hypothess Test (tatstcal Iferece) Model Valdato 4 of 5 What s t? Test I: Testg a assumpto o parameters of populato( X, X etc) based o parameters of a sample observato Test II: Testg a assumpto o dstrbuto of populato (desty fucto) based o a sample dstrbuto Geeral cheme () H 0 : ull hypothess (Assumpto o a parameter of populato s true) () H a : alteratve hypothess ( Assumpto s false) () Test H 0 agast H a based o a sample observato at the level of sgfcace ( the (-) Cofdece Iterval) Procedure for Test I populato H o : μ α: gve () (-α) C.I. -z -t α α z z t t α α / / or σ () samplg/parameters, + () sample statstcs z σ / X z, t, t s / X ( ) (v) test Are z, t, C.I? (v) Accept/Reject Ho EE6: Lect. 4 Hypothess Test ad Model Valdato 04 Tag Go Km
Eample: Hypothess o μ 5 of 5 Clam: μ = 0 Test H o : μ = 0 wth σ uow agast H a : μ 0 at α = 0.0 (sol) () amplg () () 9., s ample statstcs t - dstrbuto wth uow t t s 9. 0 X.38 /.5/ 0 (-) C.I.of /, ( ) t wth 0 ad fd sample parameters 0.005, 9. 5 t 3.5 ( )C.I: [-3.5, 3.5] -t α,( ) 3.5 f (t) t = -.38 - t α,( ) 3.5 t (v) Test f t s C.I? t -.38 t 0.005,9 3.5 Yes! (v) Doot reject Ho! EE6: Lect. 4 Hypothess Test ad Model Valdato 04 Tag Go Km
Test for Hypothess o μ 6 of 5 Two-sded (two-tal) Test Oe-sded (oe-tal) Test Hypothess Ho: μ= μ o Ha: μ μ o Ho: μ μ o Ha: μ> μ o Ho: μ μ o Ha: μ< μ o Ha Ho Ha Ho Ha Ha Ho Rego for Ho/Ha f (z) - f (z) - f (z) - -z α z α z z z -z z Accept Ho z z z / / z z z z Reject Ho z z or z z / / z z z z EE6: Lect. 4 Hypothess Test ad Model Valdato 04 Tag Go Km
Eample: Hypothess o μ o 7 of 5 Clam: H o : μ = μ o = 0.340 ad σ = 0.0 Test H o agast H a : μ μ o at α = 0.05 (sol) () amplg wth 35 ad fd sample parameters 0.343 () ample statstcs z-dstrbuto wth 0.0 X 0.343 0.340 z.77 / 0.0/ 0 () (- ) C.I. of z z / z0.05.96 ( ) C.I : [-.96,.96] (v) Test f z s C.I? z.77 z0.05.96 Yes! (v) Doot reject Ho!.96 z 0?.05 0.05 (- ) 0.975 z 0.05 f ( z) dz 0.975 z 0.975 0.05 f (z) 0.05 z [Note] () What f 0.0 z z.64 / 0.05 z.77 z.64 / Re ject H 0 () What f z z 0.05 H.64 z.77 z : (H 0.64 0 : ) at 0.05 0 Reject H 0 : 0 EE6: Lect. 4 Hypothess Test ad Model Valdato 04 Tag Go Km
Test for Hypothess o σ 8 of 5 Two-sded (two-tal) Test Oe-sded (oe-tal) Test Hypothess Ho: σ = σ o Ha: σ σ o Ho: σ σ o Ha: σ > σ o Ho: σ σ o Ha: σ < σ o Ha Ho Ha Ho Ha Ha Ho Rego for Ho/Ha f ( ) f ( ) f ( ) / / Accept Ho / / Reject Ho / or / EE6: Lect. 4 Hypothess Test ad Model Valdato 04 Tag Go Km
Eample: Hypothess o σ o 9 of 5 Clam: H o : σ = σ o = 0.50 Test H o agast H a : σ > σ o at α = 0.05 (sol) () amplg wth 5 ad fd sample parameter 0.64 () ample statstcs dstrbuto ( ) (5 ) 0.64.94 0.50 () (-)C.I of, 0.05,4 3.7 (v) Test f C.I?.94, 3.7 Yes! (v) Doot reject Ho! f ( ) 3.7.94 for Ho 0.05 If 0.0,.94 0.0,4,.. Reject H 0 EE6: Lect. 4 Hypothess Test ad Model Valdato 04 Tag Go Km
Iput Modelg from Raw Data: Desg of ta: R 0 of 5 Iput modelg from raw data observato Desg for delay tme betwee evets, sojour tme betwee state trastos Tme advace fucto (ta: R) DEV modelg Epermetal frame desg What ad how? To fd dstrbuto fucto of populato based o observato of sample data Use of Hypothess test method Test II page 3 Procedure for Test II of Hypothess Test f( ; μ,σ) =? populato f( ; μ=, σ= s) samplg () Ho: assumpto o f() wth ad s (v) goodess-of-ft test o h() f() sample () fd, s + (v) Accept/Reject Ho () hstogram h() EE6: Lect. 4 Hypothess Test ad Model Valdato 04 Tag Go Km
Assumpto o Dstrbuto wth ample Parameters of 5 Assumpto o f() s based o hstogram, h(), of observed data: eamples f () Posso f () f () Gaussa Epoetal X : arrvals/ut X : ter-arrval tme X : worg tme or delay tme or processg tme Use, s of observed data to mae resemblace betwee h() ad f() (a) Posso dstrbuto e : 0,,,... p( )! 0 : otherwse ˆ : arrval rate (b) Epoetal dstrbuto : 0 e p( ) 0 :otherwse ˆ :terarrval tme ˆ (c) Normal dstrbuto ( ) f () e wth ˆ, ˆ EE6: Lect. 4 Hypothess Test ad Model Valdato 04 Tag Go Km
Test of Goodess-of-ft: how close f() ad h() of 5 Test of Goodess-of-ft Test hypothess o dst. of populato from observed data statstcs χ - test s used χ test for dstrbuto of populato Relable oly for large sample sze: 50 Use sample parameters, s for assumpto of f() Test for ay dstrbuto of cotuous ad dscrete radom vars Test statstcs: the followg s ow ( O E ) E O: Observed data, E : Epected value calcuated by assumed desty fucto (r.v) O (observed frequecy) E = p (epected value) O E O E O E EE6: Lect. 4 Hypothess Test ad Model Valdato 04 Tag Go Km
Procedure for Goodess-of-ft Test 3 of 5 Test Procedure: same as the geeral procedure for Hypothess test ) H o : X has a dstrbuto of wth parameters ad. ) Test statstcs ( O E ) E ) (-α) C.I for,--s, where : () # of class tervals such that E >5 : # parameters estmated : level of sgfcace (v) Accept H o f,--s [Note] Gve, dfferet values may lead to dfferet testg results. Recommedato for for a gve 0 do t use test 50 5-0 00 0-0 >00 / - /5 EE6: Lect. 4 Hypothess Test ad Model Valdato 04 Tag Go Km
Eample of Iput Modelg: fdg f() 4 of 5 Fd a dstrbuto of vehcle arrvals at Epo toll gate (ol) Observato : as # of vehcles arrvg at Epo toll gate 5 m perod betwee 9:00-9:05 am. for fve worg days over 0 wees () hstogram of observed data Observed Data O 0 0 9 3 7 4 0 5 8 6 7 7 5 8 5 9 3 0 3 sum 00 hstogram 0 8 6 4 0 8 6 4 h() 0 3 4 5 6 7 8 9 0 mlar to what dstrbuto? EE6: Lect. 4 Hypothess Test ad Model Valdato 04 Tag Go Km
Eample of Iput Modelg: fdg f() co t 5 of 5 ) fd ad p( ) p( p( p( p( X e! e 0) e ) ) 3) O O 3.64 ) assumpto o dstrbuto wth Posso dstrbuto wth ˆ 3.64 3.64 e 3.64 3.64 0! 3.64! 3.64! 0 0.06 0.096 O E =p 0.6 0 9.6 9 7.4 3 7. 4 0 9. 5 8 4.0 6 7 8.5 7 5 4.4 8 5.0 9 3 0.8 0 3 0.3 0. sum 00 00 (O -E ) /E 7.80 0.5 0.80 4.4.57 0.6.6 [Note] Mae class tervals so that E >5. Total umber of classes s 7 EE6: Lect. 4 Hypothess Test ad Model Valdato 04 Tag Go Km
Eample of Iput Modelg: fdg f() co t 6 of 5 v) goodess-of-ft test at α = 0.05, s 7, s, 0.05 (O - E ) E 7.68 0.05, 5 0.05,5 7.68.. Reject H 0! f ( ), s 0.05. Let 0.005 7.68 0.005,5 6.7 tll reject H 0! f ( ) 7.68 for Ho, s 6.7 0.005 7.68 for Ho EE6: Lect. 4 Hypothess Test ad Model Valdato 04 Tag Go Km
-test for Cotuous Dstrbuto Class Itervals for -test of Goodess-of-ft Dscrete radom varables Equal wdth of tervals Eample: Posso dstrbuto Cotuous radom varables Equal probablty betwee class tervals Eample: Epoetal dstrbuto, Gaussa dstrbuto etc. Recommedato for Cotuous Dstrbuto Equal probablty for all class tervals p = p = = p p = / For -test to be relable E = p 5 E = p = / 5 / 5 f() 0 3 4 7 of 5 If = 00, / 5 = 0 EE6: Lect. 4 Hypothess Test ad Model Valdato 04 Tag Go Km
Eample: Class Itervals for Cotuous R.V. 8 of 5 Class terval for Epoetal dstrbuto f() 0 3 4 ' ' 0 0 F( ) e d' [ e ] e F( ) e e l( ) ( O E) E e 0, s Reject H f (s ) fd? Class terval for Normal dstrbuto F( ) F( ) ( ). ( ) It O E [0, ] O / [, ] O / [, 3 ] O 3 / : : : [ -, ] O / f () EE6: Lect. 4 Hypothess Test ad Model Valdato 3 X X 04 Tag Go Km
무기체계시험평가에서가설검증의적용 9 of 5 모집단 ( 양산된무기들 ) 0.80 0.74 도수 명중률 : P 75% ( 검증대상가설 ) [ 관찰 ] 0 개단위표본화과정. 각표본명중률들은난수이며이들의히스토그램은어떤분포함수모양이다.. 표본명중률난수의분포함수는아래조건이만족되면정규분표가된다. 가. 모집단명중률이정규분포일때, 혹은 나. 표본수가크질때 ( 30) 중심극한정리 3. 모집단명줄률이정규분포일경우라도 표본수가적으면표본명중률은정규분포가되지않는다 ( 이항분포, t- 분포 ) 0.85 0.7 0.69 난수형성 0.75 표본명중률 히스토그램 EE6: Lect. 4 Hypothess Test ad Model Valdato 04 Tag Go Km
명중률 : ROC( 요구성능 ) 와시험평가와의관계 ROC 제원 양산된유도무기 0 of 5 명중률 P ROC P 해석 시험평가합격조건? P ROC = N 명중 N 전체 P ROC 를만족하기위한 P? 시험평가 ⅰ) P > P 합격시험평가 ⅱ) P < P 불합격이냐? 시험평가 앙산예정 ⅲ) P α P P + α 어떤범위냐? 시험평가 명중률 P = N 시험평가 N 시험명중 시험평가 시험평가 시험평가결과 EE6: Lect. 4 Hypothess Test ad Model Valdato ( 시 ) 제품 N 시험평가 04 Tag Go Km
모비율 ( 명중률 ) 의통계적가설검정 명중률가설검정 모평균검증의특수케이스 H 0 : p p 0 H : p < p 0 유의수준 α of 5 모평균검정통계량 표본사격 ( 시험평가 ) 명중률 p 측정 T = X μ 0 σ/ 검정통계량 p p 0 T = p 0 p 0 / X = p μ 0 = p 0 σ = p 0 p 0 예 : 대표본 p 0 5 ad p 0 5 아니오 : 소표본 대표본검정 소표본검정 (z 분포 ) 정규분포 (t 분포혹은이항분포 ) f(z) - t 분포 t α, z α as f(t) - T 영역 z α H 0 : p p 0 z t 0.05, (-.796) T 7 z 0.05 (-.645) 0 T 영역 t α H 0 : p p 0 t EE6: Lect. 4 Hypothess Test ad Model Valdato 04 Tag Go Km
사례연구 : 홍상어시험평가 (/4) of 5 홍상어 ( 대잠수함유도무기 ) 개요 개발목적 개발절차 개발내용 일반경어뢰의사정거리극복하고원거리에서적잠수함을공격하기위해개발 최대 30m 의사정거리 (0 여 m 비행후물속에서잠항 ) 000년부터 009년까지약,000억원의예산을투입 국방과학연구소의주도하에 LIG 넥스원이양산 00년 차양산, 해군인도 0년 6월 차양산계약 청상어 + 로켓유도탄 가격 : 8 억 ~ 0 억원 한국형구축함 (KDX-II 급 ) 이상의함정에탑재 한국형구축함 ( 왕건함 ) 에서수직발사되는홍상어의모습 홍상어를이용한대잠수함교전시나리오예상도 EE6: Lect. 4 Hypothess Test ad Model Valdato 04 Tag Go Km
홍상어시험발사현황 (/4) 3 of 5 시험발사개요 시험발사단계 시기시험목적시험설계시험결과비고 최초시험발사 0 년 8 월홍상어양산후최초시험발사 연습탄 발 * 발실패 [] 시험발사실패로추가양산중단 차시험발사 차시험발사 0 년 0 월 ~ 03 년 월 03 년 7 월 ~ 03 년 0 월 품질향상을위한시험발사 성능검증을위한시험발사 연습탄 5 발실전탄 3 발 연습탄 발실전탄 발 8 발중 5 발명중 [] ( 연습탄 발, 실탄 발유실 ) 4 발중 3 발명중 [3] ( 실탄 발유실 ) 홍상어의전투용적합판정기준은명중률 75% 이상 총 발중 8 발명중 ( 명중률 66.7%, 이중실탄사격의명중률은 40%) 적합판정명중률달성을하지못해양산재개보류 * 연습용어뢰의경우실전용어뢰와외양이같고발사또한수중목표를표적으로진행되지만탄두부분에폭약이아닌다른센서 ( 예 : 주행정보센서 ) 가부착되어발사실패시원인규명이가능하다. [] 문화일보, http://www.muhwa.com/ews/vew.html?o=0000070877300 [] 중앙일보, http://artcle.jos.com/ews/artcle/artcle.asp?total_d=0856445 [3] YTN, http://www.yt.co.r/_l/00_0309563595 EE6: Lect. 4 Hypothess Test ad Model Valdato 04 Tag Go Km
시험평가자료기반 ROC 검정 (3/4) 4 of 5 시험평가결과 시험발사에서총 발중 8 발이명중하였다. 가설검정 검정대상 : 시험평가결과가 ROC 명중룰 75% 이상 (75% 포함 ) 을만족하는가? 가설설정 : 시험평가결과모집단의명중률이 75% 이상이다. H 0 : p 0.75 ROC 명중률 versus H : p < 0.75 α = 0.05 (95 % 신뢰구간 ) or 0.0 (99% 신뢰구간 ) =, p = 0.667, p 0 = 0.75 p 0 = 0.75 = 9 > 5, ( p 0 ) = 0.5 = 3 < 5 소표본검정 (t- 분표혹은이항분포 ) H 0 : p p 0 H : p < p 0 유의수준 α T = p p 0 p 0 p 0 / = 0.667 0.75 0.75 0.5/ = 0.667 검정통계량 T t 0.05, =.796, t 0.0, =.78 채택역 기각역 0.667.796, 0.667.78 T t α, f(t) T < t α, 유의수준 0.05, 0.0 에서가설 H 0 를채택한다 - 채택영역 축! 시험평가통과 t α T = - 0.667 t EE6: Lect. 4 Hypothess Test ad Model Valdato 04 Tag Go Km
t- 분포와이항분포이용시최소명중률비교 (4/4) 5 of 5 검정대상가설 : H 0 0. 75, H < 0.75 유의수준 : 0.05 (95% 신뢰구간 ) f(t) ROC 요구사항 ( 명중률 75% 이상 ) 을만족하기위해시험평가시요구되는최소명중횟수 t α 채택영역 t - N 시험평가 소표본검정표 (t- 분포이용 ) 최소 N 명중 시험명중률 (N 명중 /N 시험평가 00) 0 60.00 9 57.89 8 0 55.56 7 0 58.8 6 9 56.5 5 8 53.33 4 8 57.4 3 7 53.84 6 50.00 6 55.66 0 5 50.00 소표본검정표 ( 이항분포이용 ) N 시험평가 최소 N 명중 시험명중률 (N 명중 /N 시험평가 00) 0 55.00 9 57.89 8 0 55.56 7 9 5.94 6 8 50.00 5 8 53.33 4 7 50.00 3 6 46.5 5 4.67 5 45.45 0 5 50.00 EE6: Lect. 4 Hypothess Test ad Model Valdato 04 Tag Go Km