Class 27. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700

Transcription

1 Marquette Uiversity MATH 700 Class 7 Daiel B. Rowe, Ph.D. Departmet of Mathematics, Statistics, a Computer Sciece Copyright 07 by D.B. Rowe

2 Marquette Uiversity MATH 700 Agea: Recap Chapter Lecture Chapter Problem Solvig Sessio

3 Marquette Uiversity MATH 700 Recap Chapter

4 Marquette Uiversity MATH 700 0: Ifereces Ivolvig Two Populatios 0. Iferece for Mea Differece Two Depeet Samples Cofiece Iterval Proceure With Paire Differece x x (0.) s ( i ) i i ukow, a -α cofiece iterval for μ =(μ -μ ) is: i Cofiece Iterval for Mea Differece (Depeet Samples) s s t( f, / ) to t( f, / ) where f=- (0.) 4

5 Marquette Uiversity MATH 700 0: Ifereces Ivolvig Two Populatios 0. Iferece for Mea Differece Two Depeet Samples Example: Costruct a 95% CI for mea ifferece i Bra B A tire wear. 8,, 9,,, 9 i s: i 6 f 5 i t( f, / ) s ( i ) i s 5. s t( f, / ) (0.090,.7) Figure from Johso & Kuby, 0. 5

6 Marquette Uiversity MATH 700 0: Ifereces Ivolvig Two Populatios 0. Iferece for Mea Differece Two Depeet Samples 6 8,, 9,,, 9 Example: Test mea ifferece of Bra B mius Bra A is zero. Step H 0 : μ =0 vs. H a : μ 0 Step 5 Step f 5 0 t*.05 s / Step s 5. t* / 6 Step 4 t( f, / ).57 Sice t*>t(f,α/), reject H 0 Coclusio: Sigificat ifferece i trea wear at.05 level. Figures from Johso & Kuby, 0. 6

7 Marquette Uiversity MATH 700 0: Ifereces Ivolvig Two Populatios 0.3 Iferece for Mea Differece Two Iepeet Samples Cofiece Iterval Proceure With a ukow, a -α cofiece iterval for is: f Cofiece Iterval for Mea Differece (Iepeet Samples) s s s s ( x x) t( f, / ) to ( x x) t( f, / ) where f is either calculate or smaller of f, or f (0.8) Actually, this is for σ σ. Next larger umber tha s / s / s s If usig a computer program. If ot usig a computer program. 7

8 Marquette Uiversity MATH 700 0: Ifereces Ivolvig Two Populatios 0.3 Iferece Mea Differece Cofiece Iterval f Example: Itereste i ifferece i mea heights betwee me a wome. The heights of 0 females a 30 males is measure. Costruct a 95% cofiece iterval for, & ukow s s m f ( xm x f) t( f, / ) m f (.9) (.8) ( ) m f m f x x f m m s f s 0.05 t(9,.05).09 therefore 4.75 to 7.5 Figure from Johso & Kuby, 0. m 8

9 Marquette Uiversity MATH 700 0: Ifereces Ivolvig Two Populatios TuTh Iferece for Mea Differece Two Iepeet Samples Hypothesis Testig Proceure 77 values males a females x x m x f Is the height of males = height of females at α=.05? 9

10 Marquette Uiversity MATH 700 0: Ifereces Ivolvig Two Populatios 0.3 Iferece for Mea Differece Two Iepeet Samples Hypothesis Testig Proceure 77 values Step H 0 : μ f =μ m vs. H a : μ f μ m Step ( xm xf ) ( m f ) t* s s m f f 9.05 m f Step 3 Step 4 ( ) (0) t* t( f, / ).05 Step 5 Reject H 0 x x s s m f m f m f TuTh , height males height females x m x f 0

11 Marquette Uiversity MATH 700 Chapter 0: Ifereces Ivolvig Two Populatios Questios? Homework: Chapter 0 # 3, 5, 3, 5, 9, 3, 35 4, 45, 53, 57, 58, 59, 63, 83, 85, 9, 98, 99, 0, 3, 5, 7, 9, 5 33

12 Marquette Uiversity MATH 700 Lecture Chapter

13 Recall Marquette Uiversity MATH 700 9: Ifereces Ivolvig Oe Populatio 9. Iferece about the Biomial Probability of Success We talke about a Biomial experimet with two outcomes. Each performace of the experimet is calle a trial. Each trial is iepeet.,,3,...! x x 0 p P( x) p ( p) x!( x)! x 0,..., Chapter 5 = umber of trials or times we repeat the experimet. x = the umber of successes out of trials. p = the probability of success o a iiviual trial. 3

14 9: Ifereces Ivolvig Oe Populatio 9. Iferece about the Biomial Probability of Success Whe we perform a biomial experimet we ca estimate the probability of heas as Sample Biomial Probability x p'. where x is the umber of successes i trials. This is a poit estimate. Recall the rule for a CI is poit estimate Recall Marquette Uiversity MATH 700 some amout i.e. umber of H out of flips (9.3) 4

15 Marquette Uiversity MATH 700 0: Ifereces Ivolvig Two Populatios 0.4 Iferece for Differece betwee Two Proportios For Biomial, where x is umber of successes out of trials. We sai that mea( cx) cp a variace( cx) c pq. mea( x / ) p a variace( x / ) pq /. We are ofte itereste i comparisos betwee proportios p p. There is aother rule that says that if x a x are raom variables, the mea( x x) mea( x) mea( x) x x x x further, mea mea mea x x pq pq a variace. if x & x iepeet q=-p 5

16 Marquette Uiversity MATH 700 0: Ifereces Ivolvig Two Populatios 0.4 Iferece for Differece betwee Two Proportios That is where. a. i the gree box below come from If iepeet samples of size a are raw with p =P (success) a p =P (success), the the samplig istributio of p p has these properties:. mea p p p p pq pq. staar error p (0.0) p 3. approximately ormal ist if a are sufficietly large. ie I, >0 II p, q, p, q >5 III sample<0% of pop 6

17 Marquette Uiversity MATH 700 0: Ifereces Ivolvig Two Populatios 0.4 Iferece for Differece betwee Two Proportios Cofiece Iterval Proceure Assumptios for ifferece betwee two proportios p -p : The a raom observatios are selecte iepeetly from two populatios that are ot chagig Cofiece Iterval for the Differece betwee Two Proportios p - p p q p q p q p q ( p p ) z( / ) to ( p p ) z( / ) x x where p a p. (0.) 7

18 Marquette Uiversity MATH 700 0: Ifereces Ivolvig Two Populatios 0.4 Iferece for Differece betwee Two Proportios Cofiece Iterval Proceure Example: Costruct a 99% CI for proportio of female A s mius male A s ifferece pf pm. Fill i. pq 0 values f f pq m m z( / ) ( pf p m) z( / ) m 5 f m x f f 68 pf f xm xm p m x f 43 m TuTh 0 Top 5 of 6 exams. 8

19 Marquette Uiversity MATH 700 0: Ifereces Ivolvig Two Populatios 0.4 Iferece for Differece betwee Two Proportios Hypothesis Testig Proceure We ca perform hypothesis tests o the proportio H 0 : p p vs. H a : p < p pq pq pq H 0 : p p vs. H a : p > p H 0 : p = p vs. H a : p p whe p p p. Test Statistic for the Differece betwee two Proportios- 0 ( p Populatio Proportios Kow p ) ( p0 p0) z* pq x x (0.) p p p kow

20 Marquette Uiversity MATH 700 0: Ifereces Ivolvig Two Populatios 0.4 Iferece for Differece betwee Two Proportios Hypothesis Testig Proceure Test Statistic for the Differece betwee two Proportios- Populatio Proportios UKow z* (0.5) where we assume p =p a use poole estimate of proportio x x p pq pq x x p pq pp ( p p) ( p p ) 0 0 pq p p p p estimate 0

21 Marquette Uiversity MATH 700 0: Ifereces Ivolvig Two Populatios 0.4 Iferece for Differece betwee Two Proportios Is proportio of Salesma s efectives less tha Competitor s?.05 Step Step Fill i. xs x c s c Step 3 Step 4 Step 5 Figure from Johso & Kuby, 0. 3

22 Marquette Uiversity MATH 700 0: Ifereces Ivolvig Two Populatios 0.5 Iferece for Ratio of Two Variaces Two I. Samples Hypothesis Testig Proceure We ca perform hypothesis tests o two variaces H 0 : vs. H a : Assumptios: Iepeet H 0 : vs. H a : samples from ormal istributio H 0 : vs. H a : Test Statistic for Equality of Variaces F s with f a f. s * Use ew table to fi areas for ew statistic. Actually F* igore ( ) s / ( ) ( ) s / ( ) (0.6) 9

23 Marquette Uiversity MATH 700 0: Ifereces Ivolvig Two Populatios 0.5 Iferece for Ratio of Two Variaces Two I. Samples Properties of F istributio. F is o-egative f(f).6. F is osymmetrical.4 3. F is a family of ists. f =ν = -,f =ν = -. igore, ( ) ( ) ( 4), / f( F, ) / F ( )/ F μ =, = =5, = =5, =5 =0, =0 =50, = igore F 30

24 Marquette Uiversity MATH 700 0: Ifereces Ivolvig Two Populatios 0.5 Iferece for Ratio of Two Variaces Two I. Samples Hypothesis Testig Proceure Test Statistic for Equality of Variaces F s with f a f. (0.6) s * Will also ee critical values. P F F( f, f, ) Table 9 Appeix B Page 7 Figure from Johso & Kuby, 0. 3

25 f Marquette Uiversity MATH 700 0: Ifereces Ivolvig Two 0.5 Iferece Ratio of Two Variaces Example: Fi F(5,8,0.05). f f Table 9, Appeix B, Page Pops. f Figures from Johso & Kuby, 0. 3

26 Marquette Uiversity MATH 700 0: Ifereces Ivolvig Two Populatios 0.5 Iferece for Ratio of Two Variaces Two I. Samples Hypothesis Testig Proceure Oe taile tests: Arrage H 0 & H a so H a is always greater tha H 0 : vs. H a : H 0 : vs. H a : / / F* s H 0 : vs. H a : H 0 : / vs. H a : / F* s s s Reject H 0 if F* s / s > F(f,f,α). Two taile tests: put larger sample variace s i umerator H 0 : vs. H a : H 0 : / vs. H a : / if s s, ifs s Reject H 0 if F* s / s > F(f,f,α/). 33

27 Marquette Uiversity MATH 700 0: Ifereces Ivolvig Two Populatios 0.5 Iferece for Ratio of Two Variaces Two I. Samples 77 values Is variace of male heights greater tha that of females?.0 Step Fill i. 30 Step Step 3 Step 4 x x s s m f m f m f TuTh 0 x m x f Step 5 34

28 Marquette Uiversity MATH 700 Chapter 0: Ifereces Ivolvig Two Populatios Questios? Homework: Chapter 0 # 3, 5, 3, 5, 9, 3, 35 4, 45, 53, 57, 58, 59, 63, 83, 85, 9, 98, 99, 0, 3, 5, 7, 9,

29 Marquette Uiversity MATH 700 Problem Solvig Sessio 4