Chapter 12 Student Lecture Notes 12-1

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1 Chaptr 1 Studnt Lctur Nots 1-1 Businss Statistics: A Dcision-Making Approach 6 th Edition Chaptr 1 Goodnss-of-Fit Tsts and Contingncy Analysis 005 Prntic-Hall, Inc. Chap 1-1 Chaptr Goals Aftr complting this chaptr, you should b abl to: Us th chi-squar goodnss-of-fit tst to dtrmin whthr data fits a spcifid distribution St up a contingncy analysis tabl and prform a chi-squar tst of indpndnc 005 Prntic-Hall, Inc. Chap 1- Chi-Squar Goodnss-of-Fit Tst Dos sampl data conform to a hypothsizd distribution? Exampls: Ar tchnical support calls qual across all days of th wk? (i.., do calls follow a uniform distribution?) Do masurmnts from a production procss follow a normal distribution? 005 Prntic-Hall, Inc. Chap Prntic-Hall, Inc.

2 Chaptr 1 Studnt Lctur Nots 1- Chi-Squar Goodnss-of-Fit Tst (continud) Ar tchnical support calls qual across all days of th wk? (i.., do calls follow a uniform distribution?) Sampl data for 10 days pr day of wk: Sum of calls for this day: Monday 90 Tusday 50 Wdnsday 38 Thursday 57 Friday 65 Saturday 30 Sunday 19 Σ Prntic-Hall, Inc. Chap 1-4 Logic of Goodnss-of-Fit Tst If calls ar uniformly distributd, th 17 calls would b xpctd to b qually dividd across th 7 days: xpctd calls pr day if uniform Chi-Squar Goodnss-of-Fit Tst: tst to s if th sampl rsults ar consistnt with th xpctd rsults 005 Prntic-Hall, Inc. Chap 1-5 Obsrvd vs. Expctd Frquncis Monday Tusday Wdnsday Thursday Friday Saturday Sunday TOTAL Obsrvd o i Expctd i Prntic-Hall, Inc. Chap Prntic-Hall, Inc.

3 Chaptr 1 Studnt Lctur Nots 1-3 Chi-Squar Tst Statistic H 0 : Th distribution of calls is uniform ovr days of th wk H A : Th distribution of calls is not uniform Th tst statistic is (oi i) i (whr df k 1) whr: k numbr of catgoris o i obsrvd cll frquncy for catgory i i xpctd cll frquncy for catgory i 005 Prntic-Hall, Inc. Chap 1-7 Th Rjction Rgion H 0 : Th distribution of calls is uniform ovr days of th wk H A : Th distribution of calls is not uniform Rjct H 0 if (oi i) > α i α (with k 1 dgrs of frdom) 0 Do not Rjct H 0 rjct H 0 α 005 Prntic-Hall, Inc. Chap 1-8 Chi-Squar Tst Statistic (90 46) 46 k 1 6 (7 days of th wk) so us 6 dgrs of frdom: H 0 : Th distribution of calls is uniform ovr days of th wk H A : Th distribution of calls is not uniform (50 46) + 46 Conclusion: 3.05 > α so rjct H 0 and conclud that th distribution is not uniform (19 46) Prntic-Hall, Inc. Chap α.05 Do not Rjct H 0 rjct H Prntic-Hall, Inc.

4 Chaptr 1 Studnt Lctur Nots 1-4 Do masurmnts from a production procss follow a normal distribution with µ 50 and σ 15? Procss: Normal Distribution Exampl Gt sampl data Group sampl rsults into classs (clls) (Expctd cll frquncy must b at last 5 for ach cll) Compar actual cll frquncis with xpctd cll frquncis 005 Prntic-Hall, Inc. Chap 1-10 Normal Distribution Exampl Sampl data and valus groupd into classs: (continud) 150 Sampl Masurmnts tc Class lss than but < but < but < but < but < but < or ovr TOTAL Frquncy Prntic-Hall, Inc. Chap 1-11 Normal Distribution Exampl (continud) What ar th xpctd frquncis for ths classs for a normal distribution with µ 50 and σ 15? Class lss than but < but < but < but < but < but < or ovr TOTAL Frquncy Expctd Frquncy 005 Prntic-Hall, Inc. Chap 1-1? 005 Prntic-Hall, Inc.

5 Chaptr 1 Studnt Lctur Nots 1-5 Expctd Frquncis Valu lss than but < but < but < but < but < but < or ovr TOTAL P(X < valu) Expctd frquncy Expctd frquncis in a sampl of siz n150, from a normal distribution with µ50, σ15 Exampl: P(x < 30) P z < 15 P(z < ).091 (.091)(150) Prntic-Hall, Inc. Chap 1-13 Th Tst Statistic Class lss than but < but < but < but < but < but < or ovr TOTAL Frquncy (obsrvd, o i ) Expctd Frquncy, i Th tst statistic is (oi i) Rjct H 0 if > i α (with k 1 dgrs of frdom) 005 Prntic-Hall, Inc. Chap 1-14 Th Rjction Rgion 8 classs so us 7 d.f.: H 0 : Th distribution of valus is normal with µ 50 and σ 15 H A : Th distribution of calls dos not hav this distribution (oi i) Conclusion: < α so i ( ) ( 0.57) do not rjct H 0 0 Do not Rjct H 0 Businss Statistics: A Dcision-Making Approach, rjct H 0 Inc Chap Prntic-Hall, 1-15 α Prntic-Hall, Inc.

6 Chaptr 1 Studnt Lctur Nots 1-6 Contingncy Tabls Contingncy Tabls Situations involving multipl population proportions Usd to classify sampl obsrvations according to two or mor charactristics Also calld a crosstabulation tabl. 005 Prntic-Hall, Inc. Chap 1-16 Contingncy Tabl Exampl Lft-Handd vs. Gndr Dominant Hand: Lft vs. Right Gndr: Mal vs. Fmal H 0 : Hand prfrnc is indpndnt of gndr H A : Hand prfrnc is not indpndnt of gndr 005 Prntic-Hall, Inc. Chap 1-17 Contingncy Tabl Exampl Sampl rsults organizd in a contingncy tabl: (continud) sampl siz n 300: 10 Fmals, 1 wr lft handd 180 Mals, 4 wr lft handd Gndr Fmal Mal Hand Prfrnc Lft Right Prntic-Hall, Inc. Chap Prntic-Hall, Inc.

7 Chaptr 1 Studnt Lctur Nots 1-7 Logic of th Tst H 0 : Hand prfrnc is indpndnt of gndr H A : Hand prfrnc is not indpndnt of gndr If H 0 is tru, thn th proportion of lft-handd fmals should b th sam as th proportion of lft-handd mals Th two proportions abov should b th sam as th proportion of lft-handd popl ovrall 005 Prntic-Hall, Inc. Chap 1-19 Finding Expctd Frquncis 10 Fmals, 1 wr lft handd 180 Mals, 4 wr lft handd Ovrall: P(Lft Handd) 36/300.1 If indpndnt, thn P(Lft Handd Fmal) P(Lft Handd Mal).1 So w would xpct 1% of th 10 fmals and 1% of th 180 mals to b lft handd i.., w would xpct (10)(.1) 14.4 fmals to b lft handd (180)(.1) 1.6 mals to b lft handd 005 Prntic-Hall, Inc. Chap 1-0 Expctd Cll Frquncis Expctd cll frquncis: (continud) ij (i th th Row total)( j Column total) Total sampl siz Exampl: (10)(36) Prntic-Hall, Inc. Chap Prntic-Hall, Inc.

8 Chaptr 1 Studnt Lctur Nots 1-8 Obsrvd v. Expctd Frquncis Obsrvd frquncis vs. xpctd frquncis: Gndr Fmal Mal Lft Obsrvd 1 Expctd 14.4 Obsrvd 4 Expctd 1.6 Hand Prfrnc Right Obsrvd 108 Expctd Obsrvd 156 Expctd Prntic-Hall, Inc. Chap 1- Th Chi-Squar Tst Statistic Th Chi-squar contingncy tst statistic is: r c i 1 j 1 (o ) ij ij ij with d.f. (r 1)(c 1) whr: o ij obsrvd frquncy in cll (i, j) ij xpctd frquncy in cll (i, j) r numbr of rows c numbr of columns 005 Prntic-Hall, Inc. Chap 1-3 Obsrvd v. Expctd Frquncis Gndr Fmal Mal Lft Obsrvd 1 Expctd 14.4 Obsrvd 4 Expctd 1.6 Hand Prfrnc Right Obsrvd 108 Expctd Obsrvd 156 Expctd (1 14.4) 14.4 ( ) (4 1.6) ( ) Prntic-Hall, Inc. Chap Prntic-Hall, Inc.

9 Chaptr 1 Studnt Lctur Nots 1-9 Contingncy Analysis with d.f. (r -1)(c -1) (1)(1) 1 Dcision Rul: If > 3.841, rjct H 0, othrwis, do not rjct H 0 Do not rjct H α 0.05 Rjct H Prntic-Hall, Inc. Chap 1-5 Hr, < 3.841, so w do not rjct H 0 and conclud that gndr and hand prfrnc ar indpndnt Chaptr Summary Usd th chi-squar goodnss-of-fit tst to dtrmin whthr data fits a spcifid distribution Exampl of a discrt distribution (uniform) Exampl of a continuous distribution (normal) Usd contingncy tabls to prform a chi-squar tst of indpndnc Compard obsrvd cll frquncis to xpctd cll frquncis 005 Prntic-Hall, Inc. Chap Prntic-Hall, Inc.

Probability, Genetics, and Games

Probability, Genetics, and Games " Probability, Gntics, and Gams Hav you vr hard of gns? (W don t man th kind you war!) What color ar your ys? Can you curl your tongu? Your birth parnts gav you a uniqu st of gns that dtrmin such things.