14.3 comparing two populations: based on independent samples
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1 Chpter4 Nonprmetric Sttistics Introduction: : methods for mking inferences bout popultion prmeters (confidence intervl nd hypothesis testing) rely on the ssumptions bout probbility distribution of smpled popultion. (t-test, ANOVA) : sttisticl tests tht do not rely on ny underlying ssumptions bout the probbility distribution of the smpled popultion. : the brnch of inferentil sttistics devoted to distribution-free tests. : nonprmetric tests (or sttistics) bsed on the rnks of mesurements. 4.3 compring two popultions: bsed on independent smples When t-test is not pproprite, we cn use nonprmetric methods to compre two popultions. : compring the loction of center for two or more non-norml popultions. It tests the hypothesis tht the probbility distributions ssocited with the two popultions re equivlent. Wilcoxon rnk sum test: Define : the for popultion, D : the for popultion. Three reltionship between D nd D :. ypothesis: : : ( is shifted to the of D ) ( is shifted to the of D )
2 . Test sttistic: Step. ll the dt vlues (both smples) from to N, ( N = n+ n, n < n) (Note: ssign the of the rnks to ech of the tied observtions) Step. Test sttistict = Sum the rnks for the ( n ). 3. Rejection region: if nd D re not identicl; if is shifted to the left of D ; if is shifted to the right of D. Criticl vlues TL nd T U re from Tble XII, p87. Condition required for vlid rnk sum test:. the two smples re ;. the two probbility distributions from which the smples re drwn re.
3 ExmpleRUGS: At α =.5, Do the dt provide sufficient evidence to indicte shift in the probbility distributions for drug A nd B? Drug A Drug B Rection time(seconds) Rection time(seconds) Define : the probbility distribution for rection times under drug A( ) D : the probbility distribution for rection time under drug B; : : 3
4 Exmple : VENDERS: Rw mteril for n industril process is provided by two different venders. Four btches of rw mteril re purchsed from ech vender, nd the strength of the finished product (in p.s.i) is mesured for ech btch. At α =.5, do the dt provide sufficient evidence to indicte the probbility distributions of product strength for vender is shifted to left of the vender? Vender Vender
5 The Wilcoxon Rnk sum test for lrge smple ( ): ypothesis: : D nd D re identicl, nd D re not identicl, R.R: ( is shifted to the left of D, R.R: ) ( is shifted to the right of D, R.R: ) Test sttistic: n( n+ n + ) T z = nn ( n+ n+ ) Exmple: Verbl SAT scores for students rndomly selected from two different schools re listed below. Use the Wilcoxon rnk sum procedure to test the clim tht there is no difference in the scores from ech school. Use α =.5. School 55, 5, 77, 48, 75, 53, 58, 78, 6, 59, 73, 75 School 49, 44, 68, 43, 7, 59, 69, 55, 53, 63, 64, 54 5
6 4.5 compring three or more popultions: completely rndomized design Nonprmetric method lso cn used to nlyze the dt from CRD. : no ssumptions concerning the popultion probbility distribution; Rnk procedure:. Rnk observtions from smllest () to lrgest (n),. Add up the rnks for, R, j =,..., k j ypothesis: : : Test sttistic: Rejection region:, with degree of freedom. (p798) Conditions required for the vlid Kruskl-wllis test:. The k smples re,. there re mesurements in ech smple; 3. the k probbility distributions from which the smples re drwn re. 6
7 Exmple: OZONE EFFECT: To study the effects of ozone exposure on irwy resistnce, 5 helthy mle subjects were rndomly ssigned to ech of 3 exposure levels:.ppm,.6ppm,.ppm for one hour. The differences of irwy resistnce fter exposure nd before exposure for ech subject were recorded. At α =., is there evidence to conclude there exists difference in the probbility distribution mong three ozone exposure levels?. ppm ().6 ppm (). ppm (3) : : 7
8 Exmple: Vehicle miles: The U.S federl highwy dministrtion conducts nnul survey on motor vehicle trvel by type of vehicle nd publishes in highwy sttistics. Independent rndom smples of crs, buses, nd trucks provided the dt on number of miles driven lst yer, in thousnds. At α =.5, do the dt provide sufficient evidence to conclude tht difference exists in the probbility distribution of lst yer s miles mong crs, buses, nd trucks? crs buses trucks : : 8
9 4.6 Compring three or more popultions: rndomized Block design : no ssumptions concerning the popultion probbility distribution; Rnk procedure:. Rnk observtions within ;. Add up the rnks for ; R, j =,..., k ypothesis: : : j Test sttistic: : # of blocks; : # of tretments. Rejection region:, with degrees of freedom.(p798) Conditions required for vlid Friedmn test:. the tretments re ssigned to experimentl units within the blocks;. the mesurements cn be rnked within ; 3. the k probbility distributions from which the smples within ech block re drwn re. Exmple: Rection, (when drug effect is short-lived,no crryover effect; nd the drug effect vries gretly from person to person, it my be useful to employ RBD to find the effect of drug.) At α =.5, do the dt provide sufficient evidence to conclude the probbility distributions of the rection times for the three drugs differ in loction? Rection time for three drugs: subject Drug A Drug B Drug C
10 Exmple. In study compring the effects of four energy drinks on running speed, seven runners were timed (in seconds) running four miles. On ech dy, they were given single energy drink. The dt re listed below. Is there evidence of difference in the probbility distributions of the running times mong the four drinks? Use α =.. drink runner
11 4.7 Rnk Correltion In chpter, we hve introduced correltion coefficient r bsed on the observtions of two numericl vribles. SSxy r = SS SS xx yy x y x y x y... x n y n ere we introduce correltion coefficient bsed on of observtions: : it provides mesure of correltion between rnks. Where r s = SSuv SS SS u i = of the ith observtion in smple v i = of the ith observtion in smple n = uu SS ( u u )( v v ) u v uv i i i i vv ( ui)( vi) n = = ( u ) i SSuu = ( ui u ) = ui n ( v ) i SSvv = ( vi v ) = vi n Shortcut formul for r s : (no ties or less ties reltive to n ) r s =
12 Where (rnk difference of ith observtions between smple nd smple ) Note:. The vlue of r s lwys flls between,. indictes perfect positive correltion nd indictes perfect negtive correltion, 3. The closer r s flls to + or -, the the correltion between the rnks; the closer r s is to, the the correltion. Spermn s rnk correltion test: Define ρ : : ρ = ( ) : ρ ( ) Test sttistic: Rejection region: if : ρ ; if : ρ < if : ρ > Criticl vlue: rs, α from Tble XIV, p89. Note: The α vlues correspond to one-tiled test of : ρ =.
13 Condition required for vlid Spermn s Test:. The smple of experimentl units on which the two vribles re mesured is selected,. The probbility distributions of the two vribles re. Exmple: The number of bsences nd the finl grdes of 9 rndomly selected students from sttistics clss re given below. Cn you conclude tht there is correltion between the finl grde nd the number of bsences? Use α =.5. student bsence Finl grde
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