Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 10-1ne- and !"o-Sa#ple !est of $ypothesesChapter 10Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 10-%Hypothesis Testing Content&or#ulate null and alternati'e hypotheses and decision rules !ype ( and !ype (( errorsSingle Sa#ple) !esting Population *ean, !"o Sa#ple) !esting +ifference t"o #eans!ests on Proportion, p !ests on ,ariance, % -oodness-of-&it !est!est for (ndependenceWalpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 10-.What is a $ypothesis/0 hypothesis is a clai# 1assu#ption2 about a population para#eter)population #eanpopulation proportionExample:The mean monthly cell phone bill of this city is = $42Example:The proportion of a!lts in this city "ith cell phones isp = #$%Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 10-3!he 4ull $ypothesis, $0States the assu#ption 1nu#erical2 to be testedE5a#ple)!he a'erage nu#ber of !, sets in ho#es is e6ual to three12(s al"ays about a population para#eter, not about a sa#ple statistic . 7 ) $0=. 7 ) $0= . 8 ) $0=Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 10-9!he 4ull $ypothesis, $0:egin "ith the assu#ption that the null hypothesis is trueSi#ilar to the notion of innocent until pro'en guilty;efers to the status 6uo0l"ays contains , or sign*ay or #ay not be re@ected(continued)Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 10-A!he 0lternati'e $ypothesis, $1(s the opposite of the null hypothesise.g., !he a'erage nu#ber of !, sets in ho#es is not e6ual to .1 $1) 7 B . 2Challenges the status 6uo4e'er contains the , or sign*ay or #ay not be pro'en(s generally the hypothesis that the researcher is trying to pro'eWalpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu.&op!lationClaim: thepop!lationmean age is '0#()!ll Hypothesis:*E+ECT,!pposethe samplemean age is 20:- = 20,ample)!ll Hypothesis%0liCely if7 = 90/ =.s$ypothesis !esting Process.f not li/ely0 4o" select a rando# sa#ple H0: = '0 18Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 10-8Sa#pling +istribution of 8 = '0.f H0 is tr!e(f it is unliCely that "e "ould get a sa#ple #ean of this 'alue ...... then "e re@ect the null hypothesis that 7 = 90.;eason for ;e@ecting $020... if in fact this "ere the population #eanD-Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 10-EFe'el of Significance, 2efines the !nli/ely 3al!es of the sample statistic if the n!ll hypothesis is tr!e+efines re@ection region of the sa#pling distribution(s designated by , 1le'el of significance2!ypical 'alues are .01, .09, or .10(s selected by the researcher at the beginningPro'ides the critical 'alue1s2 of the test Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 10-10Fe'el of Significance and the ;e@ection ;egion$0) 7 G . $1) 7 H .0$0) 7 ? .$1) 7 I . *epresentscritical 3al!eFo"er-tail testFe'el of significance = 0Jpper-tail test!"o-tail test*e4ection region is shae K%0K%$0) 7 = .$1) 7 B .Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 10-11Errors in *aCing +ecisionsType . Error ;e@ect a true null hypothesisConsidered a serious type of error!he probability of !ype ( Error is Called le'el of significance of the testSet by researcher in ad'anceWalpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 10-1%Errors in *aCing +ecisionsType .. Error&ail to re@ect a false null hypothesis!he probability of !ype (( Error isL(continued)Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 10-1.utco#es and Probabilities5ct!al ,it!ation2ecision+o 4ot;e@ectH0)o error (1 61Type .. Error ( 7 1;e@ectH0Type . Error(1&ossible Hypothesis Test 8!tcomes $0 &alse $0 !rue9ey:8!tcome(&robability1)o Error ( 1 6 7 1Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 10-13!ype ( & (( Error ;elationship !ype ( and !ype (( errors can not happen at the sa#e ti#e !ype ( error can only occur if $0 is true !ype (( error can only occur if $0 is false (f !ype ( error probability 1 2, then !ype (( error probability 1 L 2Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 10-19&actors 0ffecting !ype (( Error0ll else e6ual,L"hen the difference bet"een hypothesiMed para#eter and its true 'alueL"henL"henNL"hennWalpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 10-1A;e'ie") 10 Steps in $ypothesis !esting1.State the null hypothesis, $0%. State the alternati'e hypotheses, $1..Choose the le'el of significance, O3.Choose the sa#ple siMe, n9.+eter#ine the appropriate statistical techni6ue and the test statistic to useA.&ind the critical 'alues and deter#ine the re@ection region1s2Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 10-1P;e'ie") 10 Steps in $ypothesis !estingP.Collect data and co#pute the test statistic fro# the sa#ple result8. Co#pare the test statistic to the critical 'alue to deter#ine "hether the test statistics falls in the region of re@ectionE.*aCe the statistical decision) ;e@ect $0 if the test statistic falls in the re@ection region10. E5press the decision in the conte5t of the proble#Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 10-18$ypothesis !ests for the *ean 9no"n :n/no"nHypothesis Tests for Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 10-1E10.9 Single Sa#ple) !est of $ypothesis for Single *ean 1N Qno"n2Con'ert sa#ple statistic 1 2 to a R test statistic 8!he test statistic is)nN7 8R=; 9no"n ; :n/no"nHypothesis Tests for Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 10-%0Critical ,alue 0pproach to !esting&or t"o tailed test for the #ean, N Cno"n)Con'ert sa#ple statistic 12 to test statistic 1Rstatistic 2+eter#ine the critical R 'alues for a specifiedle'el of significancefro# a table or co#puter+ecision ;ule) (f the test statistic falls in the re@ection region, re@ect $0 Sother"ise do not re@ect $0

8Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 10-%1$ypothesis !esting E5a#pleTest the claim that the tr!e mean < of T= sets in homes is e>!al to ?#(5ss!me ; = 0#%11-%.State the appropriate null and alternati'ehypotheses $0) 7 = .$1) 7 B .1!his is a t"o tailed test2..Specify the desired le'el of significanceSuppose that = .09 is chosen for this test3.Choose a sa#ple siMeSuppose a sa#ple of siMe n = 100 is selectedWalpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 10-%%%.0.08.1A1000.8. %.83nN7 8R ====$ypothesis !esting E5a#ple9. +eter#ine the appropriate techni6ueN is Cno"n so this is a R testA.Set up the critical 'alues&or = .09 the critical R 'alues are T1.EAP.Collect the data and co#pute the test statisticSuppose the sa#ple results are n = 100, 8 = %.831N = 0.8 is assu#ed Cno"n2So the test statistic is)(continued)Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 10-%.;e@ect $0+o not re@ect $08.(s the test statistic in the re@ection region/ = .09K%6@= 61#A$0;e@ect $0 ifR H -1.EA or R I 1.EASother"ise do not re@ect $0$ypothesis !esting E5a#ple(continued) = .09K%;e@ect $0B@= B1#A$$ere, R = -%.0 H -1.EA, so the test statistic is in the re@ection regionWalpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 10-%3 E-10. ;each a decision and interpret the result62#0SinceR = -%.0 H -1.EA,"e re@ect the null hypothesisand conclude that there is sufficient e'idence that the #ean nu#ber of !,s in JS ho#es is not e6ual to . $ypothesis !esting E5a#ple(continued);e@ect $0+o not re@ect $0 = .09K%6@= 61#A$0 = .09K%;e@ect $0B@= B1#A$Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 10-%9p-,alue 0pproach to !estingp-'alue) Probability of obtaining a test statistic #ore e5tre#e 1 ? or 2 than the obser'ed sa#ple 'alue gi'en $0 is true0lso called obser'ed le'el of significanceS#allest 'alue offor "hich $0 can be re@ected Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 10-%Ap-,alue 0pproach to !estingCon'ert Sa#ple Statistic 1e.g.,2 to !est Statistic 1e.g., R statistic 2btain the p-'alue fro# a table or co#puterCo#pare the p-'alue "ith(f p-'alueH ,re@ect $0(f p-'alue ,do not re@ect $0 8(continued)Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 10-%P#022%K% = .0%9p-,alue E5a#pleE5a#ple)$o" liCely is it to see a sa#ple #ean of %.83 1or so#ething further fro# the #ean, in either direction2 if the true #ean is = ..0/61#A$ 062#0.0%%8 %.02 P1R.0%%8 %.02 P1R= >=