ByRitvik Jainsudhanshu Kumar AshviniBasio 3tatistios3tandard Ueviation vs varianoeO.D |as auc u:it as t|at oi saulcOVa:ia:cc Dcii:cs I|c :cac a:c cist:i|utio: O.D is uo:c usciul i: cislavi:- t|c cata w|c:c as Va:ia:cc is uo:c usciul i: calculatio: o: cataat is a Probabilitv Uistribution?3Experiment: Toss a coin three times. Observe the number of heads. The possible results are: zero heads, one head, twoheads, and three heads. What is the probability distribution for the number of heads?Probabilitv Uistribution of Number of eads 0bserved in 3 1osses of a Coin4Caraoteristios of a Probabilitv Uistribution5#andom variableO#a:cou va:ia|lc a qua:titv :csulti:- i:ou a: cxc:iuc:t t|at. |v c|a:cc. ca: assuuc ciiic:c:t valucsODisc:ctc #a:cou Va:ia|lc ca: assuuc o:lv cc:tai: clca:lv sca:atcc valucs. lt is usuallv t|c :csult oi cou:ti:-souct|i:-Oo:ti:uous #a:cou Va:ia|lc ca: assuuc a: i:ii:itc :uu|c: oi valucs wit|i: a -ivc: :a:-c. lt is usuallv t|c :csult oi souc tvc oi ucasu:cuc:teatures of a Uisorete Uistribution7I|c uai: icatu:cs oi a cisc:ctc :o|a|ilitv cist:i|utio: a:c.OI|c suu oi t|c :o|a|ilitics oi t|c va:ious outcoucs is 1.00.OI|c :o|a|ilitv oi a a:ticula: outcouc is |ctwcc: 0 a:c 1.00.OI|c outcoucs a:c uutuallv cxclusivc.ean, 3.U, varianoe of a Probabilitv Uistribution8

WThe mean is a typical value used to represent the central location of a probability distribution.WThe mean of a probability distribution is also referred to as its expected vaIue.W$.D and Variance Measures the amount of spread in a distributionBinomial Probabilitv Uistribution9haracteristics o a Binomiai Probabiiity DistributionOI|c:c a:c o:lv two ossi|lc outcoucs o: a a:ticula: t:ial oi a: cxc:iuc:t.OI|c outcoucs a:c uutuallv cxclusivc. OI|c :a:cou va:ia|lc is t|c :csult oi cou:ts.O lac| t:ial is .-1.o.-1.- oi a:v ot|c: t:ial#eal Life samplesO stucv i: u:c !00! |v t|c llli:ois Dca:tuc:t oi I:a:so:tatio: co:cluccc t|at o.! c:cc:t oi i:o:t scat occua:ts uscc scat |clts.saulc oi 1! vc|iclcs is sclcctcc. W|at is t|c :o|a|ilitv t|c i:o:t scat occua:ts i: at lcast oi t|c 1! vc|iclcs a:c wca:i:- scat |cltsPoisson Probabilitv Uistribution11I|c Poisson probabiiity distribution ccsc:i|cs t|c :uu|c: oi tiucs souc cvc:t occu:s cu:i:- a scciiicc i:tc:val. I|c i:tc:val uav |c tiuc. cista:cc. a:ca. o: voluuc.O ssuutio:s oi t|c loisso: Dist:i|utio::1 I|c :o|a|ilitv is :oo:tio:al to t|c lc:-t| oi t|c i:tc:val. :! I|c i:tc:vals a:c i:ccc:cc:t.Poisson Probabilitv Uistribution12O I|c uca: :uu|c: oi succcsscs ca: |c cctc:ui:cc i: |i:ouial situatio:s |v -x. w|c:c- is t|c :uu|c: oi t:ials a:c x t|c :o|a|ilitv oi a succcss.O I|c va:ia:cc oi t|c loisso: cist:i|utio: is also cqual to - x.Poisson Probabilitv Uistribution - Lxample13ssuuc |a--a-c is :a:clv lost |v :o:t|wcst i:li:cs. uosc a :a:cou saulc oi 1.000 ili-|ts s|ows a total oi !00 |a-s wc:c lost. I|us. t|c a:it|uctic uca: :uu|c: oi lost |a-s c: ili-|t is 0.! :!001.000. li t|c :uu|c: oi lost |a-s c: ili-|t iollows a loisso: cist:i|utio: wit| :0.!. ii:c t|c :o|a|ilitv oi :ot losi:- a:v |a-s.Binomial versus Poisson151e uniform UistributionI|c u:iio:u :o|a|ilitv cist:i|utio: is c:|as t|c siulcst cist:i|utio: io: a co:ti:uous :a:cou va:ia|lc. I|is cist:i|utio: is :ccta:-ula: i: s|ac a:c is ccii:cc |v ui:iuuu a:c uaxiuuu valucs.#eal Life LxampleO out|wcst :izo:a tatc u:ivc:sitv :oviccs |us sc:vicc to stucc:ts w|ilc t|cv a:c o: cauus.|us a::ivcs at t|c :o:t| \ai: t:cct a:c ollc-c D:ivc sto cvc:v !0 ui:utcs |ctwcc: o .\. a:c 11 l.\. cu:i:- wcc|cavs. tucc:ts a::ivc at t|c |us sto at :a:cou tiucs. I|c tiuc t|at a stucc:t waits is u:iio:ulv cist:i|utcc i:ou 0 to !0 ui:utcs. low lo:- will a stucc:t tvicallv |avc to wait io: a |us l: ot|c: wo:cs w|at is t|c uca: waiti:- tiuc W|at is t|c sta:ca:c ccviatio: oi t|c waiti:- tiucs W|at is t|c :o|a|ilitv a stucc:t will wait uo:c t|a: !. ui:utcs1Caraoteristios of a Normal Probabilitv UistributionO lt is beii-shaped a:c |as a si:-lc ca| at t|c cc:tc: oi t|c cist:i|utio:. O I|c a:it|uctic uca:. uccia:. a:c uocc a:c cqual O I|c total a:ca u:cc: t|c cu:vc is 1.00. |ali t|c a:ca u:cc: t|c :o:ual cu:vc is to t|c :i-|t oi t|is cc:tc: oi:t a:c t|c ot|c: |ali to t|c lcit oi it.O lt is symmetricai a|out t|c uca:. O lt is asymptotic: I|c cu:vc -cts closc: a:c closc: to t|c axis |ut :cvc: actuallv touc|cs it. Io ut it a:ot|c: wav. t|c tails oi t|c cu:vc cxtc:c i:ccii:itclv i: |ot| ci:cctio:s.O I|c locatio: oi a :o:ual cist:i|utio: is cctc:ui:cc |v t|c uca:.u. t|c cisc:sio: o: s:cac oi t|c cist:i|utio: is cctc:ui:cc |v t|c sta:ca:c ccviatio:. .181e Normal Uistribution - amilies191e 3tandard NormalProbabilitv UistributionOI|c sta:ca:c :o:ual cist:i|utio: is a :o:ual cist:i|utio: wit| a uca: oi 0 a:c a sta:ca:c ccviatio: oi 1. Olt is also callcc t|ccist:i|utio:. Ovalucis t|c cista:cc |ctwcc: a sclcctcc valuc. ccsi-:atcc . a:c t|c oulatio: uca: u. civiccc |v t|c oulatio: sta:ca:c ccviatio:. .OI|c io:uula is.201e Lmpirioal #uleO |out o c:cc:t oi t|c a:ca u:cc: t|c :o:ual cu:vc is wit|i: o:c sta:ca:c ccviatio: oi t|c uca:.O |out 9. c:cc:t is wit|i: two sta:ca:c ccviatio:s oi t|c uca:. O l:acticallv all is wit|i: t|:cc sta:ca:c ccviatio:s oi t|c uca:.211e Lmpirioal #ule - Lxamples a:t oi its qualitv assu:a:cc :o-:au. t|c utolitc lattc:v oua:v co:cucts tcsts o: |attc:v liic. lo: a a:ticula: Dccll al|ali:c |attc:v. t|c uca: liic is 19 |ou:s. I|c usciul liic oi t|c |attc:v iollows a :o:ual cist:i|utio: wit| a sta:ca:c ccviatio: oi 1.! |ou:s. :swc: t|c iollowi:- qucstio:s.1. |out o c:cc:t oi t|c |attc:ics iailcc |ctwcc: w|at two valucs !. |out 9. c:cc:t oi t|c |attc:ics iailcc |ctwcc: w|at two valucs!. Vi:tuallv all oi t|c |attc:ics iailcc |ctwcc: w|at two valucs#eal Life LxampleOavto: Ii:c a:c #u||c: oua:v wis|cs to sct a ui:iuuu uilca-c -ua:a:tcc o: its :cw \X100 ti:c. Icsts :cvcal t|c uca: uilca-c is o.900 wit| a sta:ca:c ccviatio: oi !.0.0 uilcs a:c t|at t|c cist:i|utio: oi uilcs iollows t|c :o:ual :o|a|ilitv cist:i|utio:. lt wa:ts to sct t|c ui:iuuu -ua:a:tccc uilca-c so t|at :o uo:c t|a: + c:cc:t oi t|c ti:cs will |avc to |c :claccc. W|at ui:iuuu -ua:a:tccc uilca-c s|oulc avto: a::ou:cc!!Normal Approximation to te BinomialThe normal distribution (a continuous distribution) yields a good approximation of the binomial distribution (a discrete distribution) for large values of n. The normal probability distribution is generally a good approximation to the binomial probability distribution when nxand n(1-x ) are both greater than 5.!+Probabilitv 3amplineO :o|a|ilitv saulc is a saulc sclcctcc suc| t|at cac| itcu o: c:so: i: t|c oulatio: |ci:- stucicc |as a |:ow: li|cli|ooc oi |ci:- i:cluccc i: t|c saulc.!.etods of Probabilitv 3amplineOiulc #a:cou aulc.saulc io:uulatcc so t|at cac| itcu o: c:so: i: t|c oulatio: |as t|c sauc c|a:cc oi |ci:- i:cluccc. Ovstcuatic #a:cou auli:-.I|c itcus o: i:civicuals oi t|c oulatio: a:c a::a:-cc i: souc o:cc:. :a:cou sta:ti:- oi:t is sclcctcc a:c t|c: cvc:v t| ucu|c: oi t|c oulatio: is sclcctcc io: t|c saulc.!oetods of Probabilitv 3amplineOt:atiiicc #a:cou auli:-.oulatio: is ii:st civiccc i:to su|-:ous. callcc st:ata. a:c a saulc is sclcctcc i:ou cac| st:atuu. Olustc: auli:-.oulatio: is ii:st civiccc i:to :iua:v u:its t|c: saulcs a:c sclcctcc i:ou t|c :iua:v u:its.!etods of Probabilitv 3amplineOl: :o::o|a|ilitv saulc i:clusio: i: t|c saulc is |ascc o: t|c juc-uc:t oi t|c c:so: sclccti:- t|c saulc. OI|c sauli:- c::o: is t|cciiic:c:cc |ctwcc: a saulc statistic a:c its co::cso:ci:- oulatio: a:auctc:.!3ampline Uistribution of te 3ample eansOI|c sauli:- cist:i|utio: oi t|c saulc uca: is a :o|a|ilitv cist:i|utio: co:sisti:- oi all ossi|lc saulc uca:s oi a -ivc: saulc sizc sclcctcc i:ou a oulatio:.!9Central Limit 1eoremO lo: a oulatio: wit| a uca:a:c a va:ia:cc!t|c sauli:- cist:i|utio: oi t|c uca:s oi all ossi|lc saulcs oi sizc - -c:c:atcc i:ou t|c oulatio: will |c a:oxiuatclv :o:uallv cist:i|utcc.O I|c uca: oi t|c sauli:- cist:i|utio: cqual to a:c t|c va:ia:cc cqual to !-.!0usine te 3amplineUistribution of te 3ample ean (3iema Known)Oli a oulatio: iollows t|c :o:ual cist:i|utio:. t|c sauli:- cist:i|utio: oi t|c saulc uca: will also iollow t|c :o:ual cist:i|utio:.OIo cctc:ui:c t|c :o|a|ilitv a saulc uca: ialls wit|i: a a:ticula: :c-io:. usc.nXzou =!1Oli t|c oulatio: cocs :ot iollow t|c :o:ual cist:i|utio:. |ut t|c saulc is oi at lcast !0 o|sc:vatio:s. t|c saulc uca:s will iollow t|c :o:ual cist:i|utio:.OIo cctc:ui:c t|c :o|a|ilitv a saulc uca: ialls wit|i: a a:ticula: :c-io:. usc.n sXtu =usine te 3amplineUistribution of te 3ample ean (3iema unknown)!!I|c Qualitv ssu:a:cc Dca:tuc:t io: ola. l:c.. uai:tai:s :cco:cs :c-a:ci:- t|c auou:t oi cola i: its uu|o |ottlc. I|c actual auou:t oi cola i: cac| |ottlc is c:itical. |ut va:ics a suall auou:t i:ou o:c |ottlc to t|c :cxt. ola. l:c.. cocs :ot wis| to u:cc:iill t|c |ottlcs. : t|c ot|c: |a:c. it ca::ot ovc:iill cac| |ottlc. lts :cco:cs i:cicatc t|at t|c auou:t oi cola iollows t|c :o:ual :o|a|ilitv cist:i|utio:. I|c uca: auou:t c: |ottlc is !1.! ou:ccs a:c t|c oulatio: sta:ca:c ccviatio: is 0.+ ou:ccs. t .\. tocav t|c qualitv tcc|:icia: :a:coulv sclcctcc 1o |ottlcs i:ou t|c iilli:- li:c. I|c uca: auou:t oi cola co:tai:cc i: t|c |ottlcs is !1.! ou:ccs. ls t|is a: u:li|clv :csult ls it li|clv t|c :occss is utti:- too uuc| soca i: t|c |ottlcs Io ut it a:ot|c: wav. is t|c sauli:- c::o: oi 0.1 ou:ccs u:usualusine te 3ampline Uistributionof te 3ample ean (3iema Known) - Lxample!!Point and lnterval LstimatesOoi:t cstiuatc is t|c statistic. couutcc i:ou saulc i:io:uatio:. w|ic| is uscc to cstiuatc t|c oulatio: a:auctc:.Oco:iicc:cc i:tc:val cstiuatc is a :a:-c oi valucs co:st:uctcc i:ou saulc cata so t|at t|c oulatio: a:auctc: is li|clv to occu: wit|i: t|at :a:-c at a scciiicc :o|a|ilitv. I|c scciiicc :o|a|ilitv is callcc t|c lcvcl oi co:iicc:cc.!+Point and lnterval LstimatesOoi:t cstiuatc is t|c statistic. couutcc i:ou saulc i:io:uatio:. w|ic| is uscc to cstiuatc t|c oulatio: a:auctc:.Oco:iicc:cc i:tc:val cstiuatc is a :a:-c oi valucs co:st:uctcc i:ou saulc cata so t|at t|c oulatio: a:auctc: is li|clv to occu: wit|i: t|at :a:-c at a scciiicc :o|a|ilitv. I|c scciiicc :o|a|ilitv is callcc t|c lcvcl oi co:iicc:cc.!.aotors Affeotine Confidenoe lnterval LstimatesI|c iacto:s t|at cctc:ui:c t|c wict| oi a co:iicc:cc i:tc:val a:c.1.I|c saulc sizc. -.!.I|c va:ia|ilitv i: t|c oulatio:. usuallv cstiuatcc |v 8.!.I|c ccsi:cc lcvcl oi co:iicc:cc. 36lnterval Lstimates - lnterpretationlo: a 9. co:iicc:cc i:tc:val a|out 9. oi t|c siuila:lv co:st:uctcc i:tc:vals will co:tai: t|c a:auctc: |ci:- cstiuatcc.lso 9. oi t|c saulc uca:s io: a scciiicc saulc sizc will lic wit|i: 1.9o sta:ca:c ccviatio:s oi t|c |vot|csizcc oulatio:!Caraoteristios of te t-distribution1. lt is. li|c t|ccist:i|utio:. a co:ti:uous cist:i|utio:.!. lt is. li|c t|ccist:i|utio:. |clls|acc a:c svuuct:ical.!. I|c:c is :ot o:c t cist:i|utio:. |ut :at|c: a iauilv oi t cist:i|utio:s. ll cist:i|utio:s |avc a uca: oi 0. |ut t|ci: sta:ca:c ccviatio:s ciiic: acco:ci:- to t|csaulc sizc. -. +. I|c t cist:i|utio: is uo:c s:cac out a:c ilattc: at t|c cc:tc: t|a: t|c sta:ca:c :o:ual cist:i|utio: s t|c saulc sizc i:c:cascs. |owcvc:. t|c cist:i|utio: a:oac|cs t|c sta:ca:c :o:ual cist:i|utio:. !en to use teor 9 Uistribution for Confidenoe lnterval Computation#eal Life LxampleO ti:c ua:uiactu:c: wis|cs to i:vcsti-atc t|c t:cac liic oi its ti:cs.saulc oi 10 ti:cs c:ivc: .0.000 uilcs :cvcalcc a saulc uca: oi 0.!! i:c| oi t:cac :cuai:i:- wit| a sta:ca:c ccviatio: oi 0.09 i:c|. o:st:uct a 9. c:cc:t co:iicc:cc i:tc:val io: t|c oulatio: uca:. Woulc it |c :caso:a|lc io: t|c ua:uiactu:c: to co:clucc t|at aitc: .0.000 uilcs t|c oulatio: uca: auou:t oi t:cac :cuai:i:- is 0.!0 i:c|csPopulation Proportion- LxampleI|c u:io: :c:csc:ti:- t|c lottlc llowc:s oi uc:ica :ll is co:sicc:i:- a :oosal to uc:-c wit| t|c Icaustc:s u:io:. cco:ci:- to ll u:io: |vlaws. at lcast t|:cciou:t|s oi t|c u:io: ucu|c:s|i uust a:ovc a:v uc:-c:.:a:cou saulc oi !.000 cu::c:t ll ucu|c:s :cvcals 1.o00 la: to votc io: t|c uc:-c: :oosal. W|at is t|c cstiuatc oi t|c oulatio: :oo:tio: Dcvclo a 9. c:cc:t co:iicc:cc i:tc:val io: t|c oulatio: :oo:tio:. lasi:- vou: cccisio: o: t|is saulc i:io:uatio:. ca: vou co:clucc t|at t|c :cccssa:v :oo:tio: oi ll ucu|c:s iavo: t|c uc:-c: W|v+1at is a vpotesis? lvot|csis is a statcuc:t a|out t|c valuc oi a oulatio: a:auctc: ccvclocc io: t|c u:osc oi tcsti:-. lxaulcs oi |vot|cscs uacc a|out a oulatio: a:auctc: a:c.OI|c uca: uo:t|lv i:couc io: svstcus a:alvsts is !.o!..OIwc:tv c:cc:t oi all custouc:s at lovi:c`s |o lousc :ctu:: io: a:ot|c: ucal wit|i: a uo:t|. +!at is vpotesis 1estine?lvot|csis tcsti:- is a :occcu:c. |ascc o: saulc cvicc:cc a:c :o|a|ilitv t|co:v. uscc to cctc:ui:c w|ct|c: t|c |vot|csis is a :caso:a|lc statcuc:t a:c s|oulc :ot |c :cjcctcc. o: is u::caso:a|lc a:c s|oulc |c :cjcctcc. 43vpotesis 1estine 3teps++lmportant 1ines to #emember about 0 and 1O l0. :ull |vot|csis a:c l1. altc::atc |vot|csisO l0 a:c l1 a:c uutuallv cxclusivc a:c collcctivclv cx|austivc O l0 is alwavs :csuucc to |c t:uc O l1 |as t|c |u:cc: oi :ooi O:a:cou saulc :- is uscc to .... u0 O li wc co:clucc 'co :ot :cjcct l0'. t|is cocs :ot :cccssa:ilv uca: t|at t|c :ull |vot|csis is t:uc. it o:lv su--csts t|at t|c:c is :ot suiiicic:t cvicc:cc to :cjcct l0. :cjccti:- t|c :ull |vot|csis t|c:. su--csts t|at t|c altc::ativc |vot|csis uav |c t:uc.O lqualitv is alwavs a:t oi l0 :c.-. ~ . > . s. O = a:c ~ alwavs a:t oi l145ow to 3et up a Claim as vpotesisOl: actual :acticc. t|c status quo is sct u as l0Oli t|c claiu is |oastiul t|c claiu is sct u as l1 :wc alv t|c \issou:i :ulcs|ow uc. #cucu|c:. l1 |as t|c |u:cc: oi :ooiOl: :o|lcu solvi:-. loo| io: key vords a:c co:vc:t t|cu i:to svu|ols. ouc |cv wo:cs i:clucc. .o...1. /... |o-. o8 ..... o8. 1....- ... |o8 .|o-o.1. ctc.46Left-tail or #iet-tail 1est?eywordsInequaIitySymboIPart of:Larger (or more) than

Smaller (or less)

No more than A

t least _

Has increased

Is there difference?

Has not changed

Has "improved", "is better than". "is more effective"$ee right

W The direction of the test involving claims that use the words "has improved, "is better than, and the like will depend upon the variable being measured. W or instance, if the variable involves time for a certain medication to take effect, the words "better "improve or more effective are translated as " (less than, i.e. faster relief).W On the other hand, if the variable refers to a test score, then the words "better "improve or more effective are translated as " (greater than, i.e. higher test scores)+#eal Life exampleO aucstow: tccl oua:v ua:uiactu:cs a:c asscu|lcs ccs|s a:c ot|c: oiiicc cquiuc:t at scvc:al la:ts i: wcstc:: :cw Yo:| tatc. I|c wcc|lv :ocuctio: oi t|c \occl !!. ccs| at t|c l:cco:ia lla:t iollows t|c :o:ual :o|a|ilitv cist:i|utio: wit| a uca: oi !00 a:c a sta:ca:c ccviatio:oi 1o. #ccc:tlv. |ccausc oi ua:|ct cxa:sio:. :cw :ocuctio: uct|ocs |avc |cc: i:t:ocuccc a:c :cw culovccs |i:cc. I|c vicc :csicc:t oi ua:uiactu:i:- woulc li|c to i:vcsti-atc w|ct|c: t|c:c |as |cc: a .|o-o. i: t|c wcc|lv :ocuctio: oi t|c \occl !!. ccs|.491estine for a Population ean wit aKnown Population 3tandard Ueviation- Lxamplestep 1: state the nuii hypothesis and the aiternate hypothesis.l0. u ~ !00l1. u = !00note: keyvord in the probiem has changedjstep 2: seiect the ievei o signiicance. = 0.01 as stated in the probiemstep 1: seiect the test statistic.usc Zcist:i|utio: si:cc is |:ow:501estine for a Population ean wit aKnown Population 3tandard Ueviation- Lxamplestep +: Iormuiate the decision ruie.Reject H0 i |Z| > Z-258 . 2 notis 55 . 150 / 16200 5 . 203/2 / 01 .2 /2 /

nX --ouStep 5: ake a decision and interpret the resuIt.Because 1.55 does not faII in the rejection region, H0 is not rejected. We concIude that the popuIation mean is not different from 200. So we wouId report to the vice president of manufacturing that the sampIe evidence does not show that the production rate at the Fredonia PIant has changed from 200 per week..11vpe of Lrrors in vpotesis 1estineOIvc l l::o: O Dcii:cc as t|c :o|a|ilitv oi :cjccti:- t|c :ull |vot|csis w|c: it is actuallv t:uc.O I|is is cc:otcc |v t|c G:cc| lcttc: -Olso |:ow: as t|c si-:iiica:cc lcvcl oi a tcstOIvc ll l::o:.O Dcii:cc as t|c :o|a|ilitv oi acccti:- t|c :ull |vot|csis w|c: it is actuallv ialsc.O I|is is cc:otcc |v t|c G:cc| lcttc: 52#eal Life Lxample Population 3tandard Ueviation unknown -I|c \cla:la:c l:su:a:cc oua:v laius Dca:tuc:t :co:ts t|c uca: cost to :occss a claiu is o0. : i:cust:v coua:iso: s|owcc t|is auou:t to |c la:-c: t|a: uost ot|c: i:su:a:cc coua:ics. so t|c coua:v i:stitutcc costcutti:- ucasu:cs. Io cvaluatc t|c ciicct oi t|c costcutti:- ucasu:cs. t|c uc:viso: oi t|c laius Dca:tuc:t sclcctcc a :a:cou saulc oi !o claius :occsscc last uo:t|. I|c saulc i:io:uatio: is :co:tcc |clow. t t|c .01 si-:iiica:cc lcvcl is it :caso:a|lc a claiu is -. /.88 |o- o531estine for a Population ean wit aKnown Population 3tandard Ueviation- Lxamplestep 1: state the nuii hypothesis and the aiternate hypothesis.l0. u > o0l1. u o0note: keyvord in the probiem now less :nonjstep 2: seiect the ievei o signiicance. = 0.01 as stated in the probiemstep 1: seiect the test statistic.usc cist:i|utio: si:cc is u:|:ow:541estine for a Population ean wit aKnown Population 3tandard Ueviation- LxampleStep 5: ake a decision and interpret the resuIt.ecause -.88 does not fall in the rejection region,

is not rejected at the . significance level. We have not demonstrated that the cost-cutting measures reduced the mean cost per claim to less than $6. The difference of $3.58 ($56.42 - $6) between the sample mean and the population mean could be due to sampling error.step +: Iormuiate the decision ruie.#cjcct l0 ii t-.:1..1ests Conoernine ProportionOl:oo:tio: is t|c i:actio: o: c:cc:ta-c t|at i:cicatcs t|c a:t oi t|c oulatio: o: saulc |avi:- a a:ticula: t:ait oi i:tc:cst.O I|c saulc :oo:tio: is cc:otcc |v o a:c is iou:c |v -O I|c tcst statistic is couutcc as iollows..oAssumptions in 1estine a Population Proportion usine te z-UistributionO:a:cou saulc is c|osc: i:ou t|c oulatio:. O lt is assuucc t|at t|c |i:ouial assuutio:s ciscusscc i: |atc: o a:c uct. :1 t|c saulc cata collcctcc a:c t|c :csult oi cou:ts. :! t|c outcouc oi a: cxc:iuc:t is classiiicc i:to o:c oi two uutuallv cxclusivccatc-o:icsa succcss o: a iailu:c. :! t|c :o|a|ilitv oi a succcss is t|c sauc io: cac| t:ial. a:c :+ t|c t:ials a:c i:ccc:cc:tO I|c tcst wc will co:cuct s|o:tlv is a:o:iatc w|c: |ot| -x a:c -:1 xa:c at lcast ..O W|c: t|c a|ovc co:citio:s a:c uct. t|c :o:ual cist:i|utio: ca: |c uscc as a: a:oxiuatio: to t|c |i:ouial cist:i|utio:.1est 3tatistio for 1estine a 3inele Population Proportionnpz 1 ( x xx

=$ample proportionypothesized population proportion$ample size.1est 3tatistio for 1estine a 3inele Population Proportion - Lxampleuosc :io: clcctio:s i: a cc:tai: statc i:cicatcc it is :cccssa:v io: a ca:cicatc io: -ovc::o: to :cccivc at lcast 0 c:cc:t oi t|c votc i: t|c :o:t|c:: scctio: oi t|c statc to |c clcctcc. I|c i:cuu|c:t -ovc::o: is i:tc:cstcc i: asscssi:- |is c|a:ccs oi :ctu::i:- to oiiicc a:c la:s to co:cuct a su:vcv oi !.000 :c-istc:cc votc:s i: t|c :o:t|c:: scctio: oi t|c statc. usi:- t|c |vot|csistcsti:- :occcu:c. asscss t|c -ovc::o:`s c|a:ccs oi :cclcctio:..91est 3tatistio for 1estine a 3inele Population Proportion - Lxamplestep 1: state the nuii hypothesis and the aiternate hypothesis.l0. x > .0l1. x .0note: keyvord in the probiem o: leos:jstep 2: seiect the ievei o signiicance. = 0.01 as stated in the probiemstep 1: seiect the test statistic.usc Zcist:i|utio: si:cc t|c assuutio:s a:c ucta:c -xa:c -:1x > .601estine for a Population Proportion - LxampleStep 5: ake a decision and interpret the resuIt.The computed value of(2.8) is in the rejection region, so the null hypothesis is rejected at the .5 level. The difference of 2.5 percentage points between the sample percent (77.5 percent) and the hypothesized population percent (8) is statistically significant. The evidence at this point does not support the claim that the incumbent governor will return to the governor's mansion for another four years.step +: Iormuiate the decision ruie.Reject H0 i