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Sierra College Math 13 Spring 2009 Class 2032 nstructur Juhn Burk Erma juhniburkem n s n cum Web Page nng Urnth swerracuHEge EduSta Juhn urke Telephune am 33741425 omee nbdrs yam MW 2 3575 any M 2 4573 45 Emma Tuday Seenbns dew872 Asswgnment 8721 3 5 3 11 13 15 17 19 23 25 27 29 any 31 37 39 Next Seebbns Era Chapter 8 Hypothesis Testing erView tRuIe n dnde a gwen assumptmn t reE en ne eyenus very smaH WE f x y r prubabmty uf a pamcular ubsewed bnemde thatthe assumptmn s prubab y a Se sdbbbse We 251 a dawn by analyzmg sample data m an anempttu dwstmgmsh between resunstnat ean beeur easny by enanee eyentrme and saythatthe underlymg assumpter s prubab y false Chapter 839 Hypothesis Testing w ample Gender cnmee PruCare ndus mes E ue buys F39mk gm Expe ment mu uup eswantmg a gm use Gender cnmee F39mk sdbbbse a 52 gm b 97 gm thhuut bemg mu funnaL vynat can We undude Chapter 8 Hypothesis Testing Overview Example Claim For couples using Gender Choice Pink the proportion of girls is p gt 05 Working assumption The proportion of giris is p 05 under normal circumstances aThe sample resulted in 52 girls among 100 births so the sample proportion is p 52100 052 Chapter 8 Hypothesis Testing Overview Chapter Problems In a nationwide survey of 703 randomly selected subjects w rking 61 said that theyfound theirjob 39ng Based on these results can a newspaper publish an article with the headline Most Workers Find Jobs Through Networkingquot ii 0 a 5 32 gm 3 In a nationwide survey of 880 randomly selected drivers 56 admitted that they run red lights Is there suf cient evidence to support t e c aim that a majority of all Americans run red lights Chapter 8 Hypothesis Testing Overview The two major activities of inferential statistics are the estimation ofpopulation parameters Chapter 7 and hypothesis testing introduced here De nition n statistics a hypothesis is a claim or statement about a property of a population A hypothesis test or test ofsigni cance is a standard procedure for testing a claim about a property ofa population Chapter 8 Hypothesis Testing Overview cont For example Medicine Medical researchers claim that the mean body temperature of healthy adults is not 986 F Aircraft Safety The FAA claims that the mean weight of an airline passenger with carry on baggage is greater than the 185 lb that it was 20 years ago Quality Control When new equipment is used to manufacture aircralt altimeters the new altimeters are better because the variation in errors is reduced so that the readings are more consistent In many industries overall quality can be improved by reducing variability Informal Hypothesis Testing Researchers want to show that the mean temperature of healthy adults is NOT 986 degrees They conduct an experiment and collect data with these characteristics n 106 E 9820 s 062 and the shape of the distribution is approximately normal lfwe assumeIJ 986 then by the Central Limit Theorem know that the sample means tend to be normally distributed with 11 I 986 and ox olW sW 006 If we want to determine the interval that should contain 99 ofall sample means we can nd that the interval is 9844 9876 Method of Reasoning for Hypothesis Testing If under a given assumption the probability of a palticular observed event is very small we concludet att e assumption is probably false we reject the assumption If under a given assumption the probability of a particular observed event is NOT very small then we do not have suf cient evidence to reject the assumption we fail to reject the assumption 82 Basics of Hypothesis testing The formal components used in hypothesis testing Null hypothesis Alternative hypothesis Test statistics Critical region Signi cance level Critical value Pvalue Decision CriteriaMethods Type amp Type II errors Null and Alternative Hypotheses The null hypothesis denoted by H0 is a statement that the value of a population parameter is equal to some claimed value The null hypothesis becomes ourtest assumption The alternative or alternate hypothesis denoted by H is the statement that the parameter has a value that somehow differs 39om the null hypothesis uses lt gt or It must be true ifthe null hypothesis is false Identifying Hu and H1 Identify the speci c claim or hypothesis to be tested and express it in symbolic form Give the symbolic form that must be true when e original claim is false Let H be the symbolic expression not containing equality ie uses lt gt or Let Hg the null hypothesis be the symbolic expression that the parameter equals the xed value being considered Test Statistic The test statistic is a value computed 39om the sample data It is used in making a decision about rejecting the null hypothesis The test statistic is found by converting the sample statistic such as sample proportion or sample mean to a score of relative standing such as 2 or t Test statistic for proportion Test statistic for m ean Critical Region Significance Level Critical Value The critical region is the set ofall values of the test statistic that cause us to reject the null hypothesis The signi cance level a is the probability that the test statistic will fall in the critical region when the null hypothesis is actually true A critical value is anyvalue that separates the critical region from the values ofthe test statistic that do not lead to rejection ofthe null hypothesis Significance Level Den oted by a The probability that the test statistic will fall in the critical region when the null hypothesis is actually true Common choices are 005 001 and 010 By far the most common in fact use as the default if the signi cance level is not gIven Critical Value Value or values that separate the critical region where we reject the null hypothesis from the values ofthe test statistic that do not lead to a rejection ofthe null hypothesis Reject IIn Fail to reject IIn Cr39tical Val e z score 51 5 TwotailedRighttailed Lefttailed Tests The tails in a distribution are the extreme regions bounded by critical values Pvalue The Pvalue is the probability ofgetting a value of the test statistic that is at least as extreme as the one representing the sample data assuming the null hypothesis is true The null hypothesis is rejected if the Pvalue is very small such as 005 or less Pvalues can be found by using the procedure summarized in Figure 86 and in the tear out Formulas and Tablesquot Finding PValues 39hme mmmmmm Famulasandhblesum A 7 t e eeem esteem eeeee cl vegetated DeCIsIon CriteriaMethods Traditional Method 9 t t H Fall to reject HE f the test StaUS39UE does not fall m CR Pvalue Method Reject HE tfthe Pevalue s a Fall to reject HE f the Pevalue gt 0 Confidence Intervals Reject a dam that the populater a a meterhas a value Hhe value 5 notmcmaea mme con dence mtewat We ll tgnore tms method Traditional or Classical Method of Testlng Hypotheses 1 ldem ytheSeemclanquotorhwuthestsmheteged mdvmthnsymhuhcmrm 2 Etmthesymhohcmrmtha We Detrue Mentheo gmal clawquot ts ase D hetwu symhohc epreseensemmee Satan 3 null hyvmhests Nuhetheonetha eernetnsthe mndmon m aquamy m We mha aemmt A Setemheegnmeem lemluhasedunthesamustasmatype laror Mate Small tithemnsmuencamr jemmgatmemiareseme vheveuemuuaeneumere 5 lden ythesatitctha tsrdevan otmstegandmsSamvhngdtgnhmmn s Datermmethetst lemme wheat valueS am the Meet r gmn Draws graph and memeemete mete Emmet valuEISlt and armed r gmn 7 R2123 mmeta meme mthe armed ragmn Pal to reign mm heist gem s m m the Meet ragmn a Regaleims prevmus deusmn m stmpls men techmcaHermsand eeeresme of gmal eta m 2 PValue Method of Testing Hypotheses 1 2 lee the symbolic form that must betrue when the original claim is ialse 339 contains the condition of equality H1 is the other statement 4 selecttne signi cant level oi based on the seriousness of a typel enor Nhke ix small very common 5 ldentlfy the statistic that is relevant to this test and its sampling distribution 5 ml to nvalue statistic and pavalue 7 RejectHa irtne pvalue is less than orequal to the signi cance level n Fail to rqectHa If the pvalue is greater than n a claim Conclusions in Hypothesis Testing Always test the null hypothesis either 1 Reject HEl 2 Fail to rejectHEl Formulate the correct wording of nal conclusion Wording of the Final Conclusion ms 5 my m with it WWW s we lri the book pg 397 and in the Formulas and Tables card Sierra College Math 13 Spring 2009 Class 2432 nstructur Juhn Burke Erma juhniburkem nds ng cum Web Page mtg Urnth sterraedHege EduSta Juhn urke Terephdne am 33741425 omee huurs mam MW 2 3575 nut M 2 4573 45 Emma Tuday Seetrdns 973374 Assrgnment 97 m r 3 5 7 9 1 17 e a swarm Next After break Humewurk 3 Due Revrew 9394 Inferences About Two Means Tvvu samples are independent lm sampre vames sereeted frum dne pupu atmn are nut re ated td ursumehuvv parred ur matched thh the samples sereeted fr he upulatmn r there rs same reratrdnsmp Sn that Each vame m we sampre rs parred thh a edrrespdndmg varde m the Either paired samples Examples Independent and Matched Independent sam ples One gruup err subjects rs treated thh the chu esteruleredumng drug prwtur ere a seednd and separate gruup err subjects rs gwen a praeebd Matched pairs wergms err subjects measured befure and a erthe dret treatment Each befure vame rs matched thh the after vame 7 same persun 93 Inferences About Two Means Independent Samples Assumptions The two samples are independent Both samples are simple random samples The two sample sizes are both large n gt 30 or both samples come 39om populations having normal distributions 93 Inferences About Two Means Independent Samples Test Statistic t ifiuiu1 Degrees of Freedom df smaller ofn1 1 and n2 1 Pvalues Referto table A3 and procedure in Fig 86 Critical values Referto table A3 93 Inferences About Two Means Independent Samples Example Bonds and McGwire Home Run Distances Data Set 17 includes the distances ofthe home runs hit in recordsetting seasons by Mark McGwire and Barry Bonds Assume that we have simple random samples from large populations Use a 05 Slgmflcance level and the tradltlonal method to test the clalm that the dlstances come from populatlons Wlth dlfferent means Solve usmg the Tlr83Plu5 2SarinTe5t and StatDlsk H ypothesls e5 Testng gt Mean ewe lndependent Sampl 93 Inferences About Two Means Independent Samples Example On average do female students in the class work fewer hours per weekthan male students Use a 00 7 263s1134 n1 65 gt72 284 s2 139 n2 67 Claim in lt uz 94 Inferences About Two Means Matched Pairs Ifthere is some relationship so that each value in one sample is paired with a corresponding value in the other sample the samples are dependent matched pairs or paired samples 94 Inferences About Two Means Matched Pairs Examples When conducting an experiment to test the effectiveness of a lowfat diet the weight ofeach subject is measured once before the diet and once a erthe diet The effectiveness of an SAT coaching program is tested by giving each subject an SAT test before the program and another equivalent SAT test a erthe program The accuracy of reported weights or heights is analyzed The reported weight height is recorded and the actual weight height is then measured 94 Inferences About Two Means Matched Pairs Assumptions The sample data consist of matched pairs The samples are simple random samples The number of matched pairs is large n gt 30 or the pairs ofvalues have differences that are from a population having a distribution that is approximately normal 94 Inferences About Two Means Matched Pairs Notation d individual difference between the two values in a matched paIr u mean value ofthe differences forthe entire population of matched pairs 07 mean value ofthe differences d for the paired sample data the mean of the xy values d s standard deviation ofthe differences d forthe paired sample data n numberofpairs ofdata M 94 Inferences About Two Means Matched Pairs Test Statistic E uz L J Degrees of Freedom df n 1 t Pvalues Referto table A3 and procedure in Fig 86 Critical values Referto table A3 94 Inferences About Two Means Matched Pairs Example The table below consists of ve actual low temperatures and the corresponding low temperatures that were predicted ve da s earlier Actua1 1qu 1 75 75 urecast 1B 1E 2D 22 15 D1fference V15 721 725 1 76 e a o 05 S1gn1f1cance 1eve1 and the trad1t1ona1 method to test the darn that there 15 a d1fference betweent e actua1 ow erhperatures and e 10W temperatures thatvvere forecast ve days earher So1ve usmg the Trsspms TTest Oh the hst contammg the d1fference or ts StatD1Sk on the 011911751 W0 15 13 94 Inferences About Two Means Matched Pairs Example Page 492 16 Selfreported heights and measured heights were obtained for males aged 1216 See table below Is there suf cient evidence to support the claim that there is no difference between selfreported heights and measured heights of males aged 1216 Use a 005 signi cance level Repuneu He1ght Measured He1ght 1 112E40533520 Sierra College Math 13 Spring 2009 Class 2232 Thstruetdr udhh Burk Ermah juhniburkemmdsprmg edrh Web Page httg rhath sterraedhege EduSta Juhn urke TeTephdhe am 33741425 omee hdurs yam MW 2 3575 any M 2 4573 45 Emma Tdday Seetrdhs 874875 Assrghrheht 8741 3 5 9 8751 3 5 7 911 13 17 21 23 Next Seetrdhs 971372 84 Testing a Claim About a Mean o Known Assumptions The sarhpTe rs a srrhpre rahddrh sarhpTe 2 TheyaTue gr the pupu atmn stahdard deyratrdh a rs Wmer 7 ur we haye a gddd apprdxrrhatrdh 3 Ether ur bum drthese edhdrtrdhs rs saus ed The pupu atmn rs hurrhahy drsthbuted Dr h gt an Because thrs rs the standard l e sarh Test Statistic z FurPevalues ahd Crmcal yaTues use the standard H rmal drsthbutrdh Tame A73 and refertu Frgure are furF39eValues eaTeuTatrdhs 2 84 Testing a Claim About a Mean o Known calculate ewe shuuld be ame tn calculate u HuWEver WE may be ametd guess ah upperhrhrt uh a m whreh ease we eah use these rhethdds as rrwe knEW a r e heaTthy aduTts We may hut knDW what a s but nstmctwely we knDW rt must be has than 2 degrees and prdbamy Tessthah 1 degree Using the TI83 Plus for testing a claim about a mean aknown 1 Traditional method ZTest 2 Maine method ZTest Most common Using StatDisk for testing a claim about a mean aknown Traditional method or Pvalue method Analysis gt Hypothesis Testing gt Mean 7 One Sample m 85 Testing a Claim About a Mean 039 Unknown Assumptions i Thesambleisasimblerandomsambe The value of the population standard deviation 0 is unknown 0 Either or both otthese conditions is satistied The population is normally distributed or n gt 30 Test Statistic Z I For Pavalues and Critical Values use the studentt distribution Table A3 With df rial and refer to Figure 876 for Pavalue procedures a Important Properties of the Student t Distribution Review The Studentt drstnhutrdn is different tdr different n The Studentt drstnhutrdn hasthe sarne generai heii shape as the standard ndrrnai distributiun its Widershape re ects the reatervariabihty when s is used tn estimate 5 The Studentt drstnhutrdn has a rnean utt u The standard deviatiun utthe Studentt distributiun varies With the sarnpie SiZE and is greater than i the standard as d i nurrnai distributiun h sthe sarnpie size gets iarger the Studentt distributidn gets eidsertd the standard ndrrnai distributidn i U g the TI83 Plus for testing a claim about ean aunknawn 1 Traditional method rTTest 2 Pvalue method mestr Most common Using StatDisk for testing a claim about a mean aunknawn Traditional method or Pvalue method Anaiysrs Hyputhesis Testing a Mean 7 One Sarnpie m and Group Project Group Project up to 50 points per group member Groups of between three and ve people Groups to be chosen by Wednesday 325 Topic to be chosen and approved present a 12 paragraph plan by41 Presentations given in class Monday 54 Statistics Group Project ThehuThese enhe hTeTeciTsie ehhahce ahe Expand veuT EXPEHEHEE with the ahhTTcaiTeh eTT s1a1TsTcaT methuds veu Thu TeTTha cTasseTeTuTl tn cuiiahuvalewith ehihTshTeTeci The eTeup shuuid cehsTs1 enhTee in me s1udents ATihuueh thewuvk submitted andthE EvadEVEEENEdWiH he as a EYDUPT each s1u EM thT he accuuntahie TeT aspects uHhE hTeTeci The amp Ts TeshehsThTe TeTseTeciTTTeiheiuthie Thves1Teaie The TecuseT the hTeTeci cah he athhecTTThieTes1 n he eTeup GeciTeh T54 cemaThsmaw sueeeshehs h eth Tes1TTciTeTh isthatthis hTeTTeci Thus1 ahth ehe mihes1aiTs1TcaTahaTvsTshmceeuTesceveTeeT cTass The deiivevahieswiimciu ea Dmim E eTaT PYESEMaliEm am a WTTneh Tehen 0er mesmmnn AH eTeuh ThemheTs Thus1 he invuivEd The PVESEMatiDn shuuid Tthuee anpthTaiechansahueTah 5 e cemhmeTThTeTeciTeh TacTTTiTes mav he used Wmlm Rqnn itshuuidinciude sth1 meaia cuiiected sDescTTmTeh enhe melhud cTT ahaTvsTs rReieva ieTahhsahe s1aiTs1Tcs rStatemem eTcethusTeh sAhaTvsTs ahe se rcmiuue cTT s1udv ahe methuds The hTeTeciTehen Thus1haveamTe page nhai inciudES ah aiphaheticai iis1uHhE eTeup Them eTs ah e DYEamZEd mm mm seciTehs iaheied T immductium 2 Meihees 3 ResuTisT am a cethusTehs The vs1 SECliEIn imvuductium eTvesseme hacheTeuhe am a hTTeTsuThThaTv cTTwhanhe PYDiEEtiS ahuut The hTTTmrv uueshehihanhTs TuT T ndstu aeeTess sheuh he cTeaTTv s1ated eme Te mthE Wmducti n The SEEDnd SEEtiun Meihees dEscnbESm sumcient detaii hmvthe data has cuiiected ahe ucessu eeThahaTvahen inciudeanv iimtaliuns eT assumhiTehs maeeTeT thE hTeceeuTe aPPiiEd Th this hTeTeci The thin SEEHDVVT ResuTisT cumamsthe essehiTaT EiEmEMS ahe suhpumhe ThTeTTThaiTeTh YEiEan u tahies eT EYaPhST ahe WDYK EEnEYatEd asthe Team u h25131i51icai hTeceeuTe TheT nh Ectiun cethusTehs summaTTzesihe vesuits ahe anSWEVSt e shecmc hues1TeTh pTehesee Th the Wmducti n The hTeTeci Tehen shuuid he cTeaT cumpiete ahe cethse Sierra College Math 13 Spring 2009 Class 532 nstructur Juhn Burk Ermah juhniburkemmdsp ng edm Web Page httg math stehaedhege EduSta Juhn urke Tetephdhe am 33741425 Of ce huurs mam MW 2 3575 m M 2 4573 45 ametat Tuday Seetmhs 274 371372 Asstghmeht 2741 5 7 9 MAR 15 3721 3 5 7 9 Nuts 17 Next Seetmhs 373374 24 Statistical Graphics The mam abjectwe m usmg graphteat representatmns uf data ts td better understand the data thrddgh rDESEHpUDn eehteh shape ete rEXplu ng retatwe strengths and e Campath data frum dtttereht pupulatmns Frequency Polygon A frequency polygon use hhe segments euhheeted td pumts ucated dhem above class mtdgomz Vaues The etg ts utthe pmnts EDrrESpDndtD the etass frequenmes and the hhe segments are and end an the HEIHZEIHtal axts mass mtdpetms asthma We Emma Ogive Ah ogivets a the graph that deptcts the cumutatwe frequenmes just as the cumutatwe frequency mathhuttch hs ts cumutatwe frequenm t z a e hs m mew mm m V Dotplot A dotplot cchststs at a graph m Whmh each data vame S ptcttea as pmnturdut atchg a scate ct vames Dcts rEprESEntmg equat vames are stacked iv t a a t mu 113 m warvy Mammy w StemandLeaf Plot temandIear plot represent data by Separatmg ea h vatue mm Wu parts the stem such asthe E mustdthSD arm the Eafsuh as the hghtmest mgtt The siemrandJEafp ut shew the dtsmbutmn butmamtams 3 the thturrhattuh W the ungthat hst Raw Data Test Grades 57 72 a5 75 as as aa so as 100 Pa reto Cha rt A Pareto chart is a bar graph quot39quot quot for gualitative data with the 39quotm39quot bars arranged in order mm according to frequencies As in Jluuuu histograms vertical scales in mum Pareto charts can represent mm frequencies or relative frequencies 15mm mmquot 31qu 39 ll Halal Vehicle Fnimn iawninq Accidental Deaths by Type Pie Chart A pie chart is a graph depicting gualitative data as slices of a pie The Pareto chart does a better job representing the relative sizes of data however it can easily be altered to give a deceptive view Funnm use 19 hnlsllm a loan or ohjnl 290m 3 me me am 16 Molar MI 3500 slam Drownl39lg MM M al Parson mun 15s Fals U 2201 15 2 Scatter Diagrams A scatter diagram is a plot of paired xy data with a horizontal xaxis and a vertical yaxis The data are paired in a way that matches each value from one data set with a corresponding value from a second data set The pattern of the plotted points is helpful in determining whetherthere is some relationship between the two variables We ll use scatter plots like this in chapter 10 NICOTINE 9 TimeSeries Graph Atimeseries graph 5 usedtu represent data vames cuHemed uvemme Numbev u scveens m muwe a We hea evs Variation on the pie chart developed by Florence 39 Deaths m Evmsh Mmmy Huspna s Dunng Me M 3v Cnmean Way A representatmn uf six dy erenz manes re evanttu the march uf Napmeun s armytu Muscuvv and back m 181 24 81 3 deve uped m mm by ChanesJeseph Mmard 32 Measures of Center Dehmtmh A measure of center 5 a vame at the EEntEh ur rmdd e uf a data set We Wm euhswder the fEIHEIng measures uf center the4 M s Mean Memah Made derange Mean The arithmetic mean must mean ur average uf a setufva ues sthe measure uf centerfuund by addmg the vames and dmdmg by the tuta numberuf vames WW w m r W m m mm CaHed my Nate The mean 5 pamcmar y sensmve tn anthers Mean Example 34E 3E0 E44 E72 Mean M4 5 USE Mea e 384 Nutmethe ewem uHhe uuthev RoundOff Rule In general carry one more decimal place than is present in the original set ofvalues Round only the nal answer not intermediate values For example the mean of2 3 and 5 is 33333333 which we round to 33 Mean Example mlez 1 oweny Keyhnzm mm Ramus Mean44 Meanl 7 mm 2 z Dilan Keyhnzm mm Ramus Median he median ofa data set is the measure ofcenter that is the middle value when the original data values are arranged in order of increasing or decreasing magnitude The median is sometimes denoted as 2 pronounced xtildequot To nd the median rst sort the values lfthe number of values is odd the median is the number located in the exact middle ofthe list lfthe number ofvalues is even the median is found by computing the mean ofthe two middle numbers Note The median is largely unaffected by outliers Median Example 3 46 3 60 6 44 6 72 No exact rn ddle 7 even numberofvalues Median 3 60 6 4412 5 020 Mean EX4 5 055 Exact middle 7 rnedlan 6 440 Mean Zx5 9 384 Notice the effect oftne outlier Mode The mode of a set of data olten denoted M is the value that occurs most frequently When two values occur with the same greatest frequency each is a mode and the data set is bimoda When more than two values occurwith the same greatest 39equency the data set is said to be multimodal When no value is repeated there is no mode Mode is the only measure of central tendency that can be used with nominal data m Mode Example 5553151435 ModeisS 1 2 2234 5 6 667 9 6 BimodaI 2and6 12 36789 Nomode