Chapter 23 Summary Inferences about Means

Transcription

1 U i t 6 E x t e d i g I f e r e c e Chapter 23 Summary Iferece about Mea What have we leared? Statitical iferece for mea relie o the ame cocept a for proportio oly the mechaic ad the model have chaged. The reaoig of iferece, the eed to verify that the appropriate aumptio are met, ad the proper iterpretatio of cofidece iterval ad P-value all remai the ame regardle of whether we are ivetigatig mea or proportio. Gettig Started Now that we kow how to create cofidece iterval ad tet hypothee about proportio, it d be ice to be able to do the ame for mea. Jut a we did before, we will bae both our cofidece iterval ad our hypothei tet o the amplig ditributio model. The Cetral Limit Theorem told u that the amplig ditributio model for mea i Normal with mea μ ad tadard deviatio SD y All we eed i a radom ample of quatitative data. Ad the true populatio tadard deviatio, σ. o Well, that a problem Proportio have a lik betwee the proportio value ad the tadard deviatio of the ample proportio. Thi i ot the cae with mea kowig the ample mea tell u othig about SD( y) We ll do the bet we ca: etimate the populatio parameter σ with the ample tatitic. Our reultig tadard error i SE y We ow have extra variatio i our tadard error from, the ample tadard deviatio. o We eed to allow for the extra variatio o that it doe ot me up the margi of error ad P-value, epecially for a mall ample. Ad, the hape of the amplig model chage the model i o loger Normal. So, what i the amplig model? Goet t William S. Goet, a employee of the Guie Brewery i Dubli, Irelad, worked log ad hard to fid out what the amplig model wa. The amplig model that Goet foud ha bee kow a Studet t. The Studet t-model form a whole family of related ditributio that deped o a parameter kow a degree of freedom. We ofte deote degree of freedom a df, ad the model a t df. What Doe Thi Mea for Mea? A practical amplig ditributio model for mea o Whe the coditio are met, the tadardized ample mea follow a Studet t-model with 1 degree of freedom. We etimate the tadard error with SE y y t SE y AP Statitic Page

2 U i t 6 E x t e d i g I f e r e c e What Doe Thi Mea for Mea? (cot.) Whe Goet corrected the model for the extra ucertaity, the margi of error got bigger. o Your cofidece iterval will be jut a bit wider ad your P-value jut a bit larger tha they were with the Normal model. By uig the t-model, you ve compeated for the extra variability i preciely the right way. Studet t-model are uimodal, ymmetric, ad bell haped, jut like the Normal. But t-model with oly a few degree of freedom have much fatter tail tha the Normal. A the degree of freedom icreae, the t-model look more ad more like the Normal. I fact, the t-model with ifiite degree of freedom i exactly Normal. Fidig t-value By Had The Studet t-model i differet for each value of degree of freedom. Becaue of thi, Statitic book uually have oe table of t-model critical value for elected cofidece level. Alteratively, we could ue techology to fid t critical value for ay umber of degree of freedom ad ay cofidece level you eed. What techology could we ue? o The Appedix of ActivStat o the CD o Ay graphig calculator or tatitic program Aumptio ad Coditio Goet foud the t-model by imulatio. Year later, whe Sir Roald A. Fiher howed mathematically that Goet wa right, he eeded to make ome aumptio to make the proof work. We will ue thee aumptio whe workig with Studet t. Idepedece Aumptio: o Radomizatio Coditio: The data arie from a radom ample or uitably radomized experimet. Radomly ampled data (particularly from a SRS) are ideal. o 10% Coditio: Whe a ample i draw without replacemet, the ample hould be o more tha 10% of the populatio. AP Statitic Page

3 U i t 6 E x t e d i g I f e r e c e Aumptio ad Coditio (cot.) Normal Populatio Aumptio: o We ca ever be certai that the data are from a populatio that follow a Normal model, but we ca check the o Nearly Normal Coditio: The data come from a ditributio that i uimodal ad ymmetric. Check thi coditio by makig a hitogram or Normal probability plot. Nearly Normal Coditio: o The maller the ample ize ( < 15 or o), the more cloely the data hould follow a Normal model. o For moderate ample ize ( betwee 15 ad 40 or o), the t work well a log a the data are uimodal ad reaoably ymmetric. o For larger ample ize, the t method are afe to ue eve if the data are kewed. Oe-Sample t-iterval Whe the coditio are met, we are ready to fid the cofidece iterval for the populatio mea, μ. The cofidece iterval i y t 1 SE y where the tadard error of the mea i SE y t The critical value * deped o the particular cofidece level, C, that you pecify ad 1 o the umber of degree of freedom, 1, which we get from the ample ize. More Cautio About Iterpretig Cofidece Iterval Remember that iterpretatio of your cofidece iterval i key. What NOT to ay: o 90% of all the vehicle o Triphammer Road drive at a peed betwee 29.5 ad 32.5 mph. The cofidece iterval i about the mea ot the idividual value. o We are 90% cofidet that a radomly elected vehicle will have a peed betwee 29.5 ad 32.5 mph. Agai, the cofidece iterval i about the mea ot the idividual value. What NOT to ay: o The mea peed of the vehicle i 31.0 mph 90% of the time. The true mea doe ot vary it the cofidece iterval that would be differet had we gotte a differet ample. o 90% of all ample will have mea peed betwee 29.5 ad 32.5 mph. The iterval we calculate doe ot et a tadard for every other iterval it i o more (or le) likely to be correct tha ay other iterval. Make a Picture, Make a Picture, Make a Picture Picture tell u far more about our data et tha a lit of the data ever could. The oly reaoable way to check the Nearly Normal Coditio i with graph of the data. o Make a hitogram of the data ad verify that it ditributio i uimodal ad ymmetric with o outlier. o You may alo wat to make a Normal probability plot to ee that it reaoably traight. AP Statitic Page

4 U i t 6 E x t e d i g I f e r e c e Oe-Sample t-tet for the Mea The coditio for the oe-ample t-tet for the mea are the ame a for the oe-ample t-iterval. y We tet the hypothei H 0 : = 0 uig the tatitic t 0 1 SE y The tadard error of the ample mea i SE y Whe the coditio are met ad the ull hypothei i true, thi tatitic follow a Studet t model with 1 df. We ue that model to obtai a P-value. Sigificace ad Importace Remember that tatitically igificat doe ot mea actually importat or meaigful. o Becaue of thi, it alway a good idea whe we tet a hypothei to check the cofidece iterval ad thik about likely value for the mea. Iterval ad Tet Cofidece iterval ad hypothei tet are built from the ame calculatio. o I fact, they are complemetary way of lookig at the ame quetio. o The cofidece iterval cotai all the ull hypothei value we ca t reject with thee data. More preciely, a level C cofidece iterval cotai all of the poible ull hypothei value that would ot be rejected by a two-ided hypothei text at alpha level 1 C. o So a 95% cofidece iterval matche a 0.05 level tet for thee data. Cofidece iterval are aturally two-ided, o they match exactly with two-ided hypothei tet. o Whe the hypothei i oe ided, the correpodig alpha level i (1 C)/2. Sample Size To fid the ample ize eeded for a particular cofidece level with a particular margi of error (ME), olve thi equatio for : ME t 1 The problem with uig the equatio above i that we do t kow mot of the value. We ca overcome thi: o We ca ue from a mall pilot tudy. o We ca ue z* i place of the eceary t value. Sample ize calculatio are ever exact. o The margi of error you fid after collectig the data wo t match exactly the oe you ued to fid. The ample ize formula deped o quatitie you wo t have util you collect the data, but uig it i a importat firt tep. Before you collect data, it alway a good idea to kow whether the ample ize i large eough to give you a good chace of beig able to tell you what you wat to kow. AP Statitic Page

5 U i t 6 E x t e d i g I f e r e c e *The Sig Tet Back to Ye ad No We could tur our quatitative data ito a et of ye/o value (Beroulli trial). We ca tet a media by coutig the umber of value above ad below that value thi i called a ig tet. o The ig tet i a ditributio-free method, ice there are o ditributioal aumptio or coditio o the data. Becaue we o loger have quatitative data, we do t require the Nearly Normal Coditio. What Ca Go Wrog? Way to Not Be Normal: o Beware multimodality. The Nearly Normal Coditio clearly fail if a hitogram of the data ha two or more mode. o Beware kewed data. If the data are very kewed, try re-expreig the variable. o Set outlier aide but remember to report o thee outlier idividually. Ad of Coure: Watch out for bia we ca ever overcome the problem of a biaed ample. Make ure data are idepedet. o Check for radom amplig ad the 10% Coditio. Make ure that data are from a appropriately radomized ample. Iterpret your cofidece iterval correctly. o May tatemet that oud temptig are, i fact, miiterpretatio of a cofidece iterval for a mea. o A cofidece iterval i about the mea of the populatio, ot about the mea of ample, idividual i ample, or idividual i the populatio. AP Statitic Page