Chapter 18 - Inference about Means

Transcription

1 Chapter 18 - Iferece about Mea December 1, 2014 I Chapter 16-17, we leared how to do cofidece iterval ad tet hypothei for proportio. I thi chapter we will do the ame for mea The Cetral Limit Theorem (Agai) The Cetral Limit Theorem (Chap. 15) gave u the amplig ditributio for mea a Normal with mea µ ad tadard deviatio σ. How do you fid the populatio tadard deviatio, σ? We will ue the ample tadard deviatio,, which i the etimate for σ to get the Stadard Error of SE( x) = Goet t What ort of ditributio ca be ued? The ditributio i o loger Normal ice the Stadard Error itroduce extra variatio from. William S. Goet foud the amplig model while workig at Guie Brewery i Dubli, Irelad. The model Goet foud i commoly referred to a the Studet t ditributio. Thi model i actually a family of related ditributio that deped o a parameter kow a degree of freedom (df = 1) A amplig ditributio model for mea, whe the coditio are met, the tadardized ample mea x µ t= where SE( x) = SE( x) follow a Studet t ditributio.

2 Correctig for the extra variatio caue thi model to give u a larger margi of error which will make our iterval wider ad our P-value will be larger tha from the Normal Model. Studet t-model are uimodal, ymmetric, ad bell haped like the Normal, but t-model with oly a few degree of freedom have fatter tail tha the Normal. A the degree of freedom icreae, the t-model look more like the Normal. The t-model with ifiite ( ) degree of freedom i the Normal. Aumptio ad Coditio Plauible Idepedece: A before thi i hard to check but you hould at leat thik if idepedece i reaoable. Radomizatio Coditio: The data come from a radom ample or radomized experimet. Thi help with idepedece. 10% Coditio: Whe ample i draw without replacemet, the ample hould be o more tha 10% of the populatio. Nearly Normal Coditio: The data come from a ditributio that i uimodal ad ymmetric. The maller the ample ize (<15 or o), the more cloely the data hould follow a Normal model. For moderate ample ize (i betwee 15 ad 40 or o), the t work well a log a the data are uimodal ad ymmetric. For larger ample ize, the t model are afe to ue eve if the data are kewed.

3 Oe Sample t-iterval for the Mea: x± t 1 Example (p. 499, #24) Parkig Hopig to lure more hopper dowtow, a city build a ew public parkig garage i the cetral buie ditrict. The city pla to pay for the tructure through parkig fee. Durig a two-moth period (44 weekday), daily fee collected averaged $126, with a tadard deviatio of $15. a) What aumptio mut you make i order to ue thee tatitic for iferece? b) Write a 90% cofidece iterval for the mea daily icome thi parkig garage will geerate. c) Iterpret thi cofidece iterval i cotext. d) Explai what 90% cofidece mea i thi cotext. e) The coultat who advied the city o thi project predicted that parkig reveue would average $130 per day. Baed o your cofidece iterval, do you thik the coultat wa correct? Why?

4 18.3 Iterpretig Cofidece Iterval See p Cofidece Iterval i for a mea ad ot a idividual value A Hypothei Tet for the Mea Oe Sample t-tet for the Mea: We tet the hypothei H 0 :µ = µ 0uig the tet tatitic x µ 0 t 1= / ad the Studet t model with -1 df. Example (p. 501, #38) Savig ga Cogre regulate corporate fuel ecoomy ad et a aual ga mileage for car. A compay with a large fleet of car hope to meet the 2011 goal of 30.2 mpg or better for their fleet of car. To ee if the goal i beig met, they check the gaolie uage for 50 compay trip choe at radom, fidig a mea of mpg ad a tadard deviatio of 4.83 mpg. I thi trog evidece that they have attaied their fuel ecoomy goal?

5 18.5 Chooig the Sample Size ME t 2 t 1 = 1 = ME * z have to ue = ME 2 firt. Roud up! Example (p. 497, #14) A Eglih profeor i attemptig to etimate the mea umber of ovel that the tudet body read durig their time i college. He i coductig a exit urvey with eior. He hope to have a margi of error of 3 book with 95% cofidece. From readig previou tudie, he expect a large tadard deviatio ad i goig aume it i 10. How may tudet hould he urvey?