Consult the following resources to familiarize yourself with the issues involved in conducting surveys:

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Wesley Shelton
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Cofdece Itervals Learg Objectves: After completo of ths module, the studet wll be able to costruct ad terpret cofdece tervals crtcally evaluate the outcomes of surveys terpret the marg of error the

1 Cofdece Itervals Learg Objectves: After completo of ths module, the studet wll be able to costruct ad terpret cofdece tervals crtcally evaluate the outcomes of surveys terpret the marg of error the cotext of omal cofdece level, sample sze, ad assumptos made to costruct a cofdece terval Kowledge ad Sklls Cofdece terval Marg of error Prerequstes Calculato of percetages Arthmetc averages Normal dstrbuto Estmatg the mea of a ormal dstrbuto Before You Come to Class The Jauary 8, 2009 New York Tmes publshed a poll o Amerca s expectato of the Presdet elect Barack Obama. The poll cluded the questo What s your best guess about the Uted States fve years from ow? A total of,2 radomly selected adults respoded to the survey, ad 6% of them felt that thgs wll be better. The New York Tmes states that [] theory, 9 cases out of 20, overall results based o such samples wll dffer by o more tha three percetage pots ether drecto from what would have bee obtaed by seekg to tervew all Amerca adults. Cosult the followg resources to famlarze yourself wth the ssues volved coductg surveys: Ctato: Neuhauser, C. Cofdece Itervals Created: October 4, 2009 Revsos: Copyrght: 2009 Neuhauser. Ths s a ope access artcle dstrbuted uder the terms of the Creatve Commos Attrbuto No Commercal Share Alke Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, ad allows others to traslate, make remxes, ad produce ew stores based o ths work, provded the orgal author ad source are credted ad the ew work wll carry the same lcese. Fudg: Ths work was partally supported by a HHMI Professors grat from the Howard Hughes Medcal Isttute. Page

2 Here s a calculator that calculates the sample sze eeded for a gve marg of error, cofdece level, ad populato sze: After readg the resources, you should be able to aswer the followg questos: What s a radom sample? How s the sample for a telephoe survey selected? What are types of errors that may occur whe coductg surveys? What role do the marg of error ad the level of cofdece play reportg surveys? Ctato: Neuhauser, C. Cofdece Itervals Created: October 4, 2009 Revsos: Copyrght: 2009 Neuhauser. Ths s a ope access artcle dstrbuted uder the terms of the Creatve Commos Attrbuto No Commercal Share Alke Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, ad allows others to traslate, make remxes, ad produce ew stores based o ths work, provded the orgal author ad source are credted ad the ew work wll carry the same lcese. Fudg: Ths work was partally supported by a HHMI Professors grat from the Howard Hughes Medcal Isttute. Page 2

3 Smulato O the web pages that are metoed above you ecoutered the cocept of level of cofdece ad marg of error. We wll frst vestgate these cocepts through smulatos. Whe the NYT coducted the survey to determe the percetage of people who feel that thgs wll be better fve years from ow, they dd ot kow the percetage otherwse they would ot have eeded to coduct the survey. It should also be clear that the NYT would ot be able to call up every adult the U.S. ad asked for ther opo. Istead, the NYT called up,2 U.S. adults ad extrapolated from the sample to the etre U.S. populato. They added the clam that [] theory, 9 cases out of 20, overall results based o such samples wll dffer by o more tha three percetage pots ether drecto from what would have bee obtaed by seekg to tervew all Amerca adults. Let s smulate surveys but let s frst preted we kow the actual proporto of adults the populato feel that thgs wll be better fve years. Suppose you kew that the percetage s p00% where p s a umber betwee 0 ad. For stace, f p s 0.2, the p00% would be (0.2)(00)%=20%. We set up a smulato that smulated samples of sze 00: For each value of p betwee 0 ad, cremeted by 0.05, Fgure shows the results of 000 smulatos of samples of sze 00. The red curves from top to bottom are the 97.5th, 50th, ad 2.5th percetles. We see Fgure that the spread of the smulated arthmetc averages s largest whe p= arthmetc average p Fgure : Smulatos of arthmetc averages of samples of sze 00 for dfferet values of p (see text for further explaatos). The red curves represet the 2.5%, 50%, ad 97.5% percetles. Ctato: Neuhauser, C. Cofdece Itervals Created: October 4, 2009 Revsos: Copyrght: 2009 Neuhauser. Ths s a ope access artcle dstrbuted uder the terms of the Creatve Commos Attrbuto No Commercal Share Alke Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, ad allows others to traslate, make remxes, ad produce ew stores based o ths work, provded the orgal author ad source are credted ad the ew work wll carry the same lcese. Fudg: Ths work was partally supported by a HHMI Professors grat from the Howard Hughes Medcal Isttute. Page 3

4 Whe we estmate a proporto p, we use the arthmetc average as our estmator for p. To capture the varato, we wll defe a terval, called the cofdece terval, so that wth hgh lkelhood the true value wll be cotaed the terval. More precsely, we choose a cofdece level α, whch s ofte 95% or 99%, so that f we repeated the survey multple tmes, oly a fracto α of the cofdece tervals would ot cota the true value. The radus of the cofdece terval (.e., half ts legth) s called the marg of error. Gog back to the NYT poll, we ca ow say that the estmated proporto of adults who felt that thgs wll be better fve years was 6% that poll. The cofdece level was 95% (.e., 9 out of 20 cases), ad the marg of error was 3%. Ths meas that 95 out of 00 surveys, the result would fall betwee 58% ad 64%. Theory We eed to troduce some otato to expla how ths works. Let s deote by ( x, x2,, x ) the outcome of a survey where dvduals where asked about a ssue, ad x = f the th perso s favor of that ssue, ad x = 0 otherwse. Assume that the proporto of dvduals the etre populato who are favor of the ssue s p. The umber of people favor the sample of sze s the obtaed by addg up the s,.e., x. = The arthmetc average x = = x s the proporto of dvduals favor of the ssue. Gve a radom sample ( X, X2,, X ) from ths populato, we calculate the sample mea X Whe the sample sze s large, oe ca show that X p p( p) / = = X. s approxmately ormally dstrbuted wth mea 0 ad varace. If we deote by z α /2 the ( α / 2) percetle of a ormal dstrbuto wth mea 0 ad varace, the By rearragg the rght had sde, we obta X p P z < < z p( p) / α α/2 α/2 th Ctato: Neuhauser, C. Cofdece Itervals Created: October 4, 2009 Revsos: Copyrght: 2009 Neuhauser. Ths s a ope access artcle dstrbuted uder the terms of the Creatve Commos Attrbuto No Commercal Share Alke Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, ad allows others to traslate, make remxes, ad produce ew stores based o ths work, provded the orgal author ad source are credted ad the ew work wll carry the same lcese. Fudg: Ths work was partally supported by a HHMI Professors grat from the Howard Hughes Medcal Isttute. Page 4

5 X p α P( z α/2 < < z α/2) p( p) / = P( z p( p)/ < X p< z p( p)/ ) α/2 α/2 = P( X z p( p)/ < p< X + z p( p)/ ) α/2 α/2 = PX ( z p( p)/ < p< X + z p( p)/ ) α/2 α/2 The terval [ X z α/2 p( p)/, X + z α/2 p( p)/ ] s called the cofdece terval at the cofdece level α. There s a problem wth computg the cofdece terval sce t volves the ukow value p, whch we wll address ow. Computg the Cofdece Iterval To compute the cofdece terval, we could choose that X s the estmator of p ad, accordg to the Law of Large Numbers, arrve at the followg expresso for the cofdece terval of a proporto X for the ukow value of p. The ratoal s X X z α/2 X( X) /, X + z α/2 X( X) / p as. We thus For a 95% cofdece terval, we set z 0.05/2=.96. For a 99% cofdece terval, we set z 0.0/2= I geeral, for a α level cofdece terval, we fd the value of z α /2 by lookg up Excel =NORMINV(probablty, 0,) where probablty s equal to α/2. Fgure 2 that explas ths: Ctato: Neuhauser, C. Cofdece Itervals Created: October 4, 2009 Revsos: Copyrght: 2009 Neuhauser. Ths s a ope access artcle dstrbuted uder the terms of the Creatve Commos Attrbuto No Commercal Share Alke Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, ad allows others to traslate, make remxes, ad produce ew stores based o ths work, provded the orgal author ad source are credted ad the ew work wll carry the same lcese. Fudg: Ths work was partally supported by a HHMI Professors grat from the Howard Hughes Medcal Isttute. Page 5

6 Fgure 2: Fdg z α /2. The area uder the curve betwee.96 ad.96 s equal to α = 0.05 = Sce we do t kow the value of p whe we coduct a survey, a coservatve way to compute cofdece tervals s to replace the expresso z α /2 p( p) / by ts largest value, whch happes whe p=/2. The 95% cofdece terval s the of the form ad the marg of error s.96 / 2. X.96.96, X For a 99% cofdece terval, replace.96 by (Why?) Homework. Show graphcally that p( p) s largest whe p = /2. I ths case, p( p) = (0.5)( 0.5) = / 4 = / 2. Usg ths result, expla why t follows that the marg of error at the 95% cofdece level s calculated.96 / At the 95% cofdece level, geerate a table that lsts the mmum sample sze requred whe the marg of error s %, 3%, 5%, ad 0%. Ctato: Neuhauser, C. Cofdece Itervals Created: October 4, 2009 Revsos: Copyrght: 2009 Neuhauser. Ths s a ope access artcle dstrbuted uder the terms of the Creatve Commos Attrbuto No Commercal Share Alke Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, ad allows others to traslate, make remxes, ad produce ew stores based o ths work, provded the orgal author ad source are credted ad the ew work wll carry the same lcese. Fudg: Ths work was partally supported by a HHMI Professors grat from the Howard Hughes Medcal Isttute. Page 6

7 3. At the 99% cofdece level, geerate a table that lsts the mmum sample sze requred whe the marg of error s %, 3%, 5%, ad 0%. 4. Whe estmatg a proporto, the followg quattes play a role: sample sze, cofdece level, ad marg of error. How does the marg of error deped o the sample sze? How does the marg of error deped o the cofdece level? Lst all gredets to costruct a cofdece terval. Ctato: Neuhauser, C. Cofdece Itervals Created: October 4, 2009 Revsos: Copyrght: 2009 Neuhauser. Ths s a ope access artcle dstrbuted uder the terms of the Creatve Commos Attrbuto No Commercal Share Alke Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, ad allows others to traslate, make remxes, ad produce ew stores based o ths work, provded the orgal author ad source are credted ad the ew work wll carry the same lcese. Fudg: Ths work was partally supported by a HHMI Professors grat from the Howard Hughes Medcal Isttute. Page 7

Gene Expression Data Analysis (II) statistical issues in spotted arrays

Gene Expression Data Analysis (II) statistical issues in spotted arrays STATC4 Sprg 005 Lecture Data ad fgures are from Wg Wog s computatoal bology course at Harvard Gee Expresso Data Aalyss (II) statstcal ssues spotted arrays Below shows part of a result fle from mage aalyss