Chapter 7. How do we actually estimate the parameter, though?

Transcription

1 Key Idea Cofidece Iterval, Cofidece evel Poit timate, Margi of rror, Critical Value, Stadard rror t ditributio, Chi-Square ditributio Chapter 7 Sectio 7-: Overview All of the material i Chapter 4-6 form a foudatio of what i called iferetial tatitic. We already dealt with iferetial tatitic i Chapter 0 i a regreio ettig. Now, we will explore etimatio. Here i a outlie of the mai idea of iferetial tatitic:. There i a populatio of iteret (i.e. the group we wat to kow omethig about). We draw a radom ample of ize from the populatio. 3. We compute tatitic from the ample, 4. We ue thee tatitic to etimate imilar parameter i the populatio. For example, uppoe we wat to kow what percetage of Deio tudet ow a red car. To etimate thi, we take a radom ample of 00 tudet ad fid out what percetage of thoe 00 tudet ow red car. The, we ay that the ample percetage hould to cloe to the actual percetage of all Deio tudet who ow red car. I thi example: Populatio of Iteret: All Deio Studet Sample Size: = 00 Statitic from the Sample: The percetage of tudet ampled who ow red car Parameter i the Populatio: The percetage of all Deio tudet who ow red car How do we actually etimate the parameter, though? Geeral timatio Framework Suppoe we wat to etimate a parameter (e.g. populatio proportio, populatio average, etc.). The firt thig to otice i that it would be impoible to exactly pipoit the value with 00% accuracy without amplig every igle member of the populatio, ice there would alway be ome ucertaity. A a reult, the bet we ca do i make a gue at the true value, ad the iclude a margi of error baed o a certai level of cofidece we have i our reult. The etimate ad the margi of error form omethig called a cofidece iterval. A cofidece iterval i made of differet part.. The poit etimate i the ample tatitic (thi i our bet gue at the true parameter value give our ample).. The margi of error i added ad ubtracted from the poit etimate to make the iterval. It ca alo be ubdivided ito two part: a. A critical value from a ditributio (more to come o thi later) b. The tadard error of the poit etimate (more to come o thi a well) The cofidece iterval (CI) ha thi form: CI = (Poit timate) ± (Margi of rror) = (Poit timate) ± (Critical Value)*(Stadard rror) Of coure there i o guaratee that the true populatio parameter will be i thi iterval, o we have to make ome ort of tatemet about the chace that thi will be true. The cofidece level i the probability that the iterval actually cover the true populatio parameter. Ofte, the cofidece level i deoted ( ), where i the chace that it doe ot cover the true parameter. For example, if = 0.05, the the cofidece level i 0.95, or 95%. Thu we would ay that we are 95% cofidet that the iterval cover the parameter. Iterpretig Cofidece Iterval et coider the example from before, where we wat to etimate the percetage of Deio tudet who ow a red car. Suppoe that of our ample of ize 00, 5 tudet owed a car. Thi give a poit etimate of 5%, or 0.5 for the populatio parameter. Suppoe alo that we calculated a critical value of.645 ad a tadard error of , with a cofidece level of 95%. I thi cae, the cofidece iterval will be: CI = (Poit timate) ± (Margi of rror) = (Poit timate) ± (Critical Value)*(Stadard rror) = 0.5 ±.645* = 0.5 ±

2 Thi give the iterval (0.093, 0.087). To iterpret thi iterval, ay of the followig tatemet are equivalet:. We are 95% cofidet that the true percetage of all Deio tudet who ow red car i betwee 9.3% ad 0.87%.. If we repeatedly took differet ample of ize 00 ad computed a CI for each of thoe ample, 95% of the computed iterval would cover the true percetage of all Deio tudet who ow red car. 3. There i a 95% chace that the iterval (0.093, 0.087) cover the true percetage of all Deio tudet who ow red car. A Geeral Note Depedig o the parameter you wat to etimate, formula for the poit etimate, critical value, ad tadard error will chage. However, the format of a cofidece iterval i alway the ame. Sectio 7-: timatig a Populatio Proportio To etimate a populatio proportio, we will be uig Normal Approximatio (ee Sectio 6-5 ad 6-6 for more detail). Becaue of thi, there are 3 coditio that mut be atified for u to etimate the populatio proportio.. The ample mut be a imple radom ample.. The Biomial coditio are atified a. Fixed Number of Trial (thi i the ample ize ) b. Idepedet Trial (thi follow from the imple radom ample requiremet) c. There are two type of outcome ucce ad failure d. The probability of ucce/failure i the ame i each trial (thi hould be true if the populatio i large) 3. There are at leat 5 uccee ad 5 failure (thi guaratee ome theoretical aumptio that we hould t get ito) Some Notatio: p = the populatio proportio (ukow quatity that we wat to etimate) x pˆ = = the ample proportio of x uccee i a ample of ize qˆ = pˆ = the ample proportio of failure i a ample of ize = the z-core with a area above of Commo Value of Cofidece evel : (See figure o the right) 90% % % Computig a Cofidece Iterval for The Populatio Proportio For a ample ize of, cofidece level, ad ample proportio pˆ, the part of the cofidece iterval are: Poit timate: pˆ Critical Value: Stadard rror: pˆ qˆ So the cofidece iterval i: CI = (Poit timate) ± (Margi of rror) = (Poit timate) ± (Critical Value) (Stadard rror) = pˆ ±

3 xample gieer at a bottle-makig factory are itereted i etimatig the percetage of defective bottle maufactured by the facility i geeral. To do thi, they take a imple radom ample of 00 bottle ad fid that 8 of them are defective. Fid a 95% cofidece iterval for the percetage of defective bottle maufactured by the factory ad iterpret it. Solutio: Firt, we ote a few thig: The 3 coditio above are met x 8 The ample proportio i p ˆ = = = The cofidece level i 95% = 0.95 =, o = 0.05 The critical value i = = = (0.08)(0.9) The tadard error i = = = Therefore, the cofidece iterval i: pˆ ± 0.08 ± (.96)(0.07) 0.08 ± (0.068, 0.33) Therefore, we ca ay that we are 95% cofidet that the percetage of defective bottle maufactured i thi facility i betwee.68% ad 3.3%. Determiig Sample Size Clearly, the ize of the ample ha a impact o the cofidece iterval, ice it affect the margi of error ad the poit etimate. The quetio i what ample ize hould be choe at the tart of the tudy to give a ice cofidece iterval. Ofte, what tatiticia will do i decide they wat a mall margi of error, the fid a ample ize that give them that margi of error. xample I the example above, for pˆ = ad = 0.05 (i.e. cofidece level of 95%), fid the ample ize required for the margi of error to be le tha 0.0. Solutio: We wat the margi of error to be 0.0. Alo, we kow the margi of error i, becaue thi i the amout added ad ubtracted to the poit etimate. Therefore, figurig out i a eay a olvig the equatio = 0. 0 for! (0.08)(0.9) = = 0.0 = 0.00 = = Thu we would eed a ample ize of at leat 708 to get a margi of error that mall. Fidig A Geeral Formula for Sample Size I the geeral cae, we ca olve for a follow (here, i the deired ize of the margi of error): = = = ˆ ˆ = pq If the value of pˆ i kow, the formula i a tated above. If the value of pˆ i ukow, the ue p ˆ = = Sectio 7-3: timatig a Populatio Mea Kow To etimate a populatio mea uig the method i thi ectio, 3 coditio mut be met:. The ample i a imple radom ample. The populatio tadard deviatio i kow. 3. ither the populatio i ormally ditributed or > 30 (o we ca ue the CT) A you may have already gueed, the bet poit etimate of the populatio mea i the ample mea, x.

4 Alo, from the CT, we alo kow that x i either exactly or approximately ormally ditributed (depedig o which cae i true i the 3 rd coditio) with a mea of µ ad a tadard deviatio of For thi reao, we have the followig.. Computig a Cofidece Iterval for The Populatio Mea ( Kow) For a ample ize of, cofidece level, ad ample mea x, the part of the cofidece iterval are: Poit timate: x Critical Value: Stadard rror: So the cofidece iterval i: CI = (Poit timate) ± (Margi of rror) = (Poit timate) ± (Critical Value) (Stadard rror) = x ± xample Suppoe you wat to etimate the average price of a home i Columbu, OH. From pat iformatio, you kow that the tadard deviatio i houig price i = $00,000. You take a imple radom ample of ize = 50 ad compute a ample mea of x = $46,000. Fid a 95% cofidece iterval for the populatio mea. Solutio: Firt, we ote a few thig: The 3 coditio above are met (if you thik about it, X i ot ormally ditributed it i kewed, but > 30) The poit etimate i x = 46, 000 The cofidece level i 95% = 0.95 =, o = 0.05 The critical value i = = = ,000 The tadard error i = = Therefore, the cofidece iterval i: x ± 46,000 ± (.96)(44.4) 46,000 ± 7,78.59 (8,8.4, 73,78.59) So with 95% cofidece, we claim that the average price of a home i omewhere betwee $8,8 ad $73,78, approximately. Determiig Sample Size Now uppoe that your upervior ak you to chooe a ample large eough to get a margi of error of $0,000 or le. How do we determie the ample ize? A i the ituatio of etimatig the populatio proportio, we jut et the margi of error equal to 0,000: 00,000 00,000 = 0, = 0, = 0,000 = 9. 6 = Therefore, you eed to take a ample of at leat 385 home. Fidig A Geeral Formula for Sample Size I the geeral cae, we ca olve for a follow (here, i the deired ize of the margi of error): = = = = = Notice that thi i imilar to the formula for ample ize whe etimatig the populatio proportio.

5 We jut replace pˆ qˆ with. Note: Ofte i practice, i ukow. The commo olutio for ample ize calculatio i to tick i ome ort of etimate for. Rage Thi could be baed o a pilot tudy, or uig the Rage Rule-of-Thumb: (See p. 344 for more detail). 4 Sectio 7-4: timatig a Populatio Mea Ukow I the previou ectio, we etimated the populatio mea by uig the fact that X wa ormally ditributed with a mea of µ = µ ad a tadard deviatio of =. X X For thi reao, had a tadard ormal ditributio, where: X µ X X µ = = X Now, however, we o loger kow. How do we get aroud that? It eem a though the mot eible approach would be to jut etimate with the ample tadard deviatio. The quetio, however, i if T ha a tadard ormal ditributio, where: X µ T = Ufortuately, T doe ot have a tadard ormal ditributio, becaue we have to accout for the fact that will ot be exactly equal to the true populatio tadard deviatio. However, through tatitical theory it tur out that the ditributio of T ca be pecified. The ditributio of T i called the Studet-t ditributio, or jut the t ditributio. Recall: The reao we divided by itead of i the ample tadard deviatio formula = i= ( x i x) wa becaue of omethig called degree of freedom. I thi cae, there were oly degree of freedom becaue we already kew the ample mea x (ee Sectio 3-3 for thi dicuio). It tur out that the t ditributio chage hape depedig o the degree of freedom of. For thi reao, i fidig critical value you have to kow the degree of freedom, which i alway. A Note About the t ditributio The t ditributio i alo ymmetric, like the tadard ormal ditributio, ad ha a imilar hape. However, it i a bit arrower ad taper off le quickly to the right ad left (ee picture o the left). I computig the cofidece iterval, we will eed to ue the value t. Thi i exactly like the value from before, except for the t ditributio (ee picture o the right). To fid the value of t, we ue Table A-3 (lat page i the textbook, or p. 774). Chooe the row with Degree of Freedom = ad the colum with Area i Oe Tail = (or equivaletly, Area i Two Tail = ). The critical value i lited at the iterectio of that row ad colum. (Note: If the degree of freedom doe ot appear i the firt colum, jut chooe the cloet value i the table) timatig the Populatio Mea with Ukow

6 To etimate a populatio mea i thi ew ituatio, coditio mut be met:. The ample i a imple radom ample. ither the populatio i ormally ditributed or > 30 (o we ca ue the CT) (Notice that thee are coditio ad 3 from the cae with kow. The oly coditio that i t repeated i the coditio that i kow, which i obviouly fale i thi cae.) A before, the bet poit etimate of the populatio mea i the ample mea, x. Alo, from the dicuio above, we kow that we will be uig a t ditributio with degree of freedom. Computig a Cofidece Iterval for The Populatio Mea ( Kow) For a ample ize of, cofidece level, ad ample mea x, the part of the cofidece iterval are: Poit timate: x Critical Value: t (with degree of freedom) Stadard rror: So the cofidece iterval i: CI = (Poit timate) ± (Margi of rror) = (Poit timate) ± (Critical Value) (Stadard rror) = x ± t xample For a cla project, ome tudet are doig a urvey to fid out the average umber of hour a Deio tudet ped tudyig each week. They take a imple radom ample of 9 Deio tudet ad compute a ample mea of 5.4 hour per week with a tadard deviatio of. hour. Fid a 95% cofidece iterval for the populatio mea. Solutio: Firt, we ote a few thig: The coditio above are met ( > 30) The poit etimate i x = 5. 4 The cofidece level i 95% = 0.95 =, o = 0.05 The degree of freedom are = 90 The critical value i t = t =. 987 (Area i oe tail i 0.05, or area i two tail i 0.05) The tadard error i = = Therefore, the cofidece iterval i: x ± t 5.4 ± (.987)(0.6) 5.4 ± 0.50 (5.5, 5.65) So with 95% cofidece, we claim that the average hour per week tudied by Deio tudet i betwee 5.5 ad 5.65 hour. Determiig Sample Size Now uppoe that the tudet wat to chooe a ample large eough to get a margi of error of 0.0 or le. How do we determie the ample ize? A i the other time we etimated ample ize, we jut et the margi of error equal to 0.0:.. t = = = = = Therefore, they would eed to take a ample of at leat 569 Deio tudet. Fidig A Geeral Formula for Sample Size

7 I the geeral cae, we ca olve for a follow (here, i the deired ize of the margi of error): t t = = = = = t t t Notice agai that thi i imilar to the formula for ample ize whe etimatig the populatio proportio ad mea with kow. Chooig Betwee ad t Jut to ummarize the differece betwee whe you ue ad t, here i a ueful table. If i kow, ad: If i ukow, ad: If: The populatio i ormally ditributed ad/or 30 The populatio i ormally ditributed ad/or 30 The populatio i ot ormally ditributed ad < 30 The ue The ue t You have to ue other method ffect of Sample Size ad Cofidece evel o Cofidece Iterval Notice that i all three etimate we dicued (proportio ad mea with kow ad ukow), two thig were alway true:. I the margi of error, appear i the deomiator.. The larger get, the maller the critical value get. Thi mea that if i icreaed, the margi of error will get maller. Alo, if the cofidece level i decreaed (i.e. i icreaed), the margi of error will get maller. A maller margi of error mea a tighter cofidece iterval, which mea we are ayig the rage of value for the true populatio proportio/mea i much maller. I geeral, it i a good thig to have a maller cofidece iterval, a log a your degree of cofidece i high. Thi itroduce the idea of a trade-off i the ize of a cofidece iterval: For a fixed ample ize, to reduce the ize of a cofidece iterval, we mut reduce the cofidece level. For a fixed cofidece level, to reduce the ize of a cofidece iterval, we mut icreae the ample ize. Thu to get better cofidece iterval, you mut either reduce your level of cofidece or gather more data. Oe More Note Suppoe you are give a cofidece iterval, but ot the poit etimate or margi of error. You ca fid thee value uig jut the iterval. The poit etimate i at the ceter of the iterval, ad the margi of error i half the width of the iterval. I other word, for a cofidece iterval (a, b): b + a b a pˆ = ad (Margi of rror) = Sectio 7-5: timatig a Populatio Variace To come up with a cofidece iterval for, we have to ue yet aother differet ditributio. ( ) The quatity χ = ha what i called a Chi-Square Ditributio. (Chi i proouced kigh, like high with a k) The Chi-Square Ditributio i quite a bit differet tha the ormal or the t ditributio, becaue it i ot ymmetric. Thi ditributio alo deped o degree of freedom, ad the larger the degree of freedom are, the more ymmetric it get (ee picture below left). Similar to the ormal ad t ditributio, the critical value for the Chi-Square Ditributio are the two value that give the top ad bottom area of (ee picture below right). However, otice that thee value are o loger oppoite of each other, ice the ditributio i ot aymmetric. Therefore, they are called χ ad χ R, ad mut be foud eparately.

8 The critical value χ ad χ R ca be determied with Table A-4 (p.775 i the textbook). To fid thee i the table, firt fid the degree of freedom i the left colum. The fid the appropriate area to the right of the value (thi i for χ R ad for χ ) alog the top row. The value i at the iterectio of the degree of freedom ad area row/colum. xample: et degree of freedom be 0 ad = 0.05 (95% cofidece). The the area to the right of χ R i 0.05/ = Uig the R table, we get χ = The area to the right of χ i 0.05 = From the table, the, χ = timatig the Populatio Variace To etimate a populatio variace, coditio mut be met (otice epecially requiremet ):. The ample i a imple radom ample. The populatio mut be ormally ditributed. Computig a Cofidece Iterval for The Populatio Variace For a ample ize of, cofidece level, ad ample tadard deviatio, the cofidece iterval i: eft dpoit of the CI: Right dpoit of the CI: So the cofidece iterval i: ( ) χ R ( ) χ ( ) χ R ( ), χ xample (cotiued) Recall lat ectio example: Some tudet are doig a urvey to fid out the average umber of hour a Deio tudet ped tudyig each week. They take a imple radom ample of 9 Deio tudet ad compute a ample mea of 5.4 hour per week with a tadard deviatio of. hour (i.e. ample variace i. =.44). Fid a 95% cofidece iterval for the populatio variace. Aume the populatio i ormally ditributed. Solutio: Firt, we ote a few thig: The coditio above are met The cofidece level i 95% = 0.95 =, o = 0.05 The degree of freedom are = 90 The area above χ R i = = ad the area above χ i = = = From Table A-4, χ R = 8.36 ad χ = ( ) The left edpoit of the CI i = = =. 097 χ ( ) The right edpoit of the CI i = = =. 974 χ Therefore, the cofidece iterval i: (.097,.974) R So with 95% cofidece, we claim that the variace i hour per week tudied by Deio tudet i betwee.097 ad.974 hour. quivaletly, we could ay the tadard deviatio i hour per week tudied by Deio tudet i betwee.047 ad.405 hour. (Jut take the quare root of both edpoit) Determiig Sample Size To fid the proper ample ize for etimatig populatio variace, we caot rely o the argumet preeted for the proportio ad mea, ice there i o margi of error beig added ad ubtracted here. To fid the ample ize, we will jut ue a table (ee p. 37). Suppoe we wat the ample variace to be withi P percet of the value. Here i the ample ize required for give value of P ad cofidece level 95% ad 99%:

9 Cofidece evel 95% Cofidece evel 99% P P % 77,07 % 33,448 5% 3,48 5% 5,457 0% 805 0%,40 0% 0 0% % 97 30% 7 40% 56 40% 00 50% 37 50% 67 Alterate table are available if you prefer workig with tadard deviatio itead of variace. Suppoe we wat the ample tadard deviatio to be withi P percet of the value. Here i the ample ize required for give value of P ad cofidece level 95% ad 99%: Cofidece evel 95% Cofidece evel 99% P P % 9,04 % 33,8 5% 767 5%,335 0% 9 0% 335 0% 47 0% 84 30% 0 30% 37 40% 40% 50% 7 50% 3 xample Suppoe, at 95% cofidece, we wat the ample tadard deviatio to be withi 0% of the populatio tadard deviatio. What ample ize hould we ue? Solutio: From the table, with P = 0% ad Cofidece evel 95%, thi ample ize i = 9. Summary I cocluio, here are the formula for computig cofidece iterval of variou parameter: Cofidece Iterval for the Populatio Proportio Cofidece evel: # uccee Sample Proportio: pˆ = Critical Value: (foud i Table A-, back cover of book or p ) Cofidece Iterval: pˆ ± Sample Size Calculatio: = Cofidece Iterval for the Populatio Mea ( Kow) Cofidece evel: Sample Mea: Critical Value: x = i= x i (foud i Table A-, back cover of book or p ) Cofidece Iterval: x ± Sample Size Calculatio: =

10 Cofidece Iterval for the Populatio Mea ( ukow) Cofidece evel: Degree of Freedom: Sample Mea: x = i= x i Critical Value: t (foud i Table A-3, lat page of book or p. 774) Cofidece Iterval: x ± t Sample Size Calculatio: = t Cofidece Iterval for the Populatio Variace Cofidece evel: Degree of Freedom: Critical Value: χ ad R χ (foud i Table A-4, lat page of book or p. 775) Cofidece Iterval: ( ) χ R ( ), χ Sample Size Calculatio: (table o previou page)