100(1 α)% confidence interval: ( x z ( sample size needed to construct a 100(1 α)% confidence interval with a margin of error of w:

Transcription

1 Stat 400, ectio 7. Large Sample Cofidece Iterval ote by Tim Pilachowki a Large-Sample Two-ided Cofidece Iterval for a Populatio Mea ectio 7.1 redux The poit etimate for a populatio mea µ will be a ample mea x. a = probability of error, 1 a = cofidece level = probability that a radom iterval will capture the true value of the populatio parameter µ. Value of / for commo two-ided cofidece level / (1 % cofidece iterval: ( x ( / x / (,. ample ie eeded to cotruct a 100(1 % cofidece iterval with a margi of error of w: = / ( w I ectio 7.1, we made very importat aumptio: eve though the parameter µ wa ukow, the populatio probability ditributio wa approximately ormal ad the parameter wa kow ad wa ued i the cofidece iterval ad ample ie formula above. Example D-backgroud. I 1868, Carl Reihold Augut Wuderlich publihed hi defiitive work o cliical thermometry. I it, he give the ormal huma body temperature a 98.6º F (37º C. (He did ote that ormal temperature i a rage, decribed variatio i temperature acro 4 hour, ad etablihed F (38 C a the firt quatitative defiitio of fever. A group of reearcher ivetigated whether icreaed reliability of moder thermometer might challege thi commo kowledge ad/or whether huma phyiology may have chaged over the pat cetury. Mackowiak, P. A., Waerma, S. S., ad Levie, M. M. (199, "A Critical Appraial of 98.6 Degree F, the Upper Limit of the Normal Body Temperature, ad Other Legacie of Carl Reihold Augut Wuderlich," Joural of the America Medical Aociatio, 68, , Example D-a. Reearcher wat to ivetigate huma iteral body temperature. Baed o prior tudie, the populatio probability ditributio i approximately ormal with a tadard deviatio etimated at If they wat a 95% cofidece level, with a margi of error of o more tha 0., how large hould their ample be? awer: at leat 5 Now come a importat quetio: How ca we determie a cofidece iterval i cae where the populatio probability ditributio may ot be ormal, ad where we do t kow the value of? To awer the firt part, we ivoke the Cetral Limit Theorem which implie that X ha approximately a ormal ditributio, o matter what the hape of the ditributio of X. I cae where the populatio probability ditributio may ot be ormal, we will cotruct a cofidece iterval for which the level of cofidece i approximately 100(1 %.

2 Whe the populatio variace i ot kow (a will uually be the cae ad therefore populatio tadard deviatio i ot kow, the ratioale get a little trickier. We go back oce agai to the Cetral Limit Theorem. To awer probability quetio back i ectio 5.4 ad X µ to cotruct cofidece iterval i ectio 7.1, we worked from the liear traformatio Z =, i which oly X i a radom variable. X µ If we ow tur to Z = (ad ue radom variable S a well a radom variable X, the there i S radome i both umerator ad deomiator. However, if i ufficietly large, it will ameliorate the effect of the extra variability itroduced by uig S. The rule of thumb for ivokig the Cetral Limit Theorem wa > 30. To accout for the added variability, thi text propoe > 40 a a rule of thumb for uig the followig formula for a large-ample two-ided iterval with a cofidece level of approximately 100(1 %: ( x ( ( /, x /. Example D-b. I their 199 tudy of huma iteral body temperature, Mackowiak, Waerma ad Levie had a ample ie of 148. Give x = 98.5 ad = 0.73, cotruct a two-ided 95% cofidece iterval. awer: (98.13, What would be the effect of icreaig the level of cofidece? Would the reultig cofidece iterval be wider or arrower? Example D-c. I their 199 tudy of huma iteral body temperature, Mackowiak, Waerma ad Levie had a ample ie of 148. Give x = 98.5 ad = 0.73, cotruct a two-ided 99% cofidece iterval. awer: (98.095,

3 b Large-Sample Oe-ided Cofidece Iterval for a Populatio Mea So far, the cofidece iterval we ve coidered have bee two-ided, i.e. they have both a lower ad a upper boud. I ome cae, a reearcher will wat a oe-ided cofidece iterval, i.e. either a upper or lower boud but ot both. For example, i a tudy about a ew medicatio, doctor will be itereted i whether the ew formulatio i better tha the old. Or a mechaical egieer may wat to ivetigate a lower failure rate for compoet. 1 1 The poit etimate for a populatio mea µ will be a ample mea x. a = probability of error, 1 a = cofidece level = probability that a radom iterval will capture the true value of the populatio parameter µ. 100(1 % two-ided cofidece iterval: x ( ( / < µ < x / upper 100(1 % oe-ided cofidece boud: µ < x ( lower 100(1 % oe-ided cofidece boud: µ > Value of for commo oe-ided cofidece level. x Example B reviited: A ample ha mea = 150 ad the populatio ha kow tadard deviatio =. For a radom ample of ie 47, fid the lower boud for a oe-ided 90% cofidece iterval. awer: Example D-b reviited. I their 199 tudy of huma iteral body temperature, Mackowiak, Waerma ad Levie had a ample ie of 148. Their ample mea, x = 98.5 with = 0.73, led them to a hypothei that huma iteral body temperature i lower tha the covetioal 98.6º F. Doe a oe-ided 95% cofidece iterval baed o their ample data iclude 98.6º F? awer: µ < 98.35; No.

4 c Large-Sample Cofidece Iterval for a Populatio Proportio Whe cotructig a large-ample cofidece iterval of approximately 100(1 % for etimatig populatio parameter µ, we ued the formula ±. x / Expreed i geeric term, thi i (poit etimate of θ ± (critical value for time (etimated tadard error of the etimator. Applyig thi to populatio proportio, ad uig the tadard error formula developed i Lecture 6.1c, we get ± /. 6.1c Example B. A ew orgaiatio poll voter ad ak, Do you ited to vote for icumbet Seator Phillip E. Buter i the upcomig electio? Pollter record the followig umber. Ye Udecided No Male Female Cotruct a 95% cofidece iterval for the proportio of voter who would favor reelectig Seator Buter. Thi formula for cofidece iterval of a populatio proportio i the oe that ha bee ued i itroductory tatitic coure for a log time, ad i ofte ued i practice. If the ample ie i very large, ad if the value of the populatio proportio p i cloe to 0.5, the ± / give a reaoably good cofidece iterval. However, it ha evere limitatio. For mall value of, or for value of ad cloe to 0 or 1, the variability of the tadard error portio of the formula icreae dramatically. (See Figure 7.6 i your text. We ll backtrack a little, ad ue a proce imilar to that ued i ectio 7.1. p P / < < / ( 1 p 1 p The ext tep i replacig the iequalitie with = ad olvig for p. / = p p( 1 p p p( 1 p Ue of the quadratic formula i a iteive algebraic proce would get u to = /

5 p / / ˆ 4 ± / / / p =. A advatage of thi more complicated verio i that it addree the deficiecie of the claic formula, ad work well for almot all value of ad p, eve whe p < 10 or q < c Example B. A ew orgaiatio poll voter ad ak, Do you ited to vote for icumbet Seator Phillip E. Buter i the upcomig electio? Pollter record the followig umber. Ye Udecided No Male Female Cotruct a 95% cofidece iterval for the proportio of voter who would favor reelectig Seator Buter. ± 4