STT 315, Summer 006 Lecture 5 Materials Covered: Chapter 6 Suggested Exercises: 67, 69, 617, 60, 61, 641, 649, 65, 653, 66, 663 1 Defiitios Cofidece Iterval: A cofidece iterval is a iterval believed to iclude a ukow populatio parameter Associated with the iterval is a measure of the cofidece we have that the iterval does ideed cotai the parameter of iterest Cofidece Iterval for the Populatio Mea μ Whe is kow (1) Formula: If the sample is from a ormal populatio or the sample size is large, a (1-α )100% cofidece level iterval for μ whe the is kow is give by where zα satisfies X z, X + zα, α α P ( Z z α ) = Remark 1: The quatity z α is called the margi of error Example: A miig compay eeds to estimate the average amout of copper ore per to mied A radom sample of 50 tos gives a sample mea of 14675 pouds The populatio stadard deviatio is assumed to be 35 pouds Give a 95% cofidece iterval for the average amout of copper i the populatio of tos mied Also give a 90%, 99% cofidece iterval for the average amout of copper
STT 315, Summer 006 Remark : A good cofidece iterval should have: (1) High Cofidece Level; () Narrower Cofidece Iterval (Smaller margi of error) The legth of the above cofidece iterval = zα But, if the sample size is fixed, the higher the cofidece level, the wider the cofidece iterval; if the cofidece level is fixed, the larger the sample size, the arrower the cofidece iterval Example: Suppose you have a cofidece iterval based o a sample of size Usig the same level of cofidece, how large a sample is required to produce a iterval of oe-half the width () Theoretic foudatio of the formula Samplig distributios of the sample mea if the sample is from a ormal populatio or the sample size is large 3 Cofidece Iterval for the Populatio Mea μ Whe is ukow (1) t-distributio If the populatio is ormally distributed, the the stadardized statistic t = X μ S has a t-distributio with degrees of freedom -1, or t ~ t( 1) 04 03 0 01 00-5 -4-3 - -1 0 1 3 4 The desity fuctio of t (10)
STT 315, Summer 006 () Formula: If the sample is from a ormal populatio, a (1-α )100% cofidece level iterval for μ whe the is ukow is give by wheretα satisfies S X t, X + tα S α, P { t( 1) tα } = α Example 1: Suppose a t radom variable has degrees of freedom 10 Fid t 0 t 05, 0 05 Example : A stock market aalyst wats to estimate the average retur o a certai stock A radom sample of 15 days yields a average (aualized) retur of X =1037% ad a stadard deviatio s = 35% Assumig a ormal populatio of returs, give a 95% cofidece iterval for the average retur o this stock Remark: Wheever is ot kow ad the populatio is ormal, the correct distributio to use is the t-distributio with -1 degrees of freedom Note, however, that for large sample size, the t-distributio is approximated well by the Z distributio (stadard ormal distributio) So a large sample ( 1 α )100% cofidece iterval for μ is S X z, X + zα S α
STT 315, Summer 006 4 Large-Sample Cofidece Itervals for the Populatio Proportio Formula: A large-sample ( 1 α )100% cofidece iterval for the populatio proportio p is give by pq ˆ ˆ pˆ z pˆ, + zα pq ˆ ˆ α, where the sample proportio is equal to the umber of successes i the sample, divided by the umber of trials (the sample size), ad qˆ = 1 pˆ Example: A market research firm wats to estimate the share that foreig compaies have i the US market for certai products A radom sample of 100 cosumers is obtaied, ad 34 people i the sample are foud to be users of foreig-made products; the rest are users of the domestic products Give a 95% cofidece iterval for the share of foreig products i this market 5 Cofidece Itervals for the Populatio Variace (1) χ -distributio LetX X, 1,, X be idepedet radom variables with stadard ormal distributio, the the radom variable χ = X i i= 1 follows the chi-square distributio with degrees of freedom, or χ ~ χ ( )
STT 315, Summer 006 010 008 006 004 00 000 0 5 10 15 0 5 30 The desity fuctio of χ ~ χ (10) () Theorem: Let X X, be idepedet radom variables with ormal distributio 1,, X ( 1) s μ χ N(, ), the the radom variable degrees of freedom = has a chi-square distributio with -1 (3) Formula: A ( 1 α)100% cofidece iterval for the populatio variace (where the populatio is assumed ormal) is give by Where ( 1) s ( 1) s, χ α χ α χα satisfies P ( χ ( 1) > χα ) = α Example 1: Suppose a chi-square radom variable has degrees of freedom 10 χ 0 χ Fid, 05 0975 Example : I a automated process, a machie fills cas of coffee If the average amout filled is differet from what is should be, the machie may be adjust to correct the mea If the variace of the fillig process is too high, however, the machie is out of cotrol ad eeds to be repaired Therefore, from time to time regular checks of the variace of the fillig process are made This is doe by radomly samplig filled cas, measurig their amouts, ad computig the sample variace A radom sample of 30 cas gives a estimate
STT 315, Summer 006 s =18,540 Give a 95% cofidece iterval for the populatio variace 6 Sample Size Determiatio To decide the sample size, we eed three umbers: (1) Distace betwee the estimator ad the parameter: B () Cofidece Level: ( 1 α)100% (3) A estimatio of the populatio variace: The, the miimum required sample size i estimatig the populatio mea is zα =, B ad the miimum required sample size i estimatig the populatio proportio is = zα pq B Example 1: A market research firm wats to coduct a survey to estimate the average amout spet o etertaimet by each perso visitig a popular resort The people who pla the survey would like to be able to determie the average amout spet by all people visitig the resort to withi $10, with 95% cofidece From past operatio of the resort, a estimate of the populatio stadard deviatio is $400 What is the miimum required sample size? Example : The maufacturer of a sports car wats to estimate the proportio of people i a give icome bracket who are iterested i the model The compay wats to kow the populatio proportio p to withi 010 with 99% cofidece Curret compay records idicate that the proportio p may be aroud 05 What is the miimum required sample size for this survey?