Chapter (8) Estimation and Confedence Intervals Examples

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1 Chaptr (8) Estimatio ad Cofdc Itrvals Exampls Typs of stimatio: i. Poit stimatio: Exampl (1): Cosidr th sampl obsrvatios, 17,3,5,1,18,6,16,10 8 X i i X is a poit stimat for usig th stimator X ad th giv sampl obsrvatios. ii. Itrval stimatio: Costructig cofidc itrval Th gral form of a itrval stimat of a populatio paramtr: Poit Estimat ± Criticalvalu *Stadard rror This formula grats two valus calld th cofidc limits; - Lowr cofidc limit (LCL). - Uppr cofidc limit (UCL). Aothr way to fid th cofidc itrval w usd th cofidc 1

2 Cas1: Cofidc Itrval for Populatio Ma with kow Stadard Dviatio (ormal cas): Th cofidc limits ar: Stps for calculatig: 1. Obtai,from th tabl of th ara udr th ormal curv. Z. Calculat Z 3. L= X Z U= X Z X :Th ma stimator σ : Th stadard dviatio of th populatio. Z : Th stadard rror of th ma : Critical valu.. x Exampl (): A sampl of 49 obsrvatios is tak from a ormal populatio with a stadard dviatio of 10.th sampl ma is 55,dtrmi th 99 prct cofidc itrval for th populatio ma Solutio: X ~ N, X ~ N, σ = 10, = 49,X = 55,Cofidc lvl = 0.99, α = = 0.01 = Z =.58 Z0.01 Th cofidc limits ar: X Z ,

3 Exampl (3): - IF you hav ( , ). Basd o this iformatio, you kow that th bst poit stimat is: ˆ of th populatio ma uppr lowr ˆ 55 Cas: Cofidc Itrval for a Populatio Ma with ukow Stadard Dviatio S ˆ X t 1; Exampl (4): Th owr of Britt's Egg Farm wats to stimat th ma umbr of ggs laid pr chick. A sampl of 0 chicks shows thy laid a avrag of 0 ggs pr moth with a stadard dviatio of 8 ggs pr moth (a sampl is tak from a ormal populatio). i. What is th valu of th populatio ma? What is th bst stimat of this valu? ii. Explai why w d to us th t distributio. What assumptio do you d to mak? iii. For a 95 prct cofidc itrval, what is th valu of t? iv. Dvlop th 95 prct cofidc itrval for th populatioma. v. Would it b rasoabl to coclud that th populatio ma is 1 ggs? What about 5 ggs? Solutio: i. th populatio ma is ukow, but th bst stimat is 0,th sampl ma 3

4 ii. Us th t distributio as th stadard dviatio is ukow.howvr, assum th populatio is ormally distributd. iii. t t t ; , 19, iv. X t ; , S v. Ys, bcaus th valu of µ=1 is icludd withi th cofidc itrval stimat. No, bcaus th valu of µ=5 is ot icludd withi th cofidc itrval stimat. Exampl (5): Fid a 90% cofidc itrval for a populatio ma for ths valus: 14, 158, x s 45796, X ~ N, Solutio: t t t 1; 141, ˆ X t ; S 13, ,

5 Wh th sampl siz is larg 100, , 5, 1 5, th sampl proportio, P P ~ X = 1 N, Total umbr of succsss Total umbr of trials Th cofidc itrval for a populatio proportio: P Z P 1 P P 1 P, Th stadard rror of th proportio Exampl (6): Th owr of th Wst Ed crdit Kwick Fill Gas Statio wishs to dtrmi th proportio of customrs who us a crdit card or dbit card to pay at th pump. H survys 100 customrs ad fids that 80 paid at th pump. a. Estimat th valu of th populatio proportio. b. Dvlop a 95 prct cofidc itrval for th populatio proportio. c. Itrprt your fidigs. Solutio: a. P X b. Z Z Z 1.96 Z Z P P P Z , 0.88 c. W ar rasoably sur th populatio proportio is btw 0.7 ad o.88 prct

6 Exampl (7): Th Fox TV twork is cosidrig rplacig o of its prim-tim crim ivstigatio shows with a w family-oritd comdy show. Bfor a fial dcisio is mad, twork xcutivs commissio a sampl of 400 viwrs. Aftr viwig th comdy, 0.63 prct idicatd thy would watch th w show ad suggstd it rplac th crim ivstigatio show. a. Estimat th valu of th populatio proportio. b. Dvlop a 99 prct cofidc itrval for th populatio proportio. c. Itrprt your fidigs. Solutio: a. P 0.63 b. Z Z 0.01 Z Z Z P1 P P Z , c. W ar rasoably sur th populatio proportio is btw 0.57 ad o.69 prct. 6

7 Not: If th valu of stimatd proportio(p) ot mtiod w substitut it by o.5( as studis ad rachars rcommdd) 7

8 = ±Z σ Or = UCL LCL Th lgth of cofidc itrval= UCL LCL Th lgth of C.I= σ = Z α Th sampl siz for stimatig th populatio ma: σ = ( Z α Exampl (8): A studt i public admiistratio wats to dtrmi th ma amout mmbrs of city coucils i larg citis ar pr moth as rmuratio for big a coucil mmbr. Th rror i stimatig th ma is to b lss tha $100 with a 95 prct lvl of cofidc. Th studt foud a rport by th Dpartmt of Labor that stimatd th stadard dviatio to b $1,000. What is th rquird sampl siz? Solutio: Giv i th problm: E, th maximum allowabl rror, is $100 Th valu of z for a 95 prct lvl of cofidc is 1.96, Th stimat of th stadard dviatio is $1,000. Z ) 8

9 Exampl (9): A populatio is stimatd to hav a stadard dviatio of 10.if a 95 prct cofidc itrval is usd ad a itrval of is dsird.how larg a sampl is rquird? Solutio: Giv i th problm: E, th maximum allowabl rror, is Th valu of z for a 95 prct lvl of cofidc is 1.96, Th stimat of th stadard dviatio is10. Z Exampl (10): If a simpl radom sampl of 36 popl was usd to mak a 95% cofidc itrval of (0.57,0.67), what is th margi of rror? Solutio: uppr lowr Exampl (11): If =34, th stadard dviatio 4. What is th maximum allowabl rror Solutio: Z Th maximum allowabl rror () = 1.41 E?, 1 95% 9

10 Th margi rror for th cofidc itrval for a populatio proportio: π(1 π) = Zα Solvig "E" quatio for "" yilds th followig rsult: Or = ( Zα π(1 π) ) Zα = π(1 π) ( ) Z 1 Exampl (1): Th stimat of th populatio proportio is to b withi plus or mius o.o5, with a 95 prct lvl of cofidc. Th bst stimatio of th populatio proportio is o.15.how larg a sampl is rquird? Solutio: Z

11 Exampl (13): Th stimat of th populatio proportio is to b withi plus or mius o.10, with a 99 prct lvl of cofidc. How larg a sampl is rquird? Solutio: Z

12 Z Tabl: Ngativ Valus Z ad Tabls z

13 Z Tabl: Positiv Valus z

14 T Tabl Df

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