Confidence Intervals and Point Estimation

Transcription

1 Cofidece Iterval ad Poit Etimatio x ε < µ < x + ε ε = z σ x ε < µ < x + ε ε = t ν,, ν = 1 = z σ ε ˆp ε < p < ˆp + ε ε = z ˆp ˆq = z ε pq ( 1) < σ < ( 1), ν = 1 χ ν, χ ν, 1

2 ( x 1 x ) ε < µ 1 µ < ( x 1 x ) + ε ε = z σ σ ( x 1 x ) ε < µ 1 µ < ( x 1 x ) + ε ε = t ν, 1 p ν = m + ( p = 1 1) 1 + ( 1) ν ( x 1 x ) ε < µ 1 µ < ( x 1 x ) + ε ε = t ν, ν = roud x p ε < µ 1 µ < x p + ε ε = t ν,, ν = 1 ( ˆp 1 ˆp ) ε < p 1 p < ( ˆp 1 ˆp ) + ε ε = z ˆp 1 ˆq ˆp ˆq 1 i 1 F, 1 1, 1 < σ 1 σ < 1 i F, 1, 1 1

3 Problem 1. A ample of ize = 100 produced the ample mea of 16. Aumig the populatio tadard deviatio i 3, compute a 95% cofidece iterval for the populatio mea. A < µ < Aumig the populatio tadard deviatio of 3, how large hould a ample be to etimate the populatio mea with a margi of error ot exceedig 0.5? A. > We oberved 8 uccee i 70 idepedet Beroulli trial. Compute a 90% cofidece iterval for the populatio proportio p. A < µ < The operatio maager of a large productio plat would like to etimate the mea amout of time a worker take to aemble a ew electroic compoet. Aume that the tadard deviatio of thi aembly time i 3.6 miute. a) After obervig 10 worker aemblig imilar device, the maager oticed that their average time wa 16. miute. Cotruct a 9% cofidece iterval for the mea aembly time. A < µ < b) How may worker hould be ivolved i thi tudy i order to have the mea aembly time etimated up to ±0.5 miute with 9% cofidece? A. > Suppoe a coumer advocacy group would like to coduct a urvey to fid the proportio p of coumer who bought the ewet geeratio of a MP3 player were happy with their purchae. a) How large a ample hould they take to etimate p with % margi of error ad 90% cofidece? A. > b) The advocacy group took a radom ample of 1000 coumer who recetly purchaed thi MP3 player ad foud that 400 were happy with their purchae. Fid a 95% cofidece iterval for p. A < µ < I order to eure efficiet uage of a erver, it i eceary to etimate the mea umber of cocurret uer. Accordig to record, the ample mea ad ample tadard deviatio of umber of cocurret uer at 100 radomly elected time i 37.7 ad 9., repectively. Cotruct a 90% cofidece iterval for the mea umber of cocurret uer. A < µ < Whe ixtee cigarette of a particular brad were teted i a laboratory for the amout of tar cotet, it i foud that their mea cotet wa 18.3 mg with = 1.8 mg. Set a 90% cofidece iterval for the mea tar cotet, µ, i the populatio of cigarette of thi brad. (Aume that the amout of tar i a cigarette i ormally ditributed.) A < µ < From pat record, eaoal raifall over 16 radomly packed year yielded a mea raifall of 0.8 iche. If it ca be aumed from pat experiece that raifall durig a eao i ormally ditributed with σ =.8 iche, cotruct a cofidece iterval for the true mea raifall with 98% cofidece. A < µ <.43

4 9. A radom ample of 30 houehold wa elected a part of a tudy o electricity uage, ad the umber of kilowatt-hour (kwh) wa recorded for each houehold i the ample for the March quarter of 006. The average uage wa foud to be 375kWh. I a very large tudy i the March quarter of the previou year it wa foud that the tadard deviatio of the uage wa 81kWh. Aumig the tadard deviatio i uchaged ad that the uage i ormally ditributed, provide a expreio for calculatig a 99% cofidece iterval for the mea uage i the March quarter of A idutrial deiger wat to determie the average amout of time it take a adult to aemble a "Eay to Aemble Toy". A ample of 16 time yielded a average time of 19.9 miute, with a ample tadard deviatio of 5.73 miute. Aumig ormality of aembly time, provide a 95% cofidece iterval for the mea aembly time. 11. The recommeded retail price of a brad of deiger jea i $150. The price of the jea i a ample of 16 retailer i o average $141 with a ample tadard deviatio of 4. If thi i a "radom" ample ad the price ca be aumed to be ormally ditributed, cotruct a 95% cofidece iterval for the average ale price. 1. The average height of a radom ample of 400 people from a city i 1.75 m. It i kow that the height of the populatio are radom variable that follow a ormal ditributio with a variace of Determie the iterval of 95% cofidece for the average height of the populatio. With a cofidece level of 90%, what would the miimum ample ize eed to be i order for the true mea of the height to be le tha cm from the ample mea? 13. The mothly ale of a appliace hop are ditributed accordig to a ormal ditributio, with a tadard deviatio of $900. A tatitical tudy of ale i the lat ie moth ha foud a cofidece iterval for the mea of mothly ale with extreme of $4,663 ad $5,839. What were the average ale over the ie moth period? What i the cofidece level for thi iterval? 14. O day two of a tudy o body temperature, 106 temperature were take. Suppoe that we oly have the firt 10 temperature to work with. The mea ad tadard deviatio of thee 10 core were F ad 0.30 F, repectively. Cotruct a 95% cofidece iterval for the mea of all body temperature. 15. I a time ue tudy 0 radomly elected maager were foud to ped a mea time of.4 hour per day o paperwork. The tadard deviatio of the 0 core wa 1.30 hour. Cotruct a 98% cofidece iterval for the mea time pet o paperwork by all maager. 16. The Natioal Ceter for Educatio Statitic urveyed 4400 college graduate about the legth of time required to ear their bachelor' degree. The mea wa 5.15 year ad the tadard deviatio wa 1.68 year. Baed o the above iformatio, cotruct a 98% cofidece iterval for the mea time required to ear a bachelor'[ degree by all college tudet. 17. A fleet of 100 airplae ha a air time (time pet flyig) tadard deviatio of 14,9 hour. A ample of 3 of thee plae gave a mea air time of 49 hour. Cotruct a 90% cofidece iterval o the mea air time for thi fleet. 18. A radom ample of 50, 8 ouce cup of black "Early Rier" coffee dipeed by a ew machie gave a mea of 11.0 mg. of caffeie. It i kow from previou tudie that the tadard deviatio for 8 oz. cup of black "Early Rier" coffee dipeed by thi machie wa 7.1 mg. Cotruct a 90% cofidece iterval for the mea caffeie cotet for cup dipeed by thi machie. 19. Health iurer ad the federal govermet are both puttig preure o hopital to horte the average legth of tay (LOS) of their patiet. A radom ample of 7 hopital i oe tate had a mea LOS i 1998 of 3.8 day ad a tadard deviatio of 1. day. Cotruct a 98% cofidece iterval to etimate the populatio mea of the LOS for the tate' hopital i I a Time/CNN telephoe poll of 101 adult America, 11% of the repodet aid that Roald Rega wa a great preidet. Give a 98% cofidece iterval for the proportio of all adult America who thik that Rega wa a great preidet.

5 1. I the year 001 Youth Rik Behavior urvey doe by the U.S. Ceter for Dieae Cotrol, 747 out of 1168 female 1th grader aid they alway ue a eatbelt whe drivig. Cotruct a 95% cofidece iterval for the proportio of 1th grade female i the populatio who alway ue a eatbelt whe drivig. ˆp ε < p < ˆp + ε ε = z ˆp ˆq. Two ormally ditributed populatio,with kow populatio variace: ample ize mea variace cofidece A m = % B = ε = z σ 1 m + σ 3. Two populatio m ad large: ample ize mea variace cofidece A m = % B = ε = z 1 m + 4. Two ormally ditributed populatio m ad mall, populatio variace ukow but equal: ample ize mea variace cofidece A m = % B = ε = t m+, p 1 m + 1, where p = ( m 1) 1 + ( 1) m +

6 5. Two ormally ditributed populatio m ad mall, populatio variace ukow but uequal: ample ize mea variace cofidece A m = % B = ε = t ν, 1 m + 1 m + where ν = roud 1 m m Two large ample populatio, cofidece iterval for differece i proportio: ample ize amout cofidece A m = % B = ε = z ( p 1 q 1 ) m + p q ( )

7 7. Two depedet (iteractive) ample (matched pai idividual A B differece cofidece = 95% mea of differece = ε = t 1, 8. Ratio of ample variace: ample ize mea variace cofidece A m = % B = F, m 1, 1, 1 F, m 1, 1