n Using the formula we get a confidence interval of 80±1.64

Transcription

1 9.52 The professor of sttistics oticed tht the rks i his course re orlly distributed. He hs lso oticed tht his orig clss verge is 73% with stdrd devitio of 12% o their fil exs. His fteroo clsses verge 77% with stdrd devitio of 10%. Wht is the probbility tht the e rk of four rdoly selected studets fro orig clss is greter th the verge of four rdoly selected studets fro fteroo clss? X = rks ~ N(µ, X ~ N(73,12 X ~ N(77,10 = =4 Sice the popultio stdrd devitio is kow for both orig d fteroo clss X X ( µ µ P( X X > 0 = P( Z > 2 2 Forul: + Usig this forul we get P(Z>0.51 d redig the vlue i the Z-tble =0.31 Aswer: The probbility tht the sple e i the orig clss is lrger th the oe i the fteroo clss is A rdo sple of 25 ws drw fro orl distributio whose stdrd devitio is 5. The sple e ws 80.. Deterie the 90% cofidece itervl estite of the popultio e. 90% cofidece itervl for the popultio e α = 1-0.9=0.1 Forul: X ± Z α / 2 Usig the forul we get cofidece itervl of 80±1.64 b. Repet prt. with sple size of 100. Aswer: 80±0.82 c. Repet prt. with sple size of 400. Aswer: 80±0.41 d. Describe wht hppes to the cofidece itervl estite whe the sple size icreses. As the sple size icrese we hve ore ifortio bout the popultio. 1(5

2 10.53 The opertios ger of lrge productio plt would like to estite the verge out of tie workers tke to sseble ew electroic copoet. After observig uber of workers sseblig siilr devices, she guesses tht the stdrd devitio is 6 iutes. How lrge sple of workers should she tke if she wishes to estite the e ssebly tie to withi 10 secods? Assue tht the cofidece level is to be 90%. =6 iutes = 360 secods Usig the cofidece itervl forul we c rerrge the forul d get the followig forul to clculte : 2 Zα / 2 = W Where W= 10 secods (W equls the prt fter the ± i clcultio of CI Aswer: Usig the forul we get tht the sple size ust be of 3486 or ore. 4. Itroductio to test sttistic d p-vlue Fid the p-vlue of the followig test give tht x = 500, =50 d = 20 H : µ = H A : µ > 505 Test Sttistic X µ Z = ~ N(0,1 / Usig the forul bove we get observed Z obs of Use the Z-tble to kow the p-vlue for the observed vlue = Aswer: The probbility to observe sple e of 500 or less give tht the ull hypothesis is true = Repet prt with = 30, wht hppes? Z obs = Ad the p-vlue is =0.12 Aswer: As the popultio stdrd devitio icreses the p-vlue icreses. b. Repet prt with = 100, wht hppes? Z obs = -2.5 Ad the p-vlue is = Aswer: As the sple size icreses the p-vlue decreses (becuse we hve ore ifortio bout the popultio c. Repet prt with x = 504, wht hppes? Z obs = Ad the p-vlue is =0.36 Aswer: As the observed sple e is closer to the ull hypothesis popultio e the p-vlue icreses. 2(5

3 11.29 A busiess studet clis tht o verge MBA studet is required to prepre ore th five cses per week. To exie the cli, sttistic professor sks rdo sple of te MBA studets to report the uber of cses they prepre weekly. The results re exhibited here. C the professor coclude t the 5% sigificce level tht the cli is true, ssuig tht the uber of cses is orlly distributed with stdrd devitio is 2? X~N(µ,2 =10 α=sigificce level = 0.05 Hypothesis H 0 : µ =5 H A : µ > Clculte the sple e with forul give i erlier exercises Sple e=6.7 Forulte the test sttistic (se exercise 4 bove. Rejectio regio Z obs >Z α =Z 0.05 =1.645 The we reject the ull hypothesis Observed vlue Z obs = = / 10 Aswer: 2.68>1.645 therefore c the professor, o 5% sigificce level, coclude tht the cli is true A office ger believes tht the verge out of tie spet by office workers redig d deletig sp e-il exceeds 25 iutes per dy. To test this belief, he tkes rdo sple of 18 workers d esure the out of tie ech speds redig d deletig sp. The results re listed here. If the popultio of tie is orlly distributed with stdrd devitio of 10 iutes, c the ger ifer t the 1% sigificce level tht he is correct? X~N(µ,10 =18 α=sigificce level = 0.01 Hypothesis (5

4 H 0 : µ =25 H A : µ >25 Clculte the sple e with forul give i erlier exercises Sple e=29 Forulte the test sttistic (se exercise 4 bove. Rejectio regio Z obs >Z α =Z 0.01 =2.33 The we reject the ull hypothesis Observed vlue Z obs = 10 / 18 = 1.69 Aswer: 1.69 < 2.33 d therefore we cot reject the ull hypothesis A rdo sple of 8 observtios ws drw fro orl popultio. The sple e d sple stdrd devitio re x = 40 d s = 10.. Estite the popultio e with 95% cofidece. Sice the popultio stdrd devitio is ukow d the sple size is sll we use the Studet t distributio to clculte 95% cofidece itervl for the popultio e: Forul: X ± t α / 2, 1 Which gives CI of 40±8.36 s b. Repet prt ssuig tht you kow tht the popultio stdrd devitio is = 10 Sice we hve the popultio stdrd devitio we c use the Z-distributio d forul: X ± Z α / 2 This gives estited CI of 40±6.93 d. Expli why the itervl estite produced i prt b is rrower th tht i prt Becuse the distributio of Z is rrower th tht of the Studet t e. Repet d b with = 100, expli. Usig the t-distributio for s=10 d =100 we will get CI of 40±1.984 Usig the Z-distributio for =10 d =100 we will get CI of 40±1.96 Becuse for lrge sple size (>30 the Studet t distributio is pproxitely orl distributed. The cetrl liit theore (CLT. 4(5

5 12.27 Most owers of digitl cers store their pictures o the cer. Soe will evetully dowlod these to coputer or prit the usig their ow priters or use coercil priter. A fil-processig copy wted to kow how y pictures were stored o cers. A rdo sple of 10 digitl cer owers produced the dt give here. Estite with 95% cofidece the e uber of pictures stored o digitl cers Strt by clcultig the sple e d stdrd devitio with foruls give i erlier prcticls/the book or forul sheet Sple e=17.7 s = 9 Sice the sple size is sll d we do t kow the popultio stdrd devitio we use t- distributio to clculte the CI for the popultio e= Observed vlues 17.7 ± Which gives CI of 17.7±6.44 LCL (lower liit = UCL (upper liit = X ± t α / 2, 1 s A de of busiess school wted to kow whether the grdutes of her school used sttisticl iferece techique durig their first yer of eployet fter grdutio. She surveyed 418 grdutes d sked bout the use of sttisticl techiques. After tllyig up the resposes, she foud tht 217 used sttisticl iferece withi 1 yer of grdutio. Estite with 90% cofidece the proportio of ll busiess school grdutes who use their sttisticl eductio withi yer of grdutio. We kow tht the sple proportio ^p=x/ where x=the uber of success (i this cse uber who used sttistics d =sple size ^p=0.52 Further we kow tht the stdrd devitio for ^p is pˆ (1 pˆ To ke 90% cofidece itervl for the popultio proportio, p, use the forul: pˆ(1 pˆ pˆ ± Z α / 2 Where α=0.1 Usig our observed vlues we get CI of 0.52±0.04 for p 5(5