PRINCE SULTAN UNIVERSITY Department of Mathematical Sciences Final Examination Second Semester (072) STAT 271.
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1 PRINCE SULTAN UNIVERSITY Deparmen f Mahemaical Sciences Final Examinain Secnd Semeser (07) STAT 7 Suden Name Suden Number Secin Number Teacher Name Aendance Number Time allwed is ½ hurs. Wrie dwn yur answer in he space prvided underneah he quesin. Yu may use a prgrammable calcular and/r yur frmula shee. Use α = if n specified. Sme criical values f Z-disribuin Mark 00 Z 0.0 Z Z0. 05 Z 0. 0 Z Sme criical values f -disribuin df =5 df = Sme criical values f F-disribuin df = df =9 df = df =7 F 0.05 =4.6 F =5.59 Sme criical values f Chi-Square disribuin df =3 χ 0.05 = Quesin Mark Tal Term Mark Ou f 60 Final Exam Mark Ou f 40 Tal Mark Ou f 00 Grade
2 STAT 7 Final Examinain Secnd Semeser Quesin (0 marks) Six waer specimens aken frm a river a a specific lcain during he lw-waer seasn gave readings f 4.9, 5., 4.9, 5.0, 5.0, and 4.7 (ppm) f disslved xygen. Assume ha his daa represens a randm sample frm a nrmal disribuin wih unknwn variance σ. (I) Des he daa prvide sufficien evidence indicae ha he mean f disslved xygen cnen ( μ ) is less han 5.0 ppm? Use α = cnduc a small-sample es f hypheses. Arrange yur answer as fllws (a) The null and he alernaive hypheses [4 marks] (b) The es saisic [4 marks] (c) The rejecin regin [4 marks] (d) The cnclusin [4 marks] (II) Cnsruc a 95% small-sample cnfidence inerval fr he mean f disslved xygen cnen ( μ ). [4 marks]
3 STAT 7 Final Examinain Secnd Semeser Quesin (4 marks) (I) Tw randm samples f sizes n = 50 and n = 60 were seleced independenly frm w ppulains wih variances σ = 6 and σ = 5, respecively. The sample means f heses randm samples are X = 55and X = 53, respecively. Des his daa indicae ha here is a difference beween he means f he w ppulains? Use α=0.05. Arrange yur answer as fllws (a) The null and he alernaive hypheses [4 marks] (b) The es saisic [4 marks] (c) The rejecin regin [4 marks] (d) The cnclusin [4 marks] (II) I is claimed ha 35% f he husehlds in a cerain cunry wn a leas ne car. In a randm sample f 50 husehlds, 80 husehlds wn a leas ne car. Des his daa prvide sufficien evidence indicae ha he prprin f husehlds wih a leas ne car is differen frm 0.35? Use α=0.05. Explain and jusify yur answer. [8 marks]
4 STAT 7 Final Examinain Secnd Semeser Quesin 3 ( marks) The Graduae Recrd Examinain (GRE) scres were recrded fr fur sudens admied each f hree graduae prgrams (A, B, and C). Graduae Prgram A B C A cmpleely randmized design has been used fr his sudy. The ANOVA able f his sudy fllws ANOVA Surce f Variain SS df MS F P-value F cri Beween Grups Wihin Grups XXXXXXX XXXXXXX XXXXX Tal XXXXXXX XXXXXXX XXXXXXX XXXX (a) Cmplee he ANOVA able abve. (SST, SSE, SSTOT, [4 marks] df SSTOT, and F rai) (b) Des his daa prvide sufficien evidence indicae a difference in he mean GRE scres fr applicans admied he hree prgrams? Use α=0.05. Explain and jusify yur answer. [8 marks]
5 STAT 7 Final Examinain Secnd Semeser Quesin 4 (4 marks) T es a subjec's abiliy esimae sizes, he was shwn 9 differen bjecs and asked esimae heir lengh r diameer. The bjecs were hen measured, and he resuls were recrded in he able belw. Objec X (Acual) Y (Esimaed) Summary saisics X = 0.5, Y = 0, X =.3889, Y =. XY = 99.94, X = 99.63, Y = S XY = 49.66, S XX = 3.69, SYY = Assume he relainship beween Y and X is given by he fllwing simple linear regressin mdel Y= α + β X. The ANOVA able is given belw ANOVA df SS MS F Significance F Regressin Residual XXXXXX XXXXXX Tal XXXXXX XXXXXX XXXXXX Cefficiens Sandard Errr Sa P-value Lwer 95% Upper 95% Inercep?????? X?????? (a) Cmplee he ANOVA able abve. ( df SSR, df SSE, MSR, and MSE) [4 marks] (b) Find he leas-squares esimae f α and β. [4 marks] (c) Wrie dwn he esimaed leas-squares line (predicin equain). [ marks] (d) Use he predicin equain predic he esimaed lengh (Y) if he acual lengh is X=8. [ marks]
6 STAT 7 Final Examinain Secnd Semeser (e) Find a 95% cnfidence inerval fr β. [ marks] (f) Tes β = 0 agains β 0 (use α = 0.05). [ marks] (g) Des he daa presen sufficien evidence indicae ha Y and X are linearly relaed? Use α=0.05. Explain and jusify yur answer. [ marks] (h) Calculae he value f he cefficien f deerminain ( R ). [ marks] (i) Inerpre he value f he cefficien f deerminain ( R ) bained in par (h). [ marks] (j) Calculae he value f he cefficien f crrelain ( r ) beween X and Y. [ marks]
7 STAT 7 Final Examinain Secnd Semeser Quesin 5 (0 marks) A freeway wih fur lanes (A, B, C, and D) in each direcin was sudied see wheher drivers prefer drive n he inside lanes (lanes B and C). A al f 500 aumbiles were bserved during heavy early-mrning raffic, and he number f cars in each lane was recded. Lane A B C D Number f cars (bserved cuns) (in If here is n lane preference, hen p = p = p = p / 4 ). 3 4 = () If here is n lane preference, calculae he expeced number f cars (expeced cun) fr each lane, and fill in he fllwing able [4 marks] Lane A B C D Observed cuns ( O ) Expeced cuns ( E ) i i () Des he daa presen sufficien evidence indicae ha drivers have lane preference (i.e., es he hyphesis ha here is a lane preference) a he α = level f significance. Arrange yur answer as fllws [4 marks fr each par] (a) The null and he alernaive hypheses (b) The es saisic (c) The rejecin regin (d) The cnclusin
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