H JC Prlim Solutios 6 a + b y a + b / / dy a b 3/ d dy a b at, d Giv quatio of ormal at is y dy ad y wh. d a b () (,) is o th curv a+ b () y.9958 Qustio Solvig () ad (), w hav a, b. Qustio d.77 d d d.77 d.755 ad.755.436 l.77 l l.3676 l.9958. (arst itgr) Qustio 3 O y k+ + dy d ( + ) quatio of tagt at p is y k+ ( p) p + ( p + ) p y + k+ ( p + ) p +
H JC Prlim Solutios 6 Tagt passs through (, ) p ( p + ) p + k + + ( p) + k+ ( p + ) p + p + k( p+ ) + ( p+ ) + p + kp + kp + k + p + kp + + k p + k discrimiat (k+ ) 4 kk ( ) k+ 4 > for all k > Hc, thr ar distict valus of p. This implis thr will always b tagts to th curv that passs through (, ). Qustio 4 (i) A l d ; A l l d Sic l l, A l d ad A l l d l d A (show) (ii) 3 A dy dy y y (iii) A+ A + A3 A + A S R Qustio 5 P 5 Q h ( ) h A bas hight (5 ) 5 (show) da A ( 5 ) ( 5 ) ( 5 ) 5 d h
H JC Prlim Solutios 6 Sig tst o d A : d 5-5 5 + Sig of + - da d Slop Wh 5, A 65 is a maimum valu. A> 5 > 5+ < From GC, 5+ wh.9 or 47.9.9 47.9.9 < < 47.9 Th largst valu of is 47. Sctio B Qustio 6 (i) Obtai a list of all th shops i ach catgory i th shoppig mall from th dirctory. Us a radom samplig to slct from ach catgory a umbr which is proportioal to th umbr of shops i th catgory. For ampl, if thr ar 6 shops i th fashio catgory, slct fashio shops. A stratifid sampl of fashio shops ca b obtaid by usig a radom umbr grator to obtai distict umbrs ad th slct th shops which corrspod to th umbrs gratd. This procdur is rpatd for th rmaiig 3 catgoris. (ii) It is difficult to obtai th samplig fram i.. th umbr of shopprs i th shoppig mall, thus, it would b difficult to us a stratifid samplig. (iii) Quota samplig. Th maagr would ot b abl to obtai a radom sampl as th maagr might slct shopprs basd o his prfrc. Hc ot vryo has a qual chac of big slctd. 3
H JC Prlim Solutios 6 (i) (a) Qustio 7 Rd Bo A Blu Rd Bo B Blu Whit P(bo A blu) (ii) P(bo A blu) P(blu) 6 6 6 6 4 4 3 3 3 + 6 6 6 8 36 Rquird probability 4 3 4 4 3 4 3 53 + + 6 6 5 4 6 8 7 6 6 8 7 6 4 Qustio 8 Lt X b th umbr of spoild ggs i a carto. X ~ B(, p). P X.48 p ( p) ( p).48.48 p.77687 ( X ) P.77687.77687.365556.3656 (4 d.p) 9 X ~ B(,.77687 ).77687.4536 E(X) 4
H JC Prlim Solutios 6 P X.4536 P( X ) P( X ).544435 Rquird prob (.544435) 3.36839.368(3 s.f.) Lt Y b th umbr of cartos, out of 3, with o spoild ggs. Y ~ B(3,.48) p 4.4 > 5; ( p) 5.6; p( p) 7.488 Sic p > 5 ad (-p) > 5, Y ~ N(4.4, 7.488) approimatly. P( < Y 5) P(.5 < Y < 5.5).8757.9 (3 s.f) Qustio 9 Lt w 3, th, w 45, w 45 ad 45 Ubiasd stimat of µ is 3+ w 3 + 3.75 6 Ubiasd stimat of σ is s s w ( 45) 45 6.63356 6.63 59 6 H : µ 3 H: µ > 3 Lvl of sigificac:5% Tst Statistic: Sic 6 is sufficitly larg, so 5 s is a good stimat of Thorm, X is approimatly ormal. s X N 3, approimatly wh H is tru. X 3 Z N (,). s Rjctio rgio: z.6449 Computatio: 3.75, 6, s 6.63356 z.5598.6 p valu.358. σ ad by Ctral Limit Coclusio: Sic p valu. <.5, H is rjctd at 5% sigificac lvl. Hc thr is sufficit vidc to coclud that th machi is ot workig corrctly at th 5% sigificac lvl. Ys. Th tst is valid sic 6 is sufficitly larg, by Ctral Limit Thorm, th sampl ma lgth of a ail ( X ) is approimatly ormally distributd.
H JC Prlim Solutios 6 σ If σ., th wh H is tru, X N 3, P (prsumig machi has go wrog wh i fact it is workig corrctly). ( X a ) P > wh H is tru. a 3 P Z >.. From GC: P( Z >.3635). a 3.3635.. a 3 +.3635.33 3 + Qustio Lt B kg b th mass of a radomly chos Buttrut pumpki. B ~ N(µ, μμ 8 ) ( B µ ) P.9.9µ µ P Z µ /8 P( Z.8).788447.788 (3 s.f.) 78.8% of th Buttrut pumpkis hav mass at last.9 of th ma mass. B ~ N(.8,. ) B + B +... + B5 B 5. B ~ N.8, 5 P.9.98736.987 (3 s.f.) ( B ) Lt J kg b th mass of a radomly chos Japas pumpki. J ~ N(,.5 ) Cost of o Buttrut pumpki, X.5B E(X).5(.8) Var(X).5 (. ).65 X ~ N(,.65) Cost of o Japas pumpki, Y.67J E(Y).67().67 Var(Y).67 (.5 ).6755 Y ~ N(.67,.6755) C X + X + Y ~ N + Y + Y3 ( 9.,.33575 ) P( 8.5 < C < 9.5).68 (3 s.f.) Assum that th masss of all th pumpkis ar idpdt of o othr. 6
H JC Prlim Solutios 6 (i) Qustio (ii) r.96 (to 3 s.f.) Sic r.96 is clos to ad th poits sm to li clos to a straight li with gativ gradit ar idicatios of a strog gativ liar rlatioship btw th charg () ad th avrag umbr of vhicls trig th city ctr pr day (y). This mas that as icrass, y tds to dcras at a costat rat. (iii) Sic th valus of ar fid (or cotrolld), hc is a idpdt variabl. So th last squars rgrssio lis, y o should b usd. (iv) Th quatio of th rgrssio li of y o is y.86.738 i. y.8.7 (to 3 s.f.) (v) Wh thr is o cogstio charg i.., so th avrag umbr of vhicls which will tr th city ctr pr day is 8 (or.8 millio). Sic is out of th rag of th data 4 8,.8 millio dos ot cssarily giv th pctd avrag umbr of vhicls trig th city ctr pr day. (vi) + w r.96 (sam as th product momt corrlatio cofficit btw ad y foud i (i)) as th product momt corrlatio cofficit is uaffctd by chag of scal ad locatio. 7