l. A medical researcher is studying the relationship between age (x years) and volume each contraction of the heart. The researcher obtained the following data from a random sample of 8 patients. 3.Abiasedfour.sideddiehasfacesmarkedl,3,5andT.Therandomvariable,Yrepresentsl distribution function of X' Ftxl' is the score on the die ,rtl". iii. .ott"a. The cumulative ] given in the table oi blood (v ml) pumped by Age (x) 20 25 30 45 55 60 65 70 Volume (y) 74 76 77 72 68 67 64 62 I r: 370, S.* [You may use (a) : 2587.5, I y: 560, Z y' : 39 41 8, S,, : below. / I , 0.2 0.5 I F(r) (a) Find the probability distribution -71 0] (b) Find P(2 < x< ---.-1--_ll 5 7 0.9 i of-lK "\ 6) (2)", Calculate S,* (2) (c) (b) Write down the value (2) (c) Interpret your value ofthe corielation coefticient. (1) F(4) (1) (e) Find the equation of the regression line of y formy: orr x, giving your answer in the a 4 bx (4) Jack is a 40-year-old patient. (f) (i) c) F( .t) 4. s o't1 = t>-} , | ( xE't) -- o'J The random variable Given that P(Y (a) P(r> I^- N(4, a'?) < l7): 0.6 find 17) (1) Use your regression line to estimate the volume of blood pumped by each contraction of Jack's hearl. (ii) (b) P(ir <Y<17) ZtK b) ?AcL, e.l&,tlc.o- ho U{-rst\ - -i-'r.^a etE/ C) SDq3 C) W\P a t-o (2) ffi;6 ,-oryS (c) P(r< i rS \€(5cL.l,sQ- ho -\ t, E?+-oaltL -It.t t\)T:::,f:li* plY<tt) d)P(g>la),e4dt- to'Yo b) s., e9e ,F q, OaJt.2' €) i) rc=vo )g, i @)l Comment, giving a reason, on the reliabiliry of your estimate. 6) sbs,. , o"t O.3+ b) State, giving a reason, whether or not your value ofthe correlation coefficient supports the researcher's belief. (1) tla a\1 The resealcher believes that a linear regression model may be appropriate to describe these data. (d) of Calculate the product moment correlation coefficient for these data. rtIr,?)i.kr? Q) 4:';. ?(y<Yrt+)ro-lo ^^rl P(y<Alt<ra-) ; P(_!2) ^ = o-t.,1; lErt) o'o cj The table below shows the distances (to the nearest km) travelled to work bv the 50 i:-__l Distance (km) Frequency (l) Distance midpoint (x) 0-2 3-5 t6 I.2s t2 )zslt^ 4 6-10 l0 I -20 21-40 8 15.5 4 30.5 1t [You may use f ft:394, (_) The bar representing the distance of 3 5 ?.$ -2L=16. ' * i+rS has a width of 1.5 cm and aheight of 6 cm. of 6 - t5 Q;z's: Oz 2s L3 10 (2) Show that an estimate of the rnean distance travelled to work is 7.88 km. Ctt ig Ifr=.g, iil ^>o2 l;. tterd{'ra t-Scre 9*a F *n- ,i: Use linear interpolation to estimate the median distance travelled to work. (ii) Chlr S I fx']:6500] Calculate the width and height ofthe bar representing the distance (c) (i) ,J A, (3) (b) ,fl6ss. '=-".1t *'. A histogram has been drawn to represent these data. (a) S's 2.s emp'loyees in an office. A= ?'Jra,rz .'. h'icaa f '-9 1 \ *4*5i 1 Gr -Tei\ zTtt : Jrxr6soo-3jvz+so .gg1s.zt e,=[t ,t,riu Estimate the standard deviation ofthe distances travelled to work. (4) (d) Describe, giving a reason, the skewness ofthese data. (2) Peng starts to work in this office as the 51't employee. She travels a distance of 7.88 km to work. (e) Mthout carrying out any furdrer calculations, state, giving a reason, what effect Peng's addition to the workforce would have on your estimates ofthe (i) mean, (ll) medran. (iii) standard deviation ofthe distances travelled to work. (3) e) i) no cl,r.cr,r)t ,7.88 -- tuecrur ii) strqWf th<r^c^,c- , \=zs..S#^ Sl, ii\) rhy raJuc.*ure. g(x-::..j) ur.rc&r.rrer[ IAJ^ \ ta*r. tnr,rnera.rgg\. brX t. . ; 5. The discrete random variable Xhas the following probability distribution ,7 s-l-* x P(X:x) a 0 2 4 b a c One event at Pentor sports day is throwing a tennis ball. The distance a child throws a tennis ball is rnodelled by a normal distribution with mean 32 m and standard deviation 12 m. Any child who throws the tennis ball more than 50 m is awarded a goid certificate. (a) Show that, to 3 significant {igures, 6.68% of children are awarded a gold cerlificate. (3) where a, b and c are probabilities. \ Given that E(lO = 0.8 A silver certificate is awarded to any child who tfuows the tennis ball more than (a) find the value ofc. (b) (b) find : 5 find the value of a andthe value of 6, Var@J (c) (2) The random variable the value ofd Three children are selected at random from those who take part in the throwing a tennis ball event. (4) (c) Find the probability that 1 is awarded a gold certificate and cefiificates. Give your answer to 2 significant figures. E(r) 7u 5t Var(r) Q) (0 P(r> are awarded silver (3) P(A'io)' ?L:."B'") = ?(Z>l-S) = r-((rs) f3lL (1) (e) 2 Y- 5 -3X Find (d) metres Given that 1 9. I % of the children are awarded a siiver certificate, (2) Given also that E(X'z) d but less than 50 m. 5) P( D>d ) 0) (4) a) -:-- .@ te .l -) Ln +tr+C s t = S ta= l'8 -._ g = O,3S , - c) VCx)a 6(xl)-€(f)'g S-O't' +36 o.rql td) * + oset s o'1rs22- b) P(2, €t )'o'le\ -.zL b) 6(X")= {a*4r*+16o -- q 3o'Z =ory O.ZSa{ )'a o:6tr :o'o6br = o'r5*8 -'. -: j: i a' S-3(o'r1 G, V(S-gl)g nV(F)' 37.t11 e a) 6 (S-3x)" = ?'b r'b / +) ?(52,o), a"tc = O. \tLS i o) CrSS x 3 , -3 r s "L.l$!r z O.O66t ). O.tql o.oo1ts I 6- The Venn diagram below shows the probabilities of customers having various combinations ofa starteq main course or dessert at Polly's restaurant' S: M : D: , (c) Find (i) p(n I lrn s) (iD e1o l,lrn s'y (4) (2) (4) has one evening 63 customers are booked into Polly',s restaurant for an of{ice party. Polly asked for their starter and main course orders before they arrive' Of these 63 customers 27 ordered a main course and a stafter, 36 ordered a main course without a starter. (d) Estimate the number of desserts that these 63 customers ii) P d)! .{r tt * ( lt nr,.s') ro.s* roaT| =Lo Given that the events S and D are statistically independent Hence find the value of4. P(s^D) (o'st+p) x o'is g. o,61'- 2 c o-3' lAe z o.?.? O i ) P(o lrqre) , g:9 o.L+ ---' the event a customer has a dessert. {b) s b) the event a customer has a main course find the value ofP. fcs) r f(o) _: the event a customer has a starter. (a) c[) \ will have' (2) 3 gs . o.2l2,!l o.$V 2 / '"''J