ii) fl(alzlo)=l-f.(aoeq,), r-o.sr+q, + c'9f2s

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1 sl stq utl- 1. Patients arrive at a hospital accident and emergency department at random at a tate of 6 per hour. (a) Find the probability that, during any 90 minute period, the number of patients arriving at the hospital accident and emergency department is (D exactly 7 (ii) at least 10 (s) A patient arrives at a.m. (b) Find the probability that the next patient arrives before a.m. (3) -d---xt--=-t+ P*bte,^Jn a.,,rrrwt*an-pr-r I oarr'a- X* f!*[:il- - ii) fl(alzlo)=l-f.(aoeq,), r-o.sr+q, + c'9f2s? {&t. an-pu(t,9 U vlnl af 0 l(rrt*

2 ) The length of time, in minutes, that a customer queues in a Post Offrce is a variable, 7, with probability density function [c(81 -r'z) 0<r<9 f(r; = i 0 otherwise random wherecisaconstant' (a) Show that the value of r i, --1- (4) (b) Show that the cumulative distribution function F(r) is given by 0 f F(/) = t <0 0<r<9 t>9 (c) Find the probability that a customer will queue for longer than 3 minutes' A customer has been queueing for 3 minutes' (d) Find the probability that this customer willbe queueing for at least 7 minutes. (3) Three customers are selected at random' (e) Find the probability that exactly 2 of them had to queue for longer than 3 minutes' (3) -c-\,jo c[q 8L-La{r --\ cf rll-&bt]} =\ rt :" ) i;a ; iai( e*sia-ot- sq -r-*-*--c;q

3 ct P(u:s)--il:fJr), # a) P( t>t rffi_ l- F(a g) le =-J*--Cr;otoprcz+, toc-tan9 Lo.,r.ger #..a.,.- 3n^.-r?tr* [)'(g): egru L

4 3. A company claims that it receives s at a mean rate of 2 every 5 minutes' (a) Give two reasons why a Poisson distribution coold be a suitable model for the number of s received. (b) Using a 5oh level of significance, find the critical region for a two-tailed hypothesis that the mean number of s received in a 10 minute period piouuuitity of rejection in each tail should be as close as possible to test of the is 4. The (c) Find the actual level of significance of this test' To test this claim, the number of s received in a random 10 minute period recorded. was During this period 8 s were received' (d) comment on the company's claim in the light of this value. Justify your answer. During a randomly selected 15 minutes of play in the wimbledon Men's Toumament fnal,2 s were received by the company' Tennis (e) Test, at the l}oh level of significance, whether or not the mean rate of s received by the company during ttre WlmUtedon Men's Tennis Toumament final is lower than the mean rate received at other times. State your hypotheses clearly' (s) --s)-@ -- r'a c-e-rue-a ncr'a?orr'r'\i, lw)@ Srn"y; \cvlr1 ;u*)- ct cczrr'sh,rl.tt rtl&e--- -D-La-*-eoa.Jr-Pr.- Ho: ).=$ [Oor.rr- :(.^-f"(:t)- - 0hLJ-+so.o2l_- - - * K*'.s) g o-'-o-l)-- it-rxy*-ll:=t-o)-*o,o163-t l(x> r-r) 3 t- P(x.lu flt*lu"_o.ogt6 0(a-( u-t).-o.1tj L:_()- 01aJ+) -= o.q{+rl P(rgf ) * = e.1_+6_b ca I rc, o]-u [x r, q? - Lt- t = t -'- rr--at ";adl_"- fffi,---:a g-fr1r

5 -dj -L3;n^ads--doel'n/F.r,r.L r.a-3lg-' C..tsca{_-re-5aer fr,r,\.it rt"clr --LS-- SttrAstccrj.r*-r Sro\ wt(acr",^-l- --; -- -i- j- ndh e-r.o,q\r- -q-u*acalrn- to ebe,c,tr ntrll- qfug"r. &,rt4atrco-- fo s.:?bart c,tar-,^^- -?2per*Sa..r,*rr--- t{' : A < 6 - :-- fezrrrib. rq stalrc\ qqraf_ a.,t fleloy:_cs ( erro-t -=-.-*.ana.:r:"q,tr1-e'.rr-)e,u\r-9-,-o* Ca*st _-h{odge*u*s*j ^.,ft e{*4!3 t!--erae*"c--i-1prl(*rr, tup-r.ra_:_-sugl loutr*rf--*2*ff.rrra) d,rs- he..^.r 4"...^I. s

6 4. A cadet fires shots at atarget at distances ranging fiom 25 m to 90 m. The probability of hitting the target with a single shot isp. When firing-from a distance dm, p= ]1eO - a1 200' Each shot is fired independently. The cadet fires 10 shots from a distance of 40 m. (a) (D Find the probability that exactly 6 shots hit the target' (ii) Find the probability that at least 8 shots hit the target' (s) The cadet fires 20 shots from a distance of xm. (b) Find, to the nearest integer, the value of x if the cadet has an 800/o chance of hitting the target at least once. $) The cadet fires 100 shots from 25 m. (c) Using a suitable approximation, estimate the probability that at least 95 of these shots hit the target. (s) -e)jj- "l'= 4o I Q, 9:?5 rc ^- B( toz O, t-5_)- -- f = l* ltrii'ls r*q o'ts( 6'?( q o o.\ tt6 c> il) P(a --2_ *z*ffi256-9 ^'*61ro;o _*e**e;-+pq$ t)_" cjs356-1r- ---L( c- ;l) -= aeo --:- t-? (5,o), o'k -

7 :-al= 8*--J q S -- - c) _V s S hrb trsrm tg0stvlrs a*_2saa _ W = * rvtrsse.s troan t@shcla a:t L) -f(vzqs):""pfujs 2'$.- P{v:r_@ 2,lP ( to So TotrJon' bcstrttubu-. ln+-^ofirnar- - ) o.a.l t t1^..^c.- l&rs rs- S o.lt bb Stnro- F 0< o. S a^ll o. 1+s rs u"l L.eL\

8 5. (a) State the conditions under which the notmal distribution may be used approximation to the binomial distribution. as an A company sells seeds and claims that 55o/o of its pea seeds germinate. (b) Write down a reason why the company should not justifiz their claim by pea seeds they produce. testing all the (1) To test the company's claim, a random sample of 220 pea seeds was planted' (c) State the hypotheses for a two-tailed test of the company's claim. (1) Given that 135 of the 220 pea seeds germinated, (d) use a normal approximation to test, at the 5oh level of significance, whether or not the company's claim is justified. (7) ywzi {-. Se-Crr- S;r-rt..,.a S,e-& " -a)--p{:e.i2-rss) w Ptae>r t? (x- > 'n*).4-d_ s,tf.6pe 171 -_;o:nfr:ef11'x - l*lc-.cr^*f -JD bc-*-sla;ta.r&rr-c.-n-f -----;--refii5;ral5.:---s1.1rflevirf-*o-lrlpptr-t---gtacrc-ttxzr+-tr.:s-s-*-

9 slg_ G -r4?dk-nul( rh ((x zr3g=t)i_0fulflulj.-lil_l_ to7'f l?-l t3*', fqlb I b^.\ t"-t - \ a"-$l tl.s,r.lr b{ ^.,zef.a:- -l (Z zr-gg).- r-4(r-s3)-.-q.cs36- z.-jt. :-qedr--n ut-t--t1? otz.crs- -?U \A[,J\cr- S\r19e^A P > O'SS

10 6. The continuous random variable Xhas probability density function f(x) given by f(x) = + o(r(1? t<x<4 9 2x (x(6 otherwise (a) Find E(,rS. (4) (b) Find the cumulative distribution function F(x) for all vaiues of x. (6) (c) Find the median of X. (3) (d) Describe the skewness. Give a reason for your answer. ec).) =.Ft6.-))rc?;g "-LH - rr;t-r4--ft*i,-f{+) -{ *+ ov--, t * Le"+"+"-i

11 1_0_ J -z x- < o O-S*t oy nmurei. Ehofte,t 0Ottr"*--Sl"re^l

e) r a9".^) -FaBn^- &* B(pl hr_lma:n u

e) r a9.^) -FaBn^- &* B(*pl *hr_*lma:n* u SL SIc1- tal 1. (a) State the conditions under which the Poisson distribution may be used as an approximation to the binomial distribution. (1) A farmer supplies a bakery with eggs. The manager of the