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 bakery claims that the proportion of eggs having a double yolk is 0.009 The farmer claims that the proportion of his eggs having a double yolk is more than 0.009 (b) State suitable hypotheses for testing these claims. (1) In a batch of 500 eggs the baker records 9 eggs with a double yolk. (c) Using a suitable approximation, test atthe 5Yo level of significance whether or not this supports the farmer's claim. (s) e) r a9".^) -FaBn^- &* B(*pl *hr_*lma:n* u b) Hol A'0:0Ou)-.tk i )r>-u -oo5 cltu Po(o.0m) 7- = th an* u,rrtt^* o Jo"f*r* U'lt*--' '- c) **B(smy&ooa)-B t,-0o [+.s) 0(r", t) P_(x>s),-i-ff-eg) ;l - A*'t-? # {-t* prsrch*\s hi1!^.or o &!ct LLer^,.
The amount of flour used by a factory in a week is Ithousand kg where lhas probability density function (a) Show that the value of k is 1 Use algebraic integration to find r(v)= [og-r'l o(v(2 \// O otherwise (4) (b) the mean number of kilograms of flour used by the factory in a week, (4) (c) the standard deviation of the number of kilograms of flour used by the factory in a week, (s) at) (d) the probability that more than 1500 kg of flour will be used by the factory next week. f6p'r 4 t^jit.-.4 11 =t --lr Iu\-tt']I]\ a) his- tj z r- ali$;,i -"- tt? *t D--ei1), JiUt, J:*rfoo* - lqrn - *l"r-r, --;afeuitsolj = 6) -e(t');ji,frr)ry ---r!fl: t =ff:4r"t v (.r) = Q(S)- (r)' = +- (t)' J1+ =L* v uy=-t-gr-tlylj - t -{4J- =* :- gy=,l$ = o.j{+ "#tet'- ot'v.l s t'f ;, 0- fq:is), t- FGi):. I - ['s 3i*fl45 I al- trt-*fj';' Je.ossq i t_-
The continuous random variable 7 is uniformly distributed on the interval la,pl where B > a Given that E(fl :2 and,var(o : {, frrrd 3' (a) the value of a and the value of B, (b) P(7 < 3.4) kil sl '-) * =2--r), v(r);8 " (d, ',* - 6*-L_/tb= L,-> + u\-f (r.3,q) = 39-. s. \. g T : S'6}z3 a- --
4. Pieces of ribbon are cut to length Z cm where.l - N(7-r, 0.52) (a) Given that 30Yo of the pieces of ribbon have length more than 100 cm, find the value of p to the nearest 0.1 cm. John selects 12 pieces of ribbon at random. (b) Find the probability that fewer than 3 of these pieces of ribbon have length more than 100 cm. Aditi selects 400 pieces of ribbon at random. (c) Using a suitable approximation, find the probability that more than 127 of these pieces of ribbon will have lengih more than 100 cm. (6) 6) f ( r- >ltx)), O'3O z) eq? tq-f ) =o'3o Lt /, paraf,r*+.q* E o.s*4.'. l^= ai "+ *-"- -7- b) CC = t? pre,ta-\n'r rbhan \ll^.r,d^ ^^-r: "-t&l^ ilb(qo3) P(a<3),PL'x.ez) : Q.ZSZ8 c) /^rr.e ; lzo e'b(+oro 3) o",,np-(r-p)'- 8y r) g{ &r', N (raor-g-rr)
5. A company claims that 35o/o of its peas germinate. In order to test this claim Ann decides to plant 15 of these peas and record the number which germinate. (a) (i) State suitable hypotheses for a two-tailed test of this claim. (ii) Using a 5o/o level of significance, find an appropriate critical region for this test. The probability in each of the tails should be as close to 2.5o/o as possible. (4) (b) Axn found that 8 of the 15 peas germinated. State whether or not the company's claim is supported. Give a reason for your answer. (2) (c) State the actual significance level of this test. (1) i H,o '.? -- o.3j t{t )? fo.3s li)?(v( L) eo-ozs?{x,,tr)eos}s P&i1),0,06(+ :.p(lg u-t)*o.1i{ j. Lr_t &x gr ).= o.g l+r * _: r^_t={ 0 (z* t ) = oo tvl*? (f>-rra)r I-P{rg u-t)'uoo?s - -t [o* rcs-su trosrflsi l, 8-0,oe+ -rnd" (all- ru*lro- t!a,.- C(L :- t{&- c** t9- n(k Sr-gnr*4.t ca+j-" rr.ilf kto.gg\,l*c.u.sfle(c.- hr-c3eck nr,u4- :- c(*r,^.r Soppl^f*eA.. _d *A_SL_.r O:-ol!r2 * c_ 7). C-236_ -2.66{ - O'O t2-t+
6. A continuous random variable Xhas cumulative distribution function F(x) given by F(x) = 0 x(0 2 \(g-z*l o(x(2 20' x)2 (a) Verifli that the median ofxlies between 1.23 and 1.24 (b) Specifi tully the probability density function f(x). (c) Find the mode ofx. (d) Describe the skewness of this distribution. Justify your answer.,.\ F ( r-es) z Q 'q qs (-0'{ - Srnro- f(ad;o:{ -.' F(f -L\), 0-- 0-Q-! >,0.S -- (.28,(Oz c t:la (2) e),,) +(d=$xr); #(qi_- #l " *1r.-H.4 c)_,) 'i{-x- ao -:. f\aor -= f :S r\a"oje, :. f\oj,&*,^r*, 6lr*.r*
7. Flaws occur at random in a particular type of material at a mean rate of 2 per 50 m. (a) Find the probability that in a randomly chosen 50 m length of this material there will be exactly 5 flaws. (2) This material is sold in rolls of length 200 m. Susie buys 4 rolls of this material. (b) Find the probability that only one of these rolls will have fewer thant flaws. (6) A piece of this material of length x m is produced. Using a normal approximation, the probability that this piece of material contains fewer than26 flaws is 0.5398 (c) Find the value of x. (8) P(z=S), L-n *Ls.= O.o3bl = -il 3 =-U. f(qo,r &dr?jdna i-ke) * p(t.+):itgso = o.3r3t ( l* ootts w rll& <.[(""r^-1 ffb (4, O ;3t3O qcio'36+' Pftr:\) = 0'6r6f,,.= o,40s6
fl\ = * flooos W OCcn ffirv P, (A,-) A" = *.p(m.g.. p(m(zs.s) t rh- N(tr'/A{ Ph -. as) y? (z< as-'ld') = 0's31t 0(o't)=0's31s' -)\ / 2S.S -A' = O"\ *> Az + O.r tr - 2S.S : () -X r--a (9 loa'+\-zss: o =ffi> ^=.-l?.r) (rrl+sl X\-s).: A,=5 z) 25= -L ',a 1? L'- 625 n^r Lr2.