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1 s2 sg2- l. The manager of a leisure club is considering a change to the club rules. The club has a large mimbership and the manager wants to take the views of the members into consideration before deciding whether or not to make the change' (a)explainbrieflywhythemanagermightprefertouseasamplesurveyratherthan a census to obtain the views. (b) Suggest a suitable sampling frame (c) ldentill the sampling units. c) b) ca...-.!p../ / 1*, r,kj l-.-rlltr.^,^..1^se- rr--9 ('f1 N [ra-," ol,a]' d\^.- A/$-{ ^16-14, +b Cgx )"* Jurr-t-La.V a.".^c( C) 'tt^<- r,\^.ol^l/t.r^^ 2. Arandomsample,Yl,Xz,..., & is taken from a finite population. A statistic based on this sample. (a) Explain what you understand by the statistic L (D) Give an example of a statistic. (c) Explain what you understand by the sampling distribution of L I', is (l) Or) tf'u<rctr.a"a. 1 ttla- re\,^/-aar\ Vor.r.tcylal.a 66ry.hlt1,l'tn\ r\e,.-rv{,r,rtatte,r. prvrn^ofc,r'r. b) u\-\grvr,'!, Z* A. c.) P,a."L)t^f-^ E-^shr.b.f+ur^ c^v. o'-)- p,*^,v- S c,r-rrek-.r The continuous random variable R is uniformly distributed on the intervar d < R _< /3. Given that E(R):3 and Var(R): f. find (a) the value of aand the value ofb, (r) P(R < 6 6) L --3 2S 7 (-ofs loo ^rb= b 6g6-b O P UL<b'b) = 1t =Y; =[r- te-u] =\cd 2b- 6= to 2tr = L6 b=8 0.= -?- "

2 A magazine has a large number of subscribers who each pay a membership fee that is due on ianuary lst each year. Not all subscribers pay their fee by the due date. Based on correspondence from the subscribers, the editor ofthe magazine believes that 40o/o ofsubscribers wish io change the name of the magazine. Before making this change the editor decides to carry out a samp1. survey to obtain the opinions of the subscribers. He uses only those members who have paid their fee on time. (a) Define the population associated with the magazine. Suggest a suitable sampling frame for the survey. Identifu the sampling units. Give one advantage and one disadvantage that would have resulted from the editor using a census rather than a sample survey. As a pilot study the editor took a random sample of25 subscribers. (e) Assuming that the editor's belief is correct, find the probability that exactly 10 of these subscribers agreed with changing the name. (3) In fact only 6 subscribers agreed to the name being changed. ff) Stating your hypotheses clearly test, at the 5oh level of significance, whether or not the percentage agreeing to the change is less that the editor believes. (s) The full survey is to be carried out using 200 randomly chosen subscribers. (t) q)*(\suh&r\k/j b) SoUs.rrbefJ ui^" h$^,qf/n^&, c) fnc,nthe^t dho,\ar.?*l d) of ts - trvrrc- Con$:rrrtt nq /n"ana- ery&rnru 0rI,/ -1y1ry,.aocrrmjt (g) Again assuming the editor's belief to be correct and using a suitable approximation, find the probability that in this sample there will be least 71 but fewer than 83 subscribers who agree to the name being changed. (7) e) &*b(ls/o-.k) e) F(o: ' Ht: ((ccs (o) = f 'o'.* P<o.* (H)o *roo'6ts = o.16\\ f (tse) = >O'OS -'- n$rs,o1'r...t..*g,.'- n6l" e,lnovll tat$c,nq,to nflh^r" g) t(61 c4t( W ; 60.: G)rbC" ekrtsr,,1 1-5 S,"\PP\ruTEfL q LvNtSO/qE) +6 /( rt..< r< 8'3) c' p( ro.sg)c ( Gz.s)?(to< L( rz). 0'6{Ob- O.O gs3 = 0.gSS

3 4. Past records show that 207o of customers who buy crisps from a large supermarket buy them in single packets. During a particular day a random sample of25 customers who had bought crisps was taken and 2 ofthem had bought them in single packets. (a) Use these data to test, at the 57o level of significance, whether or not the percentage ofcustomers who bought crisps in single packets that day was lower than usual. State your hypotheses clearly. At the same supermarket, the manager thinks that the probability of a customer buying a bumper pack ofcrisps is To test whether or not this hypothesis is true the manager decides to take a random sample of300 customers. (D) Stating your hypotheses clearly, find the critical region to enable the manager to test whether or not there is evidence that the probability is different from The probability for each tail ofthe region should be as close as possible to 2.5%. (c) Write down the significance level of this test. (l) a) ol ^, B( rsro.z) Ho: P 'O'L Hr :? <O't -". r\5t' Sto)a. Frco^.,^-h :- {rrtu g,tro.rll e'cl'&e'rc'c- to "'f,e & nd-i' )- n& e^r..oy e*t}c;r,,-o^ \o s..'19er'& -[' w 6ffibl )*B(3P/ o'ss) *?' R -"' t3* /o Co) Ho. tr"( {t. : cr-(oc 3e Asu, ^1t"1 s\ul O. ozl?- o.oljto { r-l'2) i o'oq8'?- 4?q-' L) x o'oej P( q-s 3), O'o Lt:-* P( $( q) 'o'osso u-3 r6 sy s 3cD3 Q'szt "?(\r,u){d-ots f (qi u-t) +fturrq-t) Y o < Ptq s.,t-t)c( oqar / kys rs), o'?s?j Pt'.f s tj) 'o '1? 80 T -,-,X- l, l, -'- U--(L

4 5. A garden centre sells cun.r of no-lnagngtt,-sd"*.1h..ur", are bought from a supplier who uses a machine to cut canes of lenglh L where L - N(p, 0.3r). (a) Find the value ofp, to the nearest 0.1 cm, such that there is only a 50% chance that a cane supplied to the garden centre wilr have rength less than 150 cm. (4) A customer buys 10 ofthese canes from the garden centre. Findtheprobabilitythatatmost2ofthecaneshavelengthlessthanl50"-.,r, Another customer buys 500 canes. (c) Using a suitable approximation, find the probability that fewer than 35 of the canes will have lenglh less than i50 cm. 6) l- r- N(r, o'3l) P ( I < lso) = o'os?(z< '%)= 5Y- 0'3 : lr+ 9.s ^^ b) - f n, buo,o"os) P( f z) o.gsss e) f,ru 6( sod,o,os) M=np :s0) ro'(x =fl$ oa--o'df 0) = as(o'1s) 2 a3'+s t' {'rr'trt (ts, 23'+J) P(x' p(rr-.fi1 clp(t-<s1.s)s?q't#, )o?(-, r 1s) = e-g? Qg+

5 o/) From past records, a manufacturer of twine knows that faults occur in the twine at random and at a rate of 1.5 per 25 m. (a) Find the probability that in a randomly chosen 25 m length of twine there will tre exactly 4 faults. e) The twine is usually sold in balls of length 100 m. A customer buys three balls of twine. Find the probability that only one of them will have fewer than 6 faults. As a special order a ball oftwine containing 500 m is produced. (c) Using a suitable approximation, find the probability that it will contain between 23.and33 faults inclusive. L, {autt$ Pgr 2Sm /*0oCt-s)? (x, q). {'-t ('s4 -zt '- o oq+olr (-- b) $= ba{f haa te'u tlon 6 fr,^l tys c),f =ftu,rb F,( t@m {-0o(g) l!{ t$) : o'+qj+ c(q(: )n" B(Bro.r++s+) P ( g. I) G)o.qts?to.SskAz 2 0- tnto E', < tr'?"(:o) xtri.n(so,:o) t. Gr" th Der 550rn P(L:'s bsss) p( 22< t( 6q) x( (ryo<t'39:f E" $?o = Qto'61) - Q6r'3?) = Q (o et) -Lt -Q(t a+)] = o) - O.OES3. o'$36 L

6 7. The continuous random variable Xhas probabilify- density function f(r) = l5-2 t5 2<x <7.!-4. rr,.rn 9 4s' 0, otherwise. (a) Sketch f(r) forall values ofx. (b) (i) Find expressions for the cumulative distribution function, F(x), for 0 < x < 2 andfort<.rr<10. (ii)showthar for2<x<7. F(x) : i i (3) o t I (iii) Specifz F(x) for x < 0 and for x > 10. (c) Find P(X<8.2). (d) Find, to 3 significant figures, E(,t). 0(rsz f(x) =!*,tio 2(r(1?(xlo t(r) = J{try4." frrl - [(r) + J** dt, I l, * ]l +,e.il-frt f tu) = F(-,)-Jl t_h*,. S+ [h._ *]] =F&L-*) -(s-f) = f;,.-f;;$ (8) (4) = [. to{'lf = fox-' Z. J- =B-R.. f (*) c) d) P Q --/o -L*z I 3D^' )?'*+' I g*-*-!r ti4s 1 (x sb.t) -- t rc. G)--Fer*)Jr -- : t*i.tsli- L ac( o (;;') o s rs2 Y{;) 7<a-<+ (ii) = t, "3 + (.*-H)?* x-s to r(i) t > [o (iii) L) = o-9lb n lj a,'jih ar t [- q r -!{a- - rl1 *x:-fo*"]* s 4'+r.2

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