UNIVERSITY OF SWAZILAND MAIN EXAMINATION PAPER 2016 PROBABILITY AND STATISTICS ANSWER ANY FIVE QUESTIONS.

Size: px

Start display at page:

Download "UNIVERSITY OF SWAZILAND MAIN EXAMINATION PAPER 2016 PROBABILITY AND STATISTICS ANSWER ANY FIVE QUESTIONS."

Stewart Hamilton
5 years ago
Views:

1 UNIVERSITY OF SWAZILAND MAIN EXAMINATION PAPER 2016 TITLE OF PAPER PROBABILITY AND STATISTICS COURSE CODE EE301 TIME ALLOWED 3 HOURS INSTRUCTIONS ANSWER ANY FIVE QUESTIONS. REQUIREMENTS SCIENTIFIC CALCULATOR AND STATISTICAL TABLES. Page 1 of 4

2 Question 1 In a study conducted by the Department of Mechanical Engineering at a univesrsity, the steel rods supplied by two different companies were compared. Ten sample springs were made out of the steel rods supplied by each company and a measure of flexibility was recorded for each. The data are as follows: Company A: CompanyB: a) Calculate the sample mean, median, and variance for the data for the two companies. (4+4+4 Marks) b) Calculate the coefficient of variation for the two companies and comment. (8 Marks) Question 2 a) Interest centres on the life of an electronic component. Suppose it is known that the probability that the component survives for more than 6000 hours is Suppose also that the probability that the component survives no longer than 4000 hours is (i) What is the probability that the life ofthe component is less than or equal to 6000 hours? Oi) What is the probability that the life is greater than 4000 hours? (3+3 Marks) b) A regional telephone company operates three identical relay stations at different locations. During a one year period, the number of malfunctions reported by each station and the causes are shown below. Station A B c Problems with electricity supplied 2 1 Computer malfunction Malfunctioning electrical equipment Caused by other human errors Suppose that a malfunction was reported and it was found to be caused by other human errors. What is the probability that it came from station C? (14Marks) Question 3 Magnetron tubes are produced from an automated assembly line. A sampling plan is used periodically to assess quality on the lengths of the tubes. This measurement is subject to uncertainty. It is thought that the probability that a random tube meets length specification is A sampling plan is used in which the lengths of 5 random tubes are measured. Page 2 of4

3 a) Show that the probability function ofy, the number out of 5 that meet length specification, is given by the binomial discrete probability function. b) Suppose random selections are made off the line and 3 are outside specifications. Use probability function above either to support or refute the conjecture that the probability is 0.99 that a single tube meets specifications. Question 4 If a dealer's profit, in units of $5000, on a new automobile can bo looked upon as a random variable X having the density function I(x) ;:;:: 2(1 - x), 0 < x < 1 a) Find the average profit per automobile. (4 Marks) b) What is the dealer's average profit and standard deviation per automobile if the profit on each automobile is given by g(x) ;:::: X2. (6+5+5 Marks) Question 5 a) Derive the mean and variance of a probability density function of this form; b) Suppose that the service life, in years, of a hearing aid battery is a random variable having a Weibull distribution with a = 1/2 and fj = 2. (i) (ii) How long can such a battery be expected to last? What is the probability that such a battery will be operating after 2 years? Question 6 (5+5 Marks) a) A machine is producing metal pieces that are cylindrical in shape. A sample of pieces is taken and the diameters are 1.01, 0.97, 1.03, 1.04, 0.99, 0.98, 0.99, 1.01, and 1.03 centimetres. Find a 99% confidence interval for the mean diameter and variance of pieces from this machine, assuming an approximate normal distribution. b) The following measurements were recorded for the drying time, in hours, of a certain brand of latex paint: Assuming that the measurements represent a random sample from a normal population, find the 99% tolerance limits that will contain 95% of the drying times. Page 3 of4

4 Question 7 A random sample of nl = 32 specimens of cold-rolled steel give average strength Xl = 29.8 ksi, and sample standard deviation ofsl = 4. A random sample of n2 = 35 specimens of two-sided galvanized steel give average X 2 = 34.7 ksi, and S2 = a) Find the 99% confidence interval for b) Are the strengths of the two types of steel different at a = (8 Marks) (12 Marks) Page 4 of4

5 Normal Distribution Table C-1. Cumulative Probabilities ofthe Standard Normal Distribution. Entry i:> i<lfca A under the standard normal l,;urve from "" to z(/\),,>': :>:" ~ z(a) 1:. : ).04-, ()t(.0').( i9J )(1.51D ,91,50.1:10.S4'lt.5&7J : BOO ll i 'J<) 5596.s ( )9.~2' :l3S l4t Q!.\.b~ R.9.691: "1580.7Btll,$159.69S QIO :' ~:l~ 'J<,7.82.~ M.7995,1U64.7()8~ ), *51.7( S '82:; /S051, SJ40.8J65 JB I.t t.2 U. 1.4.Mll.3M ,X4:Hi.IHo(.5.MH S JU!88.906(, K S0S,t172.9, St S:; :.tUIIO.8830.S ')015.'il ; ,9162.'il7?.927(j.9292, t.s 1.6 1,7 l.8 & ,.971 :l ; <) s1j x4.9s '.l4'l5,1)591.1)( s ~ ( :515.'n2.5.!uj' ')f.i t)93.9t.'i i.lo.91:)6 \17tH !~ (j'17K 'j8JO ," , S-lIl.'AI,! S , J , : )74.99!1I 'Jlll ' ,')') ,S/'9(ttl, 'M5.99~9,996? % l! S % '> Z.997) '191<1.99BS !J<J9S J i1.999& ,9997.9'JIt& ' '.1(i, '>1.999() S.999~.999S,99% '99(j '7.99')1:1 Volume II. Appendix C: page 2

6 Chi-Square Distribution Table C-2. Percentiles of the X 2 Distribution 1~ll!r)' is x 2 (A.: /I) whoere P L\'~(v) ::;.\,2{A; ll.l} "" A ~ x~( It; II).4 )' J)(}5.010.IllS.oso.too.900.9SO.975,\ {l.o 391 D.O" 157 O.{l J 982 (H)t393 {I,CJSlI l JJ , 0.()l OJ)j(}(J 'Ut OJ IS , HI 9.35 lj.j cnm I,(1M ],t ~ l~ 1< ) 24 2~ () } ':I() 100 l D.9111) I U4 I.iJ 2-16 HO 3m 1~ J-o lli ltoj X9 IOJl \ \.1\ 12.4& 1l ) ,.11 0.)4 0.8" ~(1 }O~ (,6 -".23.) ,01 1.6J. S S fI.!Ifl 1L && fj '15 Zl.lf> 29.7f 37, S j, t ;ua S.OI HiJ o.2fi )6 \I.Z} B.91 ').$<) 10.1& 1(J.9~ n.4u tj.ll IS , &.w.48 4< [ S ,22 l z.n l.l3 3,') sn '.HI) 6.S % lui? 9.J9 10, IU<J )'()9 IH~ I UI \6.\S 1{S,9J llu9 26.SI t9 jl.74 6(l :13 Lfli 2.1(1 2.Rl lH D8 (do Ht ~ IQ , L5.{) UUI l?,77!o.!oo 2').(1$ % S:CIJ (>I.2M 73.;.!9 8~.~ IO.M 12.fr L~, IS.H 1'HI 2H) U~ ~.\l9 27~ZO 2~t H run )4.38 luti Hi]l :W.09 4Cl.26 ~I.U 6], '.'.1 9(.,S8 HH.6 JlIU IL IH)9 16.7S 11.S9 14.4S 1.6,tH l8.5s 14, '1 [ lll :M ;!O.ol~ :5, ll :14 2&.:42 2l! J S1 23./i.S HIO O.S! 3211< UI $9 ;0.19 3J.41 3~,1l S) 34.S1 37.\6 JO llai JUi7 J.t4<'l.Jb' " f>.18 4( JS ,{)4 44.t8 l6.42 3'1.J6 429~ 4~.:5~ 37.6S 4O.6j 4UL 4M3 ls.s9 4.I.n \ d. 4U '.28,ro.99 42,:56 45.n 41}.s<) " & SD fJ Tl /9.08 XUI) SS.Jll 'JO.,,, 9j.f)':! IIXl.4 11)4.2 Hi] ]2.1 IHiJ Illl 1] ').6 13: Volume II. Appendix C: page 3

7 Student's Distribution (tdistribution) Table C-4 Percentiles of the t Distribution Entry is r(a; p) \l\o11ere P{t(v) :S t(a; v}j = A ~,.. '.,,. < ~~. -, t(a~ 10') A ' J (178 6.~ " :'1: () UU 4 o::rn \ 1.\ :; 2.\l1 l.tlb S S lS O.26S : LH9 1.41S IS [" IOS OS O.l6(l 0.~4t UJ : w <l.2s O.2S ' S O.2~ 0.2~ O.l.56 O.2!56 0.2: O.2S O.2:5;'} (t.535 G :\ :53::! OS O.~JO <) ' O.'86S "1 O.S6J O.8!i9 O.S5S o.ass 0.83' o.gs(; O.iSS O.~I O.MI!I O.M~ ~ 1.071} urn f.? J OS<J l.o'ik S I.OSS t.oso '.045 l.04l 1.0.] !SO 1.34' LJ3Q l.lls 1., I.J L:31~ LH Jl I.J0e L t, :lS l l L(> I.MS ns ;:1: :U) r.9t1o Volume II, Appendix C: page 6

8 Table C-4 (Continued) Percentiles of the t Distribution v.~nf,985.9').4l92." ' J :'L {to?) U9S l :2'> l S S , J.l"} m S.9SIJ S l :\,4<;,1' S ' QK /81 1Q 2.'3S~ 2: J i :tl(}i6 J Q ! 2Jl ' MQ 2.1!()\ 3.01' \ l Z.249 l.3')7 2.00l :U,Sti Hi ;UIn '2.92' ( %5 t ()8') ' ,381 ' ? <i J 2.\89 4, ".M'; IH'l l.u? U ::U (1)\ () ' '2. 12% ( & l S '2S{) :'U CJ , <\ (X> 2.OS4 Z.l : A Volume II, Appendix C: page 7

UNIVERSITY OF SWAZILAND MAIN EXAMINATION PAPER 2015 PROBABILITY AND STATISTICS ANSWER ANY FIVE QUESTIONS.

UNIVERSITY OF SWAZILAND MAIN EXAMINATION PAPER 2015 TITLE OF PAPER PROBABILITY AND STATISTICS COURSE CODE EE301 TIME ALLOWED 3 HOURS INSTRUCTIONS ANSWER ANY FIVE QUESTIONS. REQUIREMENTS SCIENTIFIC CALCULATOR