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
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: 9.3 8.8 6.8 8.7 8.5 6.7 8.0 6.5 9.2 7.0 CompanyB: 11.09.8 9.9 10.2 10.1 9.7 11.0 11.1 10.2 9.6 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 0.42. Suppose also that the probability that the component survives no longer than 4000 hours is 0.04. (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 4 3 2 Malfunctioning electrical equipment 5 4 2 Caused by other human errors 7 7 5 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 0.99. A sampling plan is used in which the lengths of 5 random tubes are measured. Page 2 of4
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: 3.4 2.5 4.8 2.9 3.6 2.8 3.3 5.6 3.7 2.8 4.4 4.0 5.2 3.0 4.8 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
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 = 6.74. a) Find the 99% confidence interval for 111 112 b) Are the strengths of the two types of steel different at a = 0.01. (8 Marks) (12 Marks) Page 4 of4
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:. :00.01.02.0).04-,05.06.07.()t(.0').(1.1.2.3.4 5000.5398.5i9J.6179.6554..5040.54)(1.51D2.6217.6,91,50.1:10.S4'lt.5&7J.6255.6628.5120.5:511.5910.6293.6664 BOO.5557.5948.63ll.6700.5 i 'J<) 5596.s987.6368.(1736.52)9.~2'79.5319.:l3S9.5636.5675.5714.5753.6026.6064.6103.6l4t.6406.6443.6480.6517.6712.68Q!.\.b~.6379.5.6.1.R.9.691:5.1251."1580.7Btll,$159.69S0.7291.7611.7QIO.8186.6985.7:'124.7642.1~:l~.8212.7019.7357.7673.7'J<,7.82.~8.7054.1389.17M.7995,1U64.7()8~.7422.7734.802),.8189.7123.7*51.7(90.7224.74.54.7486.75.17.7S49.7764.7794.'82:;.71152./S051,8078.8106.8133.8315.SJ40.8J65 JB89 1.0 I.t t.2 U. 1.4.Mll.3M3.8849.9032.9192,X4:Hi.IHo(.5.MH69.9049.9207.S461.8686 JU!88.906(,.9222.6485.8708.K907.9082.9236.8S0S,t172.9,8925.9099.92St.11531.8749.8944.9115.9265.8554.8577.S:;99-.8621.8710.3790:.tUIIO.8830.S962.8980.8991.')015.'il ;11.9141,9162.'il7?.927(j.9292,9306.9119 t.s 1.6 1,7 l.8 &.9.9332.9452.9554.9641,.971 :l.9345.94;63.9.564.9649.971<).9351.9474.9s1j.9656.9726.9370.94x4.9s112.9664.9132.9382.'.l4'l5,1)591.1)(.71.9138.9394.9.50s.9599.96-78.~744.940(..94.8.9429.9441.9:515.'n2.5.!uj'.9.545.9608.')f.i6.9625.9633.9686.9t)93.9t.'i9.9706.9i.lo.91:)6 \17tH.9767 2.0 2.1 2.2 2.3 2.4!~712.9321.9861.9893.9918.(j'17K.9826.9864.9896.9920.9783.'j8JO.9868.9893.9922.9788.9834.9871.9901.9925.9793.9838.9875.9904.9927.97911.9842.9878.9906.9929.9803.9808.9812,"817.91146,9850.9854.91457.9S-lIl.'AI,!14.9887.9S90.9909,9911.9913.9916.993J.9932.9934.9936 2,5 2.6 2.1 2.8 2.9.9938 :9953.9965.91)74.99!1I.9940.9955.9966.9975.9'Jlll.9941.9956.9967.9'976.9982,')')43.9957,S/'9(ttl,9977.9983.9'M5.99~9,996?.9977.9984.9946.9%0.9970.991l!1.9984.994S.9949.9951.9952.9%1.9962.9963.9'>164.9971.997Z.997).9974.9979.9979.9980.'191<1.99BS.9985.9986.9986 3.0 3-.1 3.2 1.3 3.4.9987.9990.9993!J<J9S.9997.9981.999J.9993.9995.9997.991i1.999&.9994.999.5,9997.9'JIt&.9991.9994.9996.9997.9988.9992.9994.9996.'9997.9989.9992.9994.99'.1(i,9907.9939.998'>1.999().9990.9992.9992 9993.9991.9994.999S.999~.999S,99%.9996.9'99(j.9997.9997.9997.999'7.99')1:1 Volume II. Appendix C: page 2
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,\190.995 {l.o 391 D.O" 157 O.{l J 982 (H)t393 {I,CJSlI l.71 3.84 5JJ2 6.63 7.88 2, 0.()l00 0.0201 OJ)j(}(J 0.103 0211 4.61 5.99 7.311 'Ut 10.00 1 0.072 OJ IS 0.210 0.352 0,584 0.25 HI 9.35 lj.j4 12.84 4 cnm 0.297 0.484 0.111 I,(1M 7 78 9.49 11.14 ],t28 14.86 5 6 1 ~. 9 10 11 12 11 14 15 16 11 l~ 1<1 20 21 21 2) 24 2~ 26 7.1 28 29 3() 40 50 60 10 81} ':I() 100 l 0.412 0.616 D.9111) I U4 I.iJ 2-16 HO 3m 1~7 4.07 4J-o 5.14 5.70 6.lli 6.84 1.43 ltoj 8.64. 9.26 9.X9 IOJl 11.16 1\ \.1\ 12.4& 1l.12 13.79 20.71 21.99 )5.53 3.13 1.17 9.20,.11 0.)4 0.8"12 1.24 1.65 2.09 1 ~(1 }O~ 3.57 4.11.1(,6 -".23.).81 11.41 1,01 1.6J. S.26 8.90 9.S4 10.20 1fI.!Ifl 1L.52 1220 11.&& fj.56 14.26 14.'15 Zl.lf> 29.7f 37,48 45.44 53.54 61.1S 70.06 0.8j, t14 1.69 ;ua 2.70 3.25 3.32 4.40 S.OI HiJ o.2fi 6.91 1.)6 \I.Z} B.91 ').$<) 10.1& 1(J.9~ 11.69 n.4u tj.ll 13-84 14.57 IS.31 16.05 16.19 14,43 32.3&.w.48 4<.1.76 57.[ S 65.65 74,22 l.145 164 2.17 z.n l.l3 3,')4 4.57 sn '.HI) 6.S7 1.26 1.% lui? 9.J9 10,12 10.85 IU<J 12.34 1 )'()9 IH~ I UI 15.38 \6.\S 1{S,9J 11.11 llu9 26.SI 34.16 43.t9 jl.74 6(l19 69.13 17:13 Lfli 2.1(1 2.Rl 3.49 4.11 4.lH D8 (do Ht4 779 8.~5 9.31 10.09 IQ.86 11.65 12,44 1324 14.04 14.85 L5.{)6 1647 11.29 UUI 18.94 l?,77!o.!oo 2').(1$ 3169 46.% S:CIJ (>I.2M 73.;.!9 8~.~6 9.24 IO.M 12.fr2 13.36 14.68 L~,99 17.28 IS.H 1'HI 2H)6 22.31 U~ 24.71 2~.\l9 27~ZO 2~t41 29.62 30.81 32.1H run )4.38 luti 36.14.Hi]l :W.09 4Cl.26 ~I.U 6],.17 14.40 11'.'.1 9(.,S8 HH.6 JlIU IL07 12.83 IH)9 16.7S 11.S9 14.4S 1.6,tH l8.5s 14,07 16.01 18.48 20.11'1 [5.51 11.53 20.09 11.96 16.lll 19.01 2167 2:M9 1831 ;!O.ol~ 23.21 2:5,19 19.68 ll.92 24.13 2676 21.03 21.:14 2&.:42 2l!.30 22.J6 24.74 27.69 2.9.S1 23./i.S 26.12 29.14 11.32 2.HIO 27.49 3O.S! 3211<1 26.10 21UI5 32.00 34.21 27.$9 ;0.19 3J.41 3~,1l 28.87 31.S) 34.S1 37.\6 JO.14 32.65 36.19 18.53 llai 34.11 37.57 40.00 JUi7 J.t4<'l.Jb'.93 41.411 "33.92 1f>.18 4(1.29 42.80 JS.17 38.08 41,{)4 44.t8 l6.42 3'1.J6 429~ 4~.:5~ 37.6S 4O.6j 4UL 4M3 ls.s9 4.I.n 45.64 48.29 40.1.\ 43.19 46.96 49.6d. 4U4 44.46 48'.28,ro.99 42,:56 45.n 41}.s<) 52.34 43."17 46.9& SD.89 53.fJ7 55.76 59.34 63.69 66.Tl 67.50 71.42 76.15 79.49 /9.08 XUI) SS.Jll 91.95 'JO.,,, 9j.f)':! IIXl.4 11)4.2 Hi]. 9 106.6 1]2.1 IHiJ Illl 1]8.1 124.1 128.3 124.3 12').6 13:5.8 140.2 Volume II. Appendix C: page 3
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.60.70.110.85.95.975 1 0.32.'5 0.721 1.376 J.963 3.(178 6.~14 12.106 2 0.289 0.617 1.061 1.386 1.886 2.920 ".303 3 0:'1:77 0.584 0.978 1.25() 1.638 2.353 3.UU 4 o::rn 0.569 0.94\ 1.\90 1.53-:; 2.\l1 l.tlb S 0.267 0.5.59 0.920 1.1S6 1 76 2.0lS 2.571.6 O.26S 0.553 0.906 1.134 1.440 1.94:3 2.447 1 0.263 0 549 0.896 LH9 1.41S 1.895 2.365 IS ["262 0.546 0.889 1.IOS 1397 1.86 2.306 9 0.261 OS43 0.883 1.100 1.3-23 1.833 10 O.l6(l 0.~4t 0.879 UJ93 1.372 1.812 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 23 :19 30 40 64.1 120 w 0.260 0 259 0 259 <l.2s8 0.258 0.258 O.2S7 0.257 0.25'1 0.257 0.2S1 0.256 0.256 O.2~ 0.2~ O.l.56 O.2!56 0.2:56 0.256 0.256 0.255 O.2S4 0.254 O.2:5;'} 0.540 0.539 0.536 (t.535 G.534 0.534 0.533 0.5:\3 0.532 0.:53::!. 0.532 OS31 0.530 O.~JO 0.530 0.52<) 0.527 0.526 0.'24 0.816 0.873 0.870 O.'86S 0.866 0.86."1 O.S6J 0.862 0.361 0.860 O.8!i9 O.S5S o.ass 0.83'7 0.856 o.gs(; O.iSS 0.855 0.854 0.854 O.~I O.MI!I. 0.845 O.M~ 1.088 1.083 1.07 ~ 1.071} 1.074 urn 1.069 1.0f.? 1.066 1.064 1.063 J.061 1060 1.OS<J 1.058 l.o'ik 1.057 1.056 1.05S I.OSS t.oso '.045 l.04l 1.0.]6 1.363 1.356 1.3!SO 1.34'1 1.341 1.337 1.333 LJ3Q l.lls 1.,325 1.32.3 I.J21 1.319 L:31~ LH6 1.315 1.314 1.313 1.3Jl 1.310 I.J0e3 1.296 1.289 1.282 1.796 1.782 1.171 L161 1.153 t,746 1.740 1.734 1.729 1.7:lS 1.721 1.111 1.114 l.711 1.708 1.706 1.703 1.70l L(>99 1.697 1.684 1.671 1.658 I.MS 2.262 2.ns 2.201 2.179 2.160 2.145 2.111 2.120 2.110 1.101 2.093 2.086 2.080 2074 2.069 2.064 2.060 2.056 2.05;:1: 2.048 2.045 :U)42 2.021 2.000 r.9t1o 1.960 Volume II, Appendix C: page 6
Table C-4 (Continued) Percentiles of the t Distribution v.~nf,985.9').4l92.".995.9975.9995 1 15.8'95 21.205 31.821 42.434 6J.6.51 127.322 636.590 2 4.849 :'L643 6.965 {to?) 9.925 14.089 3U9S.3 3.482 3.896 4 54l 5.047 5.841 7.453 12.924 4 2.999 3.:2'>8 1.747 4.08l1. 4.604 5.598 S.610 5 2.757 3.003 3.36S 3.634 4.032 4.773 6,869 6 2.612 2.829 J.l"} 3.372 3.7m 4.317 S.9SIJ 7 2.511 2.11S 2.998 3.20l :\,4<;,1'9 4.029 5.408 8 2.449 2.634 2.10196 3.0S5 3.355 ' 3.833 5.041 9 2.398 2.574 2.821 2.9QK 3.250 3.690 4./81 1Q 2.'3S~ 2:527 2.164 2.932 3.169 3.581 4.581 J1 2328 2.49i 2.118 2879 :tl(}i6 J.497 4.437 12 2.3Q3 2.461 2.68! 2Jl36 3.055 3.428 4.3'8 13 2.232 2.436 2.MQ 2.1!()\ 3.01'2 3.172 4.22\ 14 2.264 2.4l5 2.624 2.171 2.977 3.326 4.140 15 Z.249 l.3')7 2.00l 2.746 2.947 :U,Sti 4.013 Hi 1.235 ;UIn 2.583 2.724 '2.92' 3.252 4.0(5 11 2.224 2.368 2.567 2.106 2.898 3.222 3.%5 t8 2.214 2.356 2.552 2.()8') 2.818 3.197 3.922 19 2.205 2.346 2.539 2.674 2.86' 3.174 3,381 '20 2.19? 4.336 2.528 2.<i61 2.845 3.153 3.1149 2J 2.\89 4,.318 2.518 ".M'; 2.831 3.135 3.819 22 2.183 2.320 2.508 2.639 2.IH'l l.u? 3.192 23 2.177 2.3U 2.500 2.629 2.801 ::U04 3.168 24 2.112 2.307 2.492 2.620 2.191 1.(1)\ 3.745 15 2.167 2.3()1 2.485 2.612 2.787 3.0'18 3.725 26 2.16'2. 12% 2.479 2.605 2.719 3.(167 3.701 27 2.15& 2.291 2.471 2.598 1.77l 3.057 3.690 2S 2.154 2.'2S{) 2.461 2.592 2.763 :'U147 3.674 2CJ 2.150 2.282 2.462 2.586 2.756 3.038 3.659 30 2.247 2.218 2.457 2.581 2.750 3.030 3.646 40 2.123 2,250 2.42.1 2..542 2.10<\ 2.971 3.551 60 2.099 2.223 2.390 2.504 2.660 2.915 3.460 120 2.076 2.196 2.358 2.468 2.617 2.860 3.373 (X> 2.OS4 Z.l70 2.326 2.432 2.:576 2.801 3.291 A Volume II, Appendix C: page 7