UNIVERSITY OF MORATUWA

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1 MA5100 UNIVERSITY OF MORATUWA MSC/POSTGRADUATE DIPLOMA IN FINANCIAL MATHEMATICS 009 MA 5100 INTRODUCTION TO STATISTICS THREE HOURS November 009 Answer FIVE quesions and NO MORE. Quesion 1 (a) A supplier of kerosene has a weekly demand Y possessing a probabiliy densiy funcion given by f (y) y, 1, 0, 0 y 1 1 y 1.5 elesewhere wih measuremens in hundreds of gallons. The supplier's profi is given by U = 10Y- 4. (i) Find he probabiliy densiy funcion for U. (ii) Use he answer o (i) o find E(U). (b) The life ime X of cerain elecronic componen is disribued as normal wih mean 3 hours and sandard deviaion hours. Replacing such a componen in he course of cerain job causes expensively delay, so a new componen is fied before saring he job. If he componen lass for x hours, i ensures a profi of Rs. p where, x 0 p 0 0 x 5 40 x 5 Find expeced profi. Quesion (a)environmenal Auhoriy se a maximum noise level for heavy rucks a 83 decibels (Environmen News, Ocober 008). How his limi is applied will grealy affec he indusry and he public. One way o apply he limi would be o require all rucks o conform o he noise Page 1 of 6

2 MA5100 limi. A second bu less saisfacory mehod would be o require he ruck flee mean noise level o be less han he limi. If he laer were he rule, variaion in he noise level from ruck o ruck would be imporan because a large value of would imply many rucks exceeding he limi, even if he mean flee level was 83 decibels. A random sample of six heavy rucks in 008 produced he following noise levels (in decibels): 85.4, 86.8, 86.1, 85.3, 84.8, Use hese daa o consruc a 95 % confidence inerval for he variance of he ruck noise emission readings. Inerpre your resuls. (b) In a bol facory, here are four machines A, B, C, D manufacuring 0%, 15%, 5% and 40% of he oal oupu respecively. Of heir oupus 5%, 4%, 3% and % in he same order are defecive bols. A bol is chosen a random from he facory s producion and is found defecive. Wha is he probabiliy ha he bol was manufacured by machine A or machine D?, Quesion 3 The logarihmic disribuion wih parameer ( 0 < < 1), has discree probabiliy densiy funcion x k f (x) x = 1,, 3, x (i) Find he consan k as funcion of (ii) Show ha he mean of his disribuion is k 1 (iii)by finding E(x(x 1) or oherwise, show ha Quesion 4 E(x ) k 1 A sudies of he habis of whie-ailed deer ha indicae ha hey live and feed wihin very limied ranges, approximaely 150 o 05 acres. To deermine wheher here was a difference in he ranges of deer locaed in wo differen geographical areas, fory deer were caugh, agged, and fied wih small radio ransmiers. Several monhs laer, he deer were racked and idenified and he disance y from he release poin was recorded. The mean and sandard deviaion of he disances from he release poin were as follows: Page of 6

3 Locaion 1 Sample size Sample mean 980 f 305 f Sample sandard deviaion 1140 f 963 f Populaion mean 11 4 MA5100 (a) If you have no preconceived reason for believing one populaion mean o be larger han anoher, wha would you choose for your alernaive hypohesis? Your null hypohesis? (b)would your alernaive hypohesis in par (a) imply a one- or a wo-ailed es? Explain. (c) Do he daa provide sufficien evidence o indicae ha he mean disances differ for he wo geographical locaions? Tes using 5%. Quesion 5 A ciy is considering replacing is flee of municipally owned, gasoline-powered auomobiles by elecric cars. The manufacurer of he elecric cars claims ha he ciy will experience significan savings over he life of he flee if i convers, bu he ciy has doubs. If he manufacurer is correc, he ciy will save 1 million dollars. If he new echnology is fauly, as some criics sugges, he conversion will cos he ciy $ A hird possibiliy is ha neiher siuaion will occur and he ciy will break even wih conversion. According o he recenly compleed consulan s repor, he perspecive probabiliies of hese hree evens are 0.5, 0.45 and The ciy has before i a pilo program ha if implemened would indicae he poenial cos or savings in a conversion o elecric cars. The program involves rening hree elecric cars for 3 monhs and running hem under normal condiions. The cos o he ciy of his pilo program would be $ The ciy's consulan believes ha he resuls of he pilo program would be significan bu no conclusive; she submis Table 1, a compilaion of probabiliies based on he experience of oher ciies, o suppor her conenion. Wha acions should he ciy ake if i wans o maximize expeced savings? Page 3 of 6

4 Saves money Breaks even Losses money Table 1 Savings No change A Loss MA5100 Quesion 6 Le Y 1 and Y have he join probabiliy densiy funcion given by ky1y, f (y1, y ) 0, 0 y 1 1, elsewhere 0 y 1, (i) Find he value of K ha makes his a probabiliy densiy funcion. (ii) Find he join disribuion funcion for Y l and Y. (iii)find (Y 1/, Y 3/ 4). p 1 (iv) Find he marginal probabiliy densiy funcion of y 1. Quesion 7 (a)a criminologis conduced a survey o deermine wheher he incidence of cerain ypes of crime varied from one par of a large ciy o anoher. The paricular crimes of ineres were assaul, burglary, larceny, and homicide. The following able shows he numbers of crimes commied in four areas of he ciy during he pas year. Type of Crime Disric Assaul Burglary Larceny Homicide Can we conclude from hese daa a he 0.05 level of significance ha he occurrence of hese ypes of crime is dependen upon he ciy disric? Page 4 of 6

5 MA5100 (b)in finance, an efficien marke is defined as one ha allocaes funds o he mos producive use. Business Week recenly surveyed 110 financial analyss who work for privae manufacuring firms in he effor o sell heir firms' securiies, 4 fel markes were efficien, while 31 of 75 analyss who work for brokerage houses assising in hese sales agreed ha markes were efficien. Tes wheher here appears o be a difference in he proporion of hese wo ypes of anlayss who accep he concep of marke efficiency a 5% level of significance. Quesion 8 (a) The amoun of gas required o hea a home depends on he ou-door emperaure. A sudy recorded he average daily gas consumpion of a house y ( in hundreds of cubic fee) for each monh during one heaing season. The explanaory variable x is he average number of heaing degree days per day during he monh. (One heaing degree day is accumulaed for each degree a day's average emperaure is below F). The: daa are Oc Nov Dec Jan Feb Mar Apr May Jun X Y The summary daa are x 193.9, y 50.3, x , y , x y I is hough ha linear relaionship exis beween y and x of he form y x where ~ N 0, (i) Plo he daa o confirm ha here is a linear relaionship beween he wo variable. (ii) Explain he meaning of he slope ( ) and ( ) parameers. {iii}calculae he leas squares esimaes of and, and include he esimaed line on your plo. Page 5 of 6

6 (iv) Calculae he esimaed error variance. (v) Calculae a 95% confidence inerval for. MA5100 (b) In a sudy on he relaionship beween he heigh (u) and weigh (v) of n species of monkeys a researcher colleced daa from 10 adul individuals. The summary saisics were CS(u, u) = 190.7, CS(v, v) =70.3 and CS(u,v) = Calculae he sample correlaion coefficien beween heigh and weigh and es wheher he populaion correlaion coefficien is zero. Page 6 of 6