INTRODUCTORY MATHEMATICS AND STATISTICS FOR ECONOMISTS

Transcription

1 UNIVERSITY OF EAST ANGLIA School of Ecoomics Mai Series UG Examiatio 05-6 INTRODUCTORY MATHEMATICS AND STATISTICS FOR ECONOMISTS ECO-400Y Time allowed: 3 hours Aswer ALL questios. Show all workig icludig itermediate results. Write aswers to EACH sectio i a SEARATE aswer booklet. A formula sheet is attached to this exam paper. Notes are ot permitted i this examiatio. Do ot tur over util you are told to do so by the Ivigilator. ECO-400Y Module Cotact: Dr eter Dawso, ECO Copyright of the Uiversity of East Aglia Versio

2 SECTION A: Mathematics age The Demad ad Supply equatios for a particular good are: D: Q S : Q 0 a Solve the pair of simultaeous equatios D ad S i order to obtai the equilibrium price ad quatity, * ad Q*. b Ivert the two equatios, so that they show i terms of Q. Make sure that your iverted equatios are each the equatio of a straight lie. c Sketch the two lies represeted by the equatios obtaied i b. Label the lies D ad S, ad label the market equilibrium. [3 marks] d Suppose the govermet imposes a fixed tax of 5 per uit o producers suppliers. Calculate the ew market equilibrium price ad quatity. [3 marks] e Calculate the deadweight loss that is geerated from the impositio of the tax. Let s suppose that a ecoomy is forecast to grow cotiuously so that Gross Domestic roduct GD i trillio after t years is give by: GD t 0exp0.06t a Forecast GD i years time. b After how may years is GD forecast to be 0 trillio? [4 marks] 3 A firm s total reveue fuctio is give by: TR 00l Q Its total cost fuctio is give by TC Q a Fid the value of Q which maximises profit. [8 marks] b How much profit does the firm make? [4 marks] TURN OVER

3 age 3 4 A firm s productio fuctio is give by: Q K /3 L 3/4 a Demostrate that the fuctio is homogeeous ad fid the degree of homogeeity. Commet o the returs to scale. [3 marks] b Fid the margial products of labour L ad capital K. [3 marks] M c What is the slope of the isoquat L? MK 5 A cosumer s utility fuctio is give by U 0xy goods. where x ad y are two Suppose that total icome is 96 ad the prices are for each uit of good x ad 4 for each uit of good y. a Use costraied optimisatio, specifically the Lagragia method, to fid the cosumer s demad for both goods. [8 marks] b What is the value of the Lagrage multiplier? rovide a ecoomic iterpretatio of the value. c How large is utility at the costraied maximum idetified i part a? Report your aswer to decimal places TURN OVER

4 age 4 START YOUR ANSWER TO THE NET SECTION IN A NEW BOOKLET SECTION B: Statistics 6 Twelve athletes compete i a race. After the race five ruers are selected for a drugs-test. a How may differet combiatios of five athletes are possible? b Suppose oly athletes fiishig st, d ad 3 rd have to be tested. How may differet combiatios are ow possible? A radomly chose athlete is tested for a illegal performace-ehacig drug. Suppose that the drug test is 98% accurate i the case of a user of the drug ad 90% accurate i the case of a o-user of the drug. Suppose it is kow that 0% of all athletes use this illegal drug. c The athlete is tested ad the test is positive. What is the probability that the tested athlete uses this illegal drug? [4 marks] 7 A biased coi comes up tails 5% of the time. The coi is tossed 6 times. Let be the umber of tails obtaied. Usig the biomial distributio, fid the probability of gettig: a Three tails b No tails The same coi is ow tossed 0 times. c Usig the ormal distributio approximatio, fid the probability of gettig te or more tails. [4 marks] d Is the ormal distributio a good approximatio i this istace? Briefly explai. TURN OVER

5 age 5 8 The followig jourey times i miutes to work for a radom sample of 0 idividuals livig i Norwich were recorded: a Fid the mea ad media. b Fid the variace ad stadard deviatio. [4 marks] c Fid the 99% cofidece iterval for the mea jourey time. Iterpret the result. [3 marks] d Calculate the required sample size i order for the margi of error associated with the mea to be o greater tha 0 use 99% cofidece iterval for the calculatio of the critical value. [3 marks] 9 It is claimed that the amout doated to charity varies by UK regio. To ivestigate this you fid data from a radom sample of idividuals i the East of Eglad ad compare this to a radom sample of idividuals from Lodo. The amouts doated per moth i s are recorded i the followig table: East of Eglad Lodo Sample Size 4 Mea Stadard Deviatio Data Source: Uderstadig Society, Wave. Let µ represet the populatio mea mothly doatios for East of Eglad ad µ the populatio mea mothly doatios for Lodo. a Is there evidece that the mea mothly doatios i Lodo is greater tha 95? erform a appropriate test use the 5% sigificace level. [4 marks] b Combie the two sample stadard deviatios to obtai a pooled sample stadard deviatio, Sp. c Does a compariso of the two samples reveal idividuals livig i Lodo doate more compare to idividuals livig i the East of Eglad? To aswer this coduct a -sample t-test use the 5% sigificace level. [4 marks] TURN OVER

6 age 6 0 You are iterested i the relatioship betwee hours worked per week ad hourly wage. Data for 7 idividuals i East Aglia is obtaied ad the regressio results estimated i SSS are preseted below data source: BHS, Wave 7. a Iterpret the regressio results. I your aswer make sure you cosider the sig, magitude ad sigificace of the idividual coefficiets ad commet o the goodess of fit. [8 marks] b Idetify two other variables that you thik might ifluece hours worked. END OF AER

7 age 7 ECO-400Y: Itroductory Mathematics ad Statistics for Ecoomists Formulae Sheet The Quadratic Formula If ax bx c 0 the x b b a 4ac Differetiatio Chai rule: If, y fu ad u g x the dy dy du du roduct rule: If y f x g x, let u deote f x ad v deote g x, the dy v du u dv Quotiet rule: f x If y, let u deote f x ad v deote g x, the g x dy du v u v dv

8 age 8 Descriptive statistics Mea: i. Variace: S i i. The stadard deviatio, S, is the square root of the variace. Bayes Rule A A B A A B A A B B A The combiatorial formula!!! r r C r. Biomial probabilities p p C p =0,,,...,. Mea of a biomial distributio, E = p Variace of a biomial distributio Var = p-p Cotiuity correctio If ~Biomial,p ad is large, the ~Np, p-p x x+0.5 =. p p p x Z 5 0 x x-0.5 =. p p p x Z 5 0

9 age 9 Cofidece Itervals ad Hypothesis Tests oe sample A 00-% cofidece iterval for the populatio mea,, is give by: S t, /. To test H0: =0, use: t 0. S / The test statistic t has a t- distributio uder H0. The two-sample t-test t Sp where: S S S p. The test statistic t has a t + - distributio uder H0: =.

10 age 0 Table : The stadard ormal distributio To fid the area to the right of a umber z, look dow the left had colum for the first decimal place of z. The look alog the top row for the secod decimal place. The umber read from the cetre of the table is the required area Critical values of the stadard ormal distributio Z >.8 = 0.0 Z >.645 = 0.05 Z >.960 = 0.05 Z >.36 = 0.0 Z >.576 = 0.005

11 age Table : Critical values of the t-distributio df = 0.0 = 0.05 = 0.05 = 0.0 = END OF MATERIALS

12 Feedback: ECO-400Y Fial Exam, studets sat the exam. The mark distributio was: <40 7 The average mark was 67.07% with a stadard deviatio of This is a sigificat improvemet o last year, where the average mark was i the low 50s. The percetage of studets who failed the exam was 6.04% which cotiues the tred of fallig fail rates over the last three years. It is also oticeable that whilst the average mark for Sectio A maths part remais sigificatly higher compared with Sectio B stats part, it is pleasig to see the average for Sectio B is ow i the high 50s last year the average for Sectio B was 4%. Suggested aswers to the exam paper will be made available o Blackboard i due course. Sectio A: Mathematics Questio This questio was aswered very well with most beig able to correctly icorporate the impact of the tax. Some difficulties with calculatig deadweight loss. Questio This questio was aswered very well. Almost everyoe got part a right ad most were able to re-arrage the o-liear formula i part b. Questio 3 This questio created some difficulties. art a required the use of the chai rule i order to calculate the first-order derivatives. art b required kowledge that profit is simply the differece betwee reveue ad cost. Questio 4 I geeral this questio was doe quite well. I part a the discussio of homogeeity was ot always made explicit but most were able to idetify ad demostrate icreasig returs to scale. I part b most correctly idetified the partial derivatives. I part c, the MRTS should be simplified i order to obtai full marks.

13 Questio 5 erformace o this questio was very mixed. Those that attempted it geerally did well. art b, which required the calculatio ad iterpretatio of the Lagrage multiplier created the most difficulties. Sectio B: Statistics Questio 6 Most studets correctly used the combiatorial formula for part a to calculate the required value. art b created the most difficulties. art c was geerally doe ok ad credit was give for two versios: oe usig the coditioal probability as 0.9 ad the other usig the coditioal probability of 0.. Questio 7 arts a ad b were doe well. I part c some failed to icorporate the cotiuity correctio ad i d a umber of studets did ot iclude both criteria i assessig the relevace of usig the ormal approximatio. Questio 8 arts a c were doe well. Most could correctly calculate measures of cetral tedecy ad dispersio ad costruct ad iterpret the cofidece iterval. Aswers to part d were weaker with may studets ot attemptig it. Questio 9 I parts a ad c ot all studets icluded the ull ad alterative hypotheses. Some icorrectly used two-tailed rather tha oe-tailed tests. Some got the right aswer but the icorrectly iterpreted the result. Questio 0 Aswers to this questio were, i geeral, very disappoitig. A umber of studets icorrectly iterpreted the depedet variable as wage ad the explaatory variable as hours worked, whe i fact it was the other way roud.