UNIVERSITY OF EAST ANGLIA School of Ecoomics Mai Series UG Examiatio 04-5 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
SECTION A [Mathematics] age. The Demad ad Supply equatios for a particular good are: D : Q 00. 5 S : Q 50. 5 a b c Solve the pair of simultaeous equatios D ad S i order to obtai the equilibrium price ad quatity, * ad Q*. [ marks] 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. [ marks] Sketch the two lies represeted by the equatios obtaied i b. Label the lies D ad S, ad label the market equilibrium. d Calculate the price elasticity of demad at the market equilibrium. [ marks] e f Suppose the govermet imposes a fixed tax of 4 per uit. Calculate the ew market equilibrium price ad quatity. Calculate the deadweight loss that is geerated from the impositio of the tax.. The value of a secod had car reduces expoetially with age, so its value y after t years ca be modelled by: y 0, 000exp 0. 5t a What is the value of the car after 5 years? [ marks] b The car is ow worth,000. What is the approximate age of the car?
age 3 3. A firm s demad fuctio is represeted by: Q 536 a b Fid the level of output that maximises total reveue ad verify that this is a maximum. [4 marks] Suppose we are ow told the firm s total cost fuctio is: Q TC 3 3 5Q 480Q 750 At what level of output is profit maximised? How much profit does the firm make at this level of output? [6 marks] 4. A firm s productio fuctio is give by: Q 80 0 5 0. L. K a Demostrate that the fuctio is homogeeous ad fid the degree of homogeeity. Commet o the returs to scale. b Fid the margial products of labour L ad capital K. c M What is the slope of the isoquat L? MK [ marks] TURN OVER
age 4 5. A cosumer s utility fuctio is give by goods. U x y / 5 4 / 5 where x ad y are two Suppose that total icome is 00 ad the prices are 5 for each uit of good x ad for each uit of good y. a b c Use costraied optimisatio, specifically the Lagragia method, to fid the cosumer s demad for both goods. [8 marks] What is the value of the Lagrage multiplier? rovide a ecoomic iterpretatio of the value. [ marks] How large is utility at the costraied maximum idetified i part a? [ marks]
age 5 Start your aswer to the ext sectio i a ew booklet SECTION B [Statistics] 6. Mercy s ightclub obtais the followig data o the age ad marital status of 40 customers. Marital status Sigle Married Uder 5 77 4 5 or over 8 a What is the probability a customer is married ad uder 5? [ mark] b If a customer is uder 5, what is the probability that he or she is sigle? [ mark] c Is marital status idepedet of age? Explai your aswer. [ marks] d The probability that a customer at Mercy s ightclub is ivolved i a icidet is 0.64. The probability of a customer at a rival ightclub, Flame, beig ivolved i a icidet is 0.46. The chace that there is a story i the ext editio of the local ewspaper about Mercy s is 40% ad is 60% for Flame. Suppose i the ext editio of the ewspaper there is a headlie that reads, icidet at ightclub. What is the probability that the story is about Mercy s? [4 marks] TURN OVER
age 6 7. A report by Grat Thorto suggests the mea basic salary for bosses of the largest UK compaies FTSE 00 Executives was 583,9 i 04. Assume the stadard deviatio was 45,5. Assume the populatio is ormally distributed. a What is the probability that a radomly selected boss has a salary betwee millio ad.5 millio? [ marks] b Te percet of bosses have a salary of how much or less? You obtai a radom sample of basic salary for 5 FTSE 00 Executives. c Fid the stadard error of the sample mea salary. [ marks] d What is the probability that the sample mea is greater tha 60,000? [ marks] e What is the probability that the sample mea differs from the populatio mea by more tha 00,000? 8. A radom sample of six UK maufactured cars i 04 has the followig CO emissios figures: 40 45 0 05 4 78 a Calculate the mea, variace ad stadard deviatio. [4 marks] b c Fid the 95% cofidece iterval for the populatio mea. Iterpret the result. [4 marks] Calculate the required sample size i order for the margi of error to be o greater tha 0 i either directio. [ marks]
age 7 9. The press has recetly reported extesively o the growig problem of childhood obesity i the UK. You wish to perform your ow statistical aalysis, usig data from a local school. Childre leave the school at age, at which time their weight i Kg is recorded. The data that is available is of 5 pupils who left the school i 994, ad 35 who left i 04. Descriptive statistics for the two samples are cotaied i the followig table: 994 sample 04 sample sample size 5 35 mea 43.0 48.0 stadard deviatio 6.0 0.0 Let be the populatio mea weight of 994 leavers, ad be the populatio mea weight of 04 leavers. a b c Accordig to Departmet of Health guidelies, the ideal weight for -yearolds is 44Kg. Is there evidece i the 04 sample that today s childre weigh more tha this ideal weight? Combie the two sample stadard deviatios to obtai the pooled stadard deviatio, Sp. Does a compariso of the two samples reveal that childre weigh more i 04 tha they did i 994? erform a appropriate test. [4 marks] TURN OVER
age 8 0. You are iterested i the relatioship betwee icome ad age. Usig a sample of survey data for the East of Eglad, you obtai the followig regressio results from SSS. Model Summary Mode l R R Square Adjusted R Square Std. Error of the Estimate.69 a.08.04 480.878 a. redictors: Costat, age at..006 Coefficiets a Ustadardized Coefficiets Stadardized Coefficiets Model B Std. Error Beta t Sig. Costat 985.333 885.79 3.405.00 age at..006 8.780 69.65.69.65.009 a. Depedet Variable: aual labour icome.9.006-.9.007 a Iterpret the R ad the R square. [ marks] b Iterpret the sig, magitude ad sigificace of the itercept coefficiet. c Iterpret the sig, magitude ad sigificace of the slope coefficiet. d Suppose you were told there were outliers i the icome variable. What remedies would you suggest? [ marks] END OF AER
age 9 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
age 0 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
age Cofidece Itervals ad Hypothesis Tests oe sample A 95% cofidece iterval for the populatio mea,, is give by: S t,0. 05. 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: =.
age 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..00.0.0.03.04.05.06.07.08.09.00.500.496.49.488.484.480.476.47.468.464.0.460.456.45.448.444.440.436.433.49.45.0.4.47.43.409.405.40.397.394.390.386.30.38.378.374.37.367.363.359.356.35.348.40.345.34.337.334.330.36.33.39.36.3.50.309.305.30.98.95.9.88.84.8.78.60.74.7.68.64.6.58.55.5.48.45.70.4.39.36.33.30.7.4..8.5.80..09.06.03.00.98.95.9.89.87.90.84.8.79.76.74.7.69.66.64.6.00.59.56.54.5.49.47.45.4.40.38.0.36.33.3.9.7.5.3..9.7.0.5.3..09.07.06.04.0.00.099.30.097.095.093.09.090.089.087.085.084.08.40.08.079.078.076.075.074.07.07.069.068.50.067.066.064.063.06.06.059.058.057.056.60.055.054.053.05.05.049.048.047.046.046.70.045.044.043.04.04.040.039.038.038.037.80.036.035.034.034.033.03.03.03.030.09.90.09.08.07.07.06.06.05.04.04.03.00.03.0.0.0.0.00.00.09.09.08.0.08.07.07.07.06.06.05.05.05.04.0.04.04.03.03.03.0.0.0.0.0.30.0.00.00.00.00.009.009.009.009.008.40.008.008.008.008.007.007.007.007.007.006.50.006.006.006.006.006.005.005.005.005.005.60.005.005.004.004.004.004.004.004.004.004.70.003.003.003.003.003.003.003.003.003.003.80.003.00.00.00.00.00.00.00.00.00.90.00.00.00.00.00.00.00.00.00.00 3.00.00.00.00.00.00.00.00.00.00.00 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
age 3 Table : Critical values of the t-distributio df = 0.0 = 0.05 = 0.05 = 0.0 = 0.005 3.08 6.3.7 3.8 63.66.89.9 4.30 6.97 9.93 3.64.35 3.8 4.54 5.84 4.53.3.78 3.75 4.60 5.48.0.57 3.37 4.03 6.44.94.45 3.4 3.7 7.4.90.37 3.00 3.50 8.40.86.3.90 3.36 9.38.83.6.8 3.5 0.37.8.3.76 3.7.36.80.0.7 3..36.78.8.68 3.06 3.35.77.6.65 3.0 4.35.76.5.6.98 5.34.75.3.60.95 6.34.75..58.9 7.33.74..57.90 8.33.73.0.55.88 9.33.73.09.54.86 0.33.73.09.53.85.3.7.08.5.83.3.7.07.5.8 3.3.7.07.50.8 4.3.7.06.49.80 5.3.7.06.49.79 6.3.70.06.48.78 7.3.70.05.47.77 8.3.70.05.47.76 9.3.70.04.46.76 30.3.70.04.46.75 40.30.68.0.4.70 50.30.68.0.40.68 60.30.67.00.39.66 70.9.67.99.38.65 80.9.66.99.37.64 90.9.66.99.37.63 00.9.66.98.36.63 5.9.66.98.36.6 50.9.65.98.35.6 00.9.65.97.35.60.8.64.96.33.58 END OF MATERIALS