ECONOM ICS EC 219 SA M PLING AND INFERENCE One and One HalfH ours (1 1 2 H ours) Answerallparts ofquestion 1,and ONE other question. M athem aticaland Statisticaltables are provided. An approved calculatormay be used. Section A :C om pulsory Question [60% ofthe totalm ark;allparts carry equalw eight] 1. a. Fora stratified population when simple random sam ples are chosen in each ofthe strata,define the stratified sam pling m ean. Show thatitis an unbiased estim atorofthe population m ean,and find its variance. b. Fam ily Spending 1994-95, reported from the Fam ily Expenditure Survey, show s the average w eekly expenditure of all the sam pled households on fares and travelcosts (otherthan motoring)to be 6.64 w ith percentage standard error 4.9. W hatcan you estim ate for the UK population from this information,and whatcan you say aboutboth the precision and the accuracy ofyourestim ate?
c. Setup a Likelihood Ratio Testto test H 0 :µ = µ 0 against H 1 :µ µ 0 fora normalpopulation with know n variance σ 2. W hatvalue should be chosen forthe constantifthe testis to have a type Ierrorof 0.05? d. A firm m easures the outputper hour of three daily shifts on each of five days ofthe w eek. Use the two way analysis ofvariance m odelto testfordifferences in outputperhourbetw een the differentshifts and betw een the different days of the w eek ifthe sum s of squares are as follow s: Betw een shifts SS = 83.5 Betw een days SS = 281.0 Total = 467.0 Explain and interpretyourconclusions clearly. e. Explain how the linear regression technique can be used to estimate a nonlinearrelationship. Give one exam ple ofsuch a m odeland give an interpretation ofits coefficients. f. W hatis a dum m y variable? Explain how you w ould interpret the coefficientofa dummy explanatory variable in a regression m odel.
Section B :A nsw er ONE ofthe follow ing questions [40% ofthe totalm ark] 2. a. Show,using the form ula forthe variance ofa sam ple m ean,that N n var ( p) = P( 1 P) / n N 1 where p is the sam ple proportion, P is the population proportion, n is the sam ple size and N is the population size. [40% ] If P is unknow n,w hatform ula is used to estim ate var ( p) and w hy? [20% ] b. A listcontains 1000 nam es and addresses. A simple random sam ple is to be chosen and the addresses checked. If we wish to estimate the proportion of w rong addresses in the listto w ithin 0.04 of the true proportion with 95% certainty,whatsam ple size is required? [20% ] Suppose a sam ple of 144 nam es and addresses is chosen and found to include 18 w rong addresses. Estim ate the overall proportion of addresses thatare wrong,and find 95% confidence lim its. [20% ]
3. A population has been divided into two strata prior to the selection of a stratified random sam ple. The sam pling costs are expected to be of the form C = w here c i and n i are respectively the unit cost and the cn i i num bersam pled in the ith stratum. Show how to choose the sam ple in order to minimise the variance ofthe stratified sam pling m ean ifthere is a specified budgetforthe sam pling costs, C. [25% ] Estimates of the total num bers in the strata, their variances and the unit sam pling costs are given below,togetherw ith the sam pling budget: Stratum N i 2 S i c i 1 500 81 5 2 1000 144 14 C = 1000 Using the principle of optim alallocation,choose the num bers to be sam pled from each stratum. [25% ] Afterthe sam ple is taken itis found thatthe variances ofthe values in strata 1 and 2 are in fact 35 and 180 respectively. Ifthese values had been know n prior to sam pling,how w ould the num bers sam pled have been affected, and whatdifference w ould this have made to the precision ofthe stratified sam pling m ean? [50% ]
4. a. Explain w hatis m eantby cluster sam pling and discuss its advantages and disadvantages. [30% ] b. Suppose there are M equalsized clusters in the population each with L elem ents and w e selecta simple random sam ple of m clusters to enum erate. Show w hat form ulae should be used to estim ate the population m ean and the variance ofthe clustersam pling m ean. [40% ] c. The 75 branches ofa studenttravelcom pany each em ploy 12 people. A random sam ple of 10 branches are selected,and allthose em ployed ateach are asked how farthey travelled on holiday in the lastyear. The m ean distance travelled for the em ployees at each of the sam pled branches are: 1065 1230 2911 3154 3232 3456 3891 4086 5567 6430 Find 95% confidence lim its forthe m ean distance travelled on holiday by allem ployees ofthe com pany. [30% ] 5. Setup the one way analysis ofvariance m odeland find leastsquares estimators forits param eters. [25% ] Show how the totalsum ofsquares forthe m odelcan be expressed as the sum ofothersum s ofsquares. [25% ] Define the m ean squares used in the teststatistic of the m odeland find the expectation ofeach. Define the teststatistic and justify its construction. [50% ]
6. Surveys of fam ily incom e and consum ption are carried outin Leicester and Loughborough. Y ou specify the follow ing m odel: y i = β 1 + β 2 x i + u i where y = log consum ption, x = log incom e and u is a random disturbance satisfying E(u x)= 0. The follow ing summary m easures are calculated from the two sam ples: Leicester Loughborough n 20 20 x 1 1 y 8 9 2 ( xi x) 12 10 ( xi x)( yi y) 10 12 y 2 y 10 18 ( i ) (i) How w ould you interpretthe coefficient β 2 in this m odel? [20% ] (ii) Estimate β 1 and β 2 foreach ofthe two sam ples separately. [20% ] (iii) Calculate the standard errorof $ b 2 separately forthe two sam ples. [20% ] (iv) If $ LE b 2 and $ LO b 2 show that: are the Leicesterand Loughborough estimates of β 2, LE LO LE LO ( b2 b2 ) = ( b2 ) + ( b2 ) var $ $ var $ var $ provided the two sam ples are independently draw n. [20% ] (v) Conduct a test of the hypothesis that Leicester and Loughborough consum ers have the sam e value of β 2. [20% ]