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Sp09_Mid2A_key - EC 41 UCLA Spring 2009 Name(print lZ...

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Unformatted text preview: EC 41, UCLA Spring 2009 Name (print) lZ ' Ve/L'51W/l A Midterm #2 — 5/13/09 TA: Name & Section Time - The normal table and usefiil formulas are on the last page of this exam. - Only pens, pencils and erasers may be used, this is a closed book, closed note, exam. - Students may use a calculator, but nothing that can access the internet. - Write non—integer answers to 3 significant digits, e.g., 333, or 3.33 or .0333 - This exam consists of 10 True/False (20 points), 10 short answer (40 points) and 4 longer questions (40 points) - Clearly write answers on this exam. No points are awarded for illegible answers. — Be prepared to show a photo ID during the exam (e.g., UCLA ID) - You may leave when finished. Do not disrupt those still taking the exam. I. Circle T for True or F for False (2 points each) 1 T r F The p-value from Excel regression output assumes a two—sided alternate hypothesis, HA Zfilor F A confidence interval gets more narrow (smaller) as the number of observations in a sample is increased. 3®or F A lower p-value will provide better evidence in favor of rejecting the null hypothesis. 5 4) T o€l The number of ways two people can be chosen from a group of five is: 5C2 = [2] = 20 5) T 06} If returns on two investments are uncorrelated, p = 0, it is theoretically possible to form a portfolio with no risk (standard deviation and variance of the returns on the portfolio = 0). @r F The distribution of a sample mean of a NON—normal population with finite stande deviation will approach a normal distribution as the number of observations becomes large enough. 7) T 0(6) The standard deviation of the sum of two random variables is the sum of the two individual standard deviations IF the correlation between them is 0. 8) T 0I€l For a specific “95% confidence interval,” 95% of the population means, it, are in this interval. 9) T o F ‘ Benford’s Law (used to check accuracy of financial documents) implies that the probability of the first digit of numbers in financial documents is equally likely to be any integer from 1 to 9. 10) T Olé) For a particular positive sample mean, )6 , a null hypothesis may be rejected in favor of a two—sided alternative HA: M 75 0 but not in favor of a one sided alternative HA: uA > 0 . It 2 Mulls gt” 1) In a population of workers, 30% earn exactly $10 and 70% earn exactly $20 per hour. a) What is the average wage of workers in this population? Egaé l/l " 0 l M p (s)ml(7)zc) : 2+H (ll/15 Wm a“ II. Brie y, c ear , y answer the allowing 1 questions p . eac b) What is the population standard deviation of wage for these workers? it Ate-171W 478/47” 0" l7? Lls8 ., . ’Aéli’ll t 71(8) '= ”(37le : Ql 2) Consider two events, A = a person has lung cancer; and B = person grew up on Los Angeles. P(A) =.02 (= 2 in 100); P(BIA) 880; P(BlAC ) 2.20 Find the following: ”\ , A a) Probability of having lung cancer given one grew up in Los Angeles: P(AlB) = Péllri) MAM) 6954222) : DH. Pll5l43P P(tl +P®M BPQ‘) gin) +£2¥qg : .leJr 1% ,, 3) The table below shows number of auto insurance customers in each category. Realize total number of cum 2000 and drivers may file a claim after an accident that requires repairs. Filed Claim Did Not File Claim Female: 100 1,000 ”00 Male: 100 8.00 6MP /8. 5-13-13 Tag 2 0‘70 ......... a) The conditional probability P(Filed Claim | F emale)/ I D D Dqflo w b) The conditional probability P(Female 1 Filed Claim) 4) Suppose the effect of an $50 Billion open market purchase of bonds from commercial banks by the Fed, on the reserves of the commercial banking system can be written as a geometric series: 50.00 + 45.00 + 40.50 + 36.45 + 32.805 + (the sum of different “rounds” of deposit creation) a) What is the implied required reserve ratio, I‘D? [recall ' crease in deposits for each “round” = (l- rD)] 8v[.a0.;.q't.92+,93+\ >> MM => oar b) What is the total increase in Bank Reserves caused b the above open market purchase by the Fed? N l . ,. . ) -~— \ .2 . —— b0 1161' ‘30 ([03 New, 5) X and Y represent the return on two individual investments. ux = 2 o x = 5 uy = 2 (S y = 5 Z = a portfolio, half invested in X and Y each, so Z = .5X + .5Y . a)2 What is the standard deviation of the retmn on the portfolio if true correlation between the returns, (81 5 +(8J s to (281119352 (8‘38 CT Liz-89:33 8312318125 b) What is the standard deviation of the return on the portfolio if true correlation between the returns, pxy = 1? QB“ 123+ 208818195" : (28§+lZ.S: 2g A 6) The mean return to any one insurance policy is $100 and standard deviation is $1,000. Assume returns on any one policy are independent of returns on any other policy (losses are independent). (r : [000 (TI: [ng a 300 a) What is the expected total return to the insurance company of 5 policies? $00316?) b) What is the standard deviation of the total return on 5 policies? - i . ‘~ . _' (\2 g 21 2 , 2- _ 1 J 'Bl'wég " ‘gDL‘O [7110 Jim 0' HI” rug‘ ‘ 5‘ a. x f; [01905700 9:1 m i \_/'\_r\._/ ' ‘ 5 > S0d0)DDv N 7) Suppose a Simple random sample, srze IS t en; the true population standard deviation, 0, is know to equal 4; but the true mean u is unknown; and the sample mean, X = 40. [9/0 1‘ f9 ‘3 0i: a) A 95% confidence interval for the sample mean is: 0‘ >222; {a :> alarms—i b) If H0: ax = 39 and Ha: ux 7f 39, the implied p-value is: 2 Silltvi [3 tictb:2E(Z >lq953‘i /) f5 7 [I‘P(Z< L53) ”/é , t 2 P(z “4% ~// .8) A random variable X can take on three di erent v ues with the following probabilities: w .bEZlingvhfizah ~—Lz’+o 442:4 G"? J» (2412-; 209:02 Plant? ibhhzm? + 2623’- : ,b-+.2+.8 (.4, a) What is the mean of X? 9) Suppose the probability of receiving an order from any one particular sales call during a given period is exactly equal to .15 for any sales call. What is the probability that: a) the first three sales calls result in two orders? ‘ 7‘2'.,_f_i ”Ami. ,4, (gm (so , acmmbr $097275 5 b) the first order is fiom the 3rd sales call (so the first two calls do not resul '- orders)? ($933 ($5 095: i 108375 9’ 10) A deck of 52 cards has 4 aces. a) What is the probability that two cards randomly drawn with replacement . egg); 2%: pastime b) What is the probability that two cards randomly drawn without replacement are b u . a a ,3, .. _ 625‘: 2sz ‘“ ioniszazm Ifl. Clearly answer the following questions. Show your work (10 points each, 40 total) A I) For an auto insurance company, three outcomes are possible for any policy (lIlCiEEHCd in thousands of dollars) are: a as) , ~ - bigloss -60 .02 ‘01 Eb Dl f‘ D9($-l* tqD[ZJ smallloss -5 .08 = ~l/Z ~ 'Ll + 1.5) :_ [‘2 no loss +2 .90 a) The expected return from one policy, E(X) = U): = ,2 7, ' A, Calculate the standard deviation of the return fiom one policy = ox = U 77.5,: g, 8 i (find variance first) ’3 @2: 49/2650 v.23?— + £969,232 + 906? —.all "72.49159 + 2. 1537. + Zfllb : 7755 1)) If they have 10 cu ers what is the total (aggregate) expected net return from all 10 customers? /0 [2) F ’L c) Assume the outcomes of different customers are independent. Find the stande deviation of the total aggregate return onthese 10 customers 361; 77g {5; jfl‘tlfifib: 27,5496 N 920 can aJJ rt the true mean p is unknown; and the sample mean, f = 52. 3) Find a 95% confidence interval for the mean value of X. iiZJ,G71/; b) Consider the Null Hypothesis, Ho: 1.10 = 50 and a two-sided i) What is the z-calculated of'an observed sample mean X = 52? Zfla __ int/(p 52"" 50 I 15' ' f/m :' B/q l ii) What is the z*-critical when H0: #0 = 50 ; H“: 1.1,, #3 50.a11d significance level, 0'. = .05? . __ ”P? - 3‘" 2‘3:de 0L , .06 Zap;I ' 2,02; @ iii) What is the p-value of this hypothesis text? Do you reject H0? \{95 P—ualw :2 M2 > (2%?!) : 2(0122): 0) Consider the Null Hypothesis, Ho: 1,10 = 50 and a one sided alternative, IL: 11,, 3* 50. i) What is the z-calculated of'an observed sample mean X = 52? I i _ LL ii) What is the z*-critical when Ho: Ho = 5 ' z ' _ ' canoe level, a = .05? 1‘, 1 51 del Z a; = 1 iii) What is the p-value ofthis hypothesis text? Do you reject H0? Pain @ > 2 15—5 5 I D l 2 2— 3) a) In a widget factory 10% of widgets are defective. Ten widgets are randomly chosen. Let X = number defective in this sample of 10. Use the binomial distribution to find the probability that exactly three are defective in this sample. PG<B:3) : (’30) (10336610J7;[email protected])(001}(Ji752t36q)= $573625: (7 b) Suppose a sample of 100 widgets is taken. Use the Normal approximation to the Binomial with continuity correction to find the probability that i) 11 or fewer in the sample are de ective; ii) 172 or fewer in the s ple are defective; iii) exactly 12 are defective. (A (3 (x84 [(335 PX” < [153: i)? <‘ ”gig —; P(Z< f5\ : @Z M‘vi’lp : mom: 10 = éi;’:"(3ic,'<125=‘ .sgiwk “W amt/073m” G’AWB \ (V ) PCZ< 7; iwfléfifl : W; 3 [171' 3 Mm; {2\ :4 PGLS'OLAK 12-543 —. 15;?th r ‘ 4) Use the true percent return for two difierent investments given below. X=Inv1(%) Probability Y=Inv2(%) Probability 3 .7 2 .6 5 .3 6 ., a) Find the mean of the random variable X variance ( e 8 Li :, , , $133+ - M5") r1; . 7' [3 v3.41“ ,‘3 @‘ié)z : néaéJt (EU-91>) > r54 2 n l + l I S : g ‘ ab 7‘ a b) Find the mean of the random variable variance% ) 1123+ it @A t i-2 +2 ‘I oy‘ : .b (rat-91+ .4 (has? ~— i536 7.; 32L! z a- 84/ Consider a portfolio half comprised of investment 1 and half comprised of investment 2 0) Assume the returns are independent so the true correlation between X and Y, pxy = 0. ar"=;e~ffisgt+.s23. ' : .211»? . , p t z (w) .b flag/56$ For the portfolio find: variance fl\ and standard deviation ii i' ‘7 :l L 08 > (1) Now assume true correlation betweende Y, pxy = .5 I m i age-scgllWfisMyv m {”447 = {A 124% 2724 2, t etzqsiq.9ss>§s ’" -— 1 For the portfolio find: variance I . 102. 1 and standard deviation ( (a. Z 7 2 I e) Now assume true correlation between X and Y, pXY = 1 “7+3” 7.206744777 1.438053 ‘ i ‘ L For the portfolio find: variance and standard deviation 1- L! l \ ...
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