Probability and Statistics
Probability and Statistics ECON 209
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This 9 page Class Notes was uploaded by Aliya Schumm PhD on Wednesday October 28, 2015. The Class Notes belongs to ECON 209 at Vassar College taught by Staff in Fall. Since its upload, it has received 31 views. For similar materials see /class/230541/econ-209-vassar-college in Economcs at Vassar College.
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Date Created: 10/28/15
Econ 209 1 Review Problems 927znnz Do not do starred Problems Letz be a standatd nozma1tandom vanab1e a Fmd P z gt 12 bF1ndP711ltZlt12 c PZ gt a 7 44s 3 Fmd a d PZgta e 3030 Fmd a Letx be a nonna1 xandom mab1e vnth mean 50 and standatd dmanon 5 a Fmd P x gt 52 b Fmd P 52lt x lt 55 cPXgtAr 15 FmdA Twenty petcent otthe new cats that comeoffthe assenb1y hne have majoz detects Betote a11thtee mmectoxs Inspectots nevet nd nonaxistmt ptob1nns a What 15 the pzobabihty that a detectwe mwm be shspped b What 15 the ptobabthty that a cazwhich 15 shspped 15 pettect c1ass 15 chosen at xandom and two petsons ate se1ected om It a What 15 ptobabthty that the tstpetson chosen 15 a women woman c1ass 3 As a quahty Gonna measute she tomne1y mmects one Item om each shipment Ltthe Item b 1n otdet to be mote ceetam shedemdes to Inspect a second 1mm om each shxpmmt Itthe tst Item Inspected 15 pettect what 15 ptobabthty that the second 15 detectwe snowman we the my Hume 10 What 15 the plobablhty that he makes at 1east one sa1e on a gwm day the swItchboazd What 15 the plobablhty that m any gwm mmute the swItchboazd 15 ovet1oaded In 10 what 15 the mobabihty that thete 15 a two mmute um mteeva1 with no ca11s times 1 aces ones come up 5 l rth mamaquot u al How populataon oonslsnng of so mm and 40 women Fmd the mlopetsons ate selected mm a 5m alant llypetgeomemo mean and vanance of the numbetorwomen cho equally matched What asthe pzobablhly that the wold senes W111 end aftet 4 games aftet 5 games aftez games aftez7 games Dif cult x N N2016 Find EM and E100 10x 2 spades ve hearts and 1 club and ldlamond Show lfA as andependent ofB then B as andependent ofA UseaVenn dlagamtoshowthat 3A g 5 5 EU 0 oxtoshow thatthas statement as nottnae votetwould have oonectly ptedaoed the wannet 3 votets 9 vctexs IDD votets n m th mm 30 and at sea 20 An aat seatohas beang planned The ptobabalaty thatthe seatch masses the aasplane as 30 anthe mountaans zoantheptaane and so at sea a Betotethe seatoh as started up nothang What as the plobablhty that the massang plane as an the mountaans house the paotute Wmdow lleas tzyangto guessthe sex ottheothet ch d lle teasons as follows the famlly as elthet 1313130 GB 01 GO Sanoehe famlly types Intwo o ut So he teasons that the pzobablhty that the unseen ch d as a gtl as 23 Do you agxee Y 100000 wavn oeetaanty Calculate themean and vananoe otthe galn an each case Whlch would you choose Why Econ 2H9 Review Problems 9272nn2 Solutions 1 us Let z be a standazd nonnal zamom vanable a Pltzgt12gt57pwltzlt12gt757384971151 b F11ltzlt12P0ltzlt11F0ltzlt12364338497492 Pzgta74483Poltzlta70517a7013 d Pltzgta3030Pzgta1515Pultzlta3385 10 Letx be a nonnal zandom vanable wath mean so and standazd dmanon 5 x750 527 50 5 12 5 52750 55750 b P52ltxlt55 PTlt2ltT P4ltzlt134115541859 Pa gt 52 7 Pz gt 4 7 571554 7 3446 A750 PxgtAPzgt 5 15 ASU 5 104 A505104552 S our shipped D car dzfectxve PS D 7 333 7 027 LetD 7 dance class second a s 7 stansncs Class M 7 woman chosen zst W2 7woman chosen W1 PDn W1 P Sn W1 7 PDPW1D PSPW1S 7 121 12127 34 PWln W2 PDn Wln W2 PSn Wln W2 WW W 7 1211 1212919 7 825 39 3 4 quot PW1n W2 DPD 13mm W2 PWln W2 DPD PWln W2 SPS 1mm 2 809 11121291912 39 LetA17 as Item palm A2 7 secmd 1mm pnfect s 7 shipment peszt 3 PA1SPS 18 P 311 7 777 7787 PA1SPS PA1SPS 18 62 b 7 PA2 nA1 PA2 nA1mpm PA2 nA1S PS HA2 W 7 PA1 PA1 0X8 462 ms mm 052 Ea bx 211 bxPx Z aPx 2 beOc 7 a2 Px 172 x1300 7 a 17 Let Fi Probability that he fails on the ith call Patleastonesale 71 1n Fin F3n F40 F5 1 PF1PF2PF3PF4PFSFln Mn Fan F4 7 17 7 1 7784 Poisson A 5 Pxgt 10J70082 0034 0013 0005 000270 0136 Exponential A 7 s Pzgt 2 7 2749 7 2quot 7 0000454 6 7 5 x Nmmt17 If 7 0014160 27 7 204 P95 lt x lt 105 7 P220 lt z lt 259 7451547485170103 5 A test pnsmve D 7 d1sease plesent PD 01 PAD 95 PAD39 02 a PM PADPD Pamyum 95X 01 0299 0293 Expense osmves 293 11 ofp b POD PADPD 9501 32 P05 0293 PA DP 05 01 PDIA 00515 Expemea ofmxsses 005159707 50 N 7 DD 1 7 4D 11 760 x inumbex Bfwtymm chosen Em 2 1040100gt4 M 100710 Ina 1 7 l001104010050100 218 The pmbab uythax me ALtuam w39ms 1n m smegma 2 116 0625 5 71250 The pmbabmrychax the emhezteam w1ns 1n out snmghtr 2062 4 rst fouzgames andthmwm Lhe fthgamerLJ 1 231212 125 The pmbablhty that nthezteam w1ns1n ve games 2125 2250 By s1m1lazzeasnmngthepmbab y Dfasm game smes 2 123122123125 and the pmbabmty era seven game smes 15 2Q 12312312 3125 x N2016 E02 01 Ex 1s207 416 E10010x 2x 10010Ex25x1100200 832 1132 15115113113 ii PAn B PAPB 16 PAB 7 PA PBA 7 W W7 133 17 True xx n71 p 7 n73 p 7 sax n79 p 7 7332 n7mo p 7 9 72 Fox the n 7 100 case you can use the normal appxoxmnon to the Mmmm letx WWW 39 H mm m Then 7 Nmzmz17 71 N5o24 505 7 60 Pgtc gt 505 7 Pz gt Pz gt 7 194 75 4738 79738 19 LetM P and s 7 respectively theevmtsthattheplane was lost m the mountams the pmme and the sea Lem A 7 the emeLhaLLhe searchmlssestheplane LeLF 7 the evmt manhe Search mmsup nothmg Then PM 05 PP 03 1903702 05 a PM PMn F PMgtPAMgt 53J 1 PM DWWWB PMnF AIM 0 MW PltFgt PltMgtPltwgtPltPgtPltA4PgtPltsgtPltA4sgt 5 2 5 3 3Zgt Z8gt NoteLhaLLhe evenLF changes bemweenb and c 37 Let B the event that a boy is seen thru the window PGG PBG PGB P0313 1 4 PBiGG 0 PBBG 12 PBGB 12 PBBB 1 P0313 B PB t BB PBBPBBB 7 P03 PBBPBBB PBGPBBG PGBPBGB 14 1 14 mu4 1214 12 1 let RA 7 the return on opportunity A RE the return on opportunity B Eon 50000012 100000012 250000 of 7750000212 750000212 562500000000 a 750000 ERE 100000 Di 0 In P101301 A ha 5 the higher exp ected return but project B is iess n sky Economics 209 Fall 2002 Review 11 a A sample of sizen 25 is drawn from a normal population with unknown mean and standard deviation 10 If the sample mean is 756 nd a 95 confidence interval for the population mean b a 99 con dence interval a A sample of size n 25 is drawn from a normal population with unknown mean and standard deviation If the sample mean is 756 and the sample standard deviation is 10 find a 95 con dence interval for the population mean b a 99 con dence interval Why are the confidence interval which you found in 2 wider than those which you found in 1 The following sample is drawn from a normal population 793266984 Find a 95 con dence interval for the population mean The following sample is drawn from a normal population 34 37 33 40 Find a 99 confidence interval for the population mean Which is the most efficient estimator ofthe mean ofa population a The sample mean from a sample of size 100b The sample mean from a sample of size 40 or c The mean of the sample means from aand b Which is the most efficient estimator ofthe proportion of persons in apopulation who favor some form of gun control a The proportion favoring gun control in a sample of size 100 b The proportion favoring it in a sample ofsize 20 or c The mean ofthe sample proportions in a and b Which is the more ef cient estimator of the mean of a population of size N 100 a The sample mean from a sample of size 50 drawn with replacement b The sample mean from a sample of size 25 drawn without replacement or c The average of the two samples In a sample of 100 Democrats 55 preferred Bradley to Gore Find a 95 confidence interval for the proportion of Democrats in the population who supp ort Bradley Find a 95 upper confidence limit for the population mean given the sample in question 5 A rm is testing a new production process Under the old method the proportion of defective units was 20 The following test is used One hundred test units are produced and inspected 12 are defective Test the hypothesis that the new process is no different from the old A coin is ipped 100 times 41 heads occur a Test the hypothesis 1391 50 against the alternative that H gt 50 b How would your answer change if the alternative hypothesis was 1391 5O 20 of the jury age population in a certain Alabama county are African Americans However only 30 African Americans appear among the last 400 persons called forjury duty Does this evidence suggestthat the jury selection process discriminates against African Americans Clearly state the null and alternative hypotheses and the results ofthe hypothesis test Use a 5 signi cance level A cereal manufacturer claims that the average amount ofsugar in a one pound box ofcereal is at most50 grams with a standard deviation of 10 grams In order to test that claim tl following test is de ned A sample of2 5 boxes ofcereal is selected and the me an nount of sugar X is calculated If X gt X5 the null hypothesis is rejected What should the value of X C be if the significance level ofthe testisto be 5 You should clearly state what the null and alternative hypotheses are A public opinion polling organization claims that99 of the time its estimates ofthe proportion ofpersons favoring a particular candidate will be within three percentage points ofthe actual pro portion ofpersons in the population favoring that candidate How large a sample is necessary to justify that claim Explain the reduction factor N nN 1 in words A tobacco company claims that its cigarettes contain an average ofless than one milligrams of nicotine per cigarette The standard deviation ofthe amount of nicotine contained in a cigarette is 05 milligrams In order to test the claim the following test is carried out A sample of25 cigarettes is selected and tested Ifthe average amount of nicotine in the sampled cigarettes exceeds 11 miligrams the companies claim is rejected a What are the null and alternative hypotheses for this test Is this a one sided or a two sided test Why is this appropriate bFind the significance level ofthis test c How many cigarettes should be sampled ifthe significance level ofthis test is to be 01 A firm is testing a new production process Under the old method the proportion of defective units was 20 The following test is used One hundred units are produced with the new process and inspected If 12 or fewer are defective it is concluded that the new process is superior to the old process a What is the significance level ofthe test b How many chips should you produce ifthe significance level of this test is A firm which produces SAT preparation courses claims that 80 ofthe persons who take their course achieve a score of at least 600 on the math SAT In order to test that claim you construct the following test A random sample of 100 persons who have taken the test is selected and the proportion P who achieved a math score of 600 is calculated If the proportion achieving that score isless than a critical value Pc the claim of the firm is rejected What should P6 be ifthe significance level ofthis test is to be 01 A sample of size 25 is drawn from a normal population The sample mean is 72 and the sample variance is 64 a Test the hypothesis that the population mean is 68 against the alternative hypothesis that the population mean is greater than 68 b How would your answer change if the alternative hypothesis was that the population mean was different than 68 You are the president of a firm which manufactures hard drives for computers You are considering coming out with a new model ofhard drive The model which you are now producing fails on the average 1 of the time in the first mo nth Your chief engineer assures you that this model will fail less frequently that The following test is designed 1000 hard drives are produced and tested If less than 5 fail in the first month it is concluded that the new model is superior Find the significance level of the test How would the following information affect your choice of a significance level a The chief engineer is willing to stake his reputation on the fact that the new drive is superior and you have reason to trust his judgement b The changover to the production of the new model is very costly Two makes of typewriter are to be tested Six typists are randomly chosen and tested on machine A Six more typists are tested on typewriter B The results are typewriter A typewriter B 52 82 60 70 68 76 70 90 80 82 74 a Would you conclude that one typewriter is faster You may assume that typing speeds are normally distributed b How would your answer change ifthe two machines were tested with the same typists In order to investigate whether men or women are more likely to favor gun control independent samples of 100 men and 100 women are polled The results are that the proportion of men favoring gun control is 60 and the proportion ofwomen is 70 Test the hypothesis that there is no difference between ofmen and women supporting gun control Samples of size 100 and 200 are drawn respectively from two normal populations A and B The results are A sample size 100 sample mean 64 sample variance 30 B sample size 200 sample mean 67 sample variance 49 Test the hypothesis that the population means are the same