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# Statistics ECO 105

DC

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This 10 page Class Notes was uploaded by Khalid Hagenes on Sunday October 11, 2015. The Class Notes belongs to ECO 105 at Davidson College taught by David Martin in Fall. Since its upload, it has received 90 views. For similar materials see /class/221259/eco-105-davidson-college in Economcs at Davidson College.

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Date Created: 10/11/15

Review 1 Economics 204 Statistics Dr Martin Spring 2000 You have 50 minutes for this review Please allocate your time appropriately If you don t know the answer to one part of a question but you need the answer to work a subsequent part then choose a value to use so that you can demonstrate your abilities on that subsequent part Simply write that you are assuming an answer to the part that you couldn t solve To receive full credit for an answer you must show your work A correct final number without an adequate demonstration of the steps leading to that number is not worth full credit You might receive partial credit for an incorrect or an incomplete answer if you show your work or an appropriate diagram Also I can t grade an answer on what you intended to write I can grade an answer only on what you actually write Turn in your formula sheet with your review Please write your name the complete honor pledge and sign the pledge Printed name lt7 Honor pledge Signature The review is worth 100 points distributed as follows Question 1 15 points Question 2 25 points Question 3 30 points Question 4 30 points Question 1 15 points total each part is worth 5 points a Prove that the expected value of the deviation from the mean equals zero Erxw got m an 2 for 474 ng I364 MK Mx 67 MX39 xhj 0 5339 Clearly explain why the mean and variance for a population of grouped data are calculated as approximations Iffm 07571 cfk I mV MK 2M m VJ VI4901474 79quot 0 c4 N amp 77 79 7 17quot G Jfl 0quot ZMM 139A X l c Assume that you have calculated the expected monetary value of two actions A and B to be EMVA100 and EMVB 60 Neither A nor B is dominated by the other action The three relevant states of nature are Federal Reserve raises interest ratesquot Federal Reserve leaves interest rates constant and Federal Reserve lowers interest ratesquot Clearly explain why the expected value of perfect information about which state of nature will occur will be less than the 40 difference between the two expected monetary values zt MQ gs AM VacW Mj 565 M 7W 7 9479 your aLcILro39n Wf gj lw l msfaQU you owl01 69 1427494 gum cu Wed 7 glorify 1 7 49 93 l9 1 a7 ana 76571 5 n474rlt faJ Igr 7 9 C OJG J 07f HA V r J 77394 lgj f 39e 3919quot 4 quotO 4 Question 2 25 points total each part is worth 5 points Assume that you have the following two samples of data Sample A 5437126 I 3 1 Sample B 7431454 I 3 v q l K 6 7 7 57 7 a Calculate the mean of Sample A x19 Ex15 7 Y T39quot7 b Calculate the median of Sample A n 4 7 x T 7 77139 44 397 7 c Calculate the interquartile range for Sample A 36quot 0 34 6 724 IQz Lzsl 25 2g L Jomug L d Calculate the variance for Sample A 3 L Zx1 0 5LMMii9 7 x W 7 9 e Clearly explain why Sample B has a smaller variance than Sample A does even though the two samples have the same sample size mean median and range 7 auva ZMAJawev a VZpm 75 4 mum 5 4 23 x mm mw an 7 ma a41 7 Aa 11 4 gm 9Yquot 8 arm7 6 A 4 Accpr39ttJ VY 1quot 6 Vania Flalay 39 397 F C VLJ g PKG5N4 7 2516742 3 Fgvt Question 3 30 points total each part is worth 5 points J 5 76quot4 t I r c 3 quotJFM 5N4 6amJ4aJa17Mj Karripu and Poonai the two sweetest cats in the world like good music When the music of the band Barenaked Ladies is playing in a room the probability is 4 that the cats are not in the room The music of Barenaked Ladies is playing in the room or the cats are in the room 70 percent of the time The music of Barenaked Ladies is playing in the room half of the time a If the music of Barenaked Ladies is not playing in the room what is the probability that the cats are not in the room Z WL aI d D EM 43 c I LJ s 1413 WA b What is the probability that the cats are in the room and the music of Barenaked Ladies is not playing in the room at the same time Wan274 Wearc2573 53 1 1 4 4 c What is the probability that the cats are in the room and the music of Barenaked Ladies is playing in the room at the same time loCn3 VA PEtA Fcwvz rye 13 A a d What is the probability that the music of Barenaked Ladies is not playing in the room or the cats are not in the room I M3N4 uE Fc 3WL 3 27 from J e What is the probability that the music of Barenaked Ladies is playing in the room if you know that the cats are in the room or that the music ot Burermked Ladies is playing in the room p l0EV4 BAIL ac g LVC39J s U FlaV4 J7 Flavi 11 FwyUM quot 217517 f Are the two variables the music of Barenaked Ladies playing in the room and the cats being in the room statistically independent Show your work F74 Pcn3VL9cngxu39lt 3 1 3 4 I a5m c 00 FV4Pc5 SJgo8VLIC 3 W0 397 If quot397 577IJ7e A Question 4 30 points total Cindy s Shoe Shop has 6 salespeople 3 males and 3 females that can work a 3person shi on Saturdays Cindy randomly assigns her salespeople to work the 3person Saturday shift with the exception being that at least one salesperson must be a female Let the random variable X represent the number of males that work a Saturday shi a 124points Find the probability mction for X ark J A may X O 3C0 x 3 xi 1 r 3 o 17 K2 3C x 3 3x37 I 97 Q 7 7 XL 3CL X 2C 1 3X2 9 7 b 5 points Calculate the expected value of X EDltJ Z7 Z lg L s onj c 5 points Clearly interpret the value that you calculated in part b Harps Mawj Saflf jSJ w gt 74 maca WaysVA fit J 99 d 5 points Calculate the variance of X EZK MXL leHx1 09z 29p 03 j 41 27 J 0 may L 1369 o937 039 3 90 e 5 points Calculate the stande deviation of X 0 fajg 5 7077 Review 2 Economics 204 Statistics Dr Martin Spring 2000 You have fty minutes for this review Please allocate your time appropriately If you don t know the answer to one pan of a question but you need that answer to work a subsequent part then use an assumed value that will allow you to demonstrate your abilities on that subsequent part Simply write that you are assuming an answer to the part that you couldn t solve To receive rll credit for an answer you must show your work A correct nal number without an adequate demonstration of the steps leading to that number is not worth rll credit You might receive partial credit for an incorrect or an incomplete answer if you show your work or an appropriate diagram Also I can t grade an answer on what you intended to write I can grade an answer only on what you actually write Turn in your formula sheet and the photocopies of the statistical tables with your review Please don t write on the statistical tables Please print your name write the complete honor pledge worth a possible two points of extra credit and Sign the honor pledge Printed Name 2 Honor Pledge Signature This review is worth 100 points distributed as follows Question 1 25 points 39 Question 2 25 points Question 3 25 points Question 4 25 points Question 1 25 points total each part is worth 5 points Assume that the random variable X is normally distributed with mean 10 and standard deviation 5 a For what value of X is the cumulative density function equal to 3 D mm leh F a 515 310 62 7377 b What is the probability that X is greater than 13 I3Io f x7ndquot gt gt 639 717 T 17 9393 c What is the probability that X falls between 8 and l3 3quotquot 3 10 75 5 4 X 3 FgfUFxJ I s C 2 9 g l yjj t 71r7 6rrsg I 5 39 d Without calculating the relevant cumulative densities clearly explain how and why your answer to part c would change if the standard deviation increased to 10 0f5 gain 3 wadJ dv gt 9034 A a 0 m eweW M dime 511 WayL ll c CJ 7 y lo 13 70 quote cjvlnuD u e The normally distributed random variable Y has standard deviation equal to 5 but has mean not equal to 10 it is also true that the probability that Y falls between 8 and l3 is the same as you found for X in part c What is the mean of Y 39IJ SJMM VI quot 7 quotG 9 7quot3 I F why 4 gal 719 gaycJ w I Question 2 25 points total each part is worth 5 points Assume that you have a sample of 50 observations ofa random variable for which the sample skewness equals 2 and the sample kurtosis equals 2 Also assume that you wish to test the null hypothesis that the variable is normally distributed against the alternative that it is not normally distributed 3 Using the critical value for the 5 signi cance level test the null hypothesis L v 1 L393L H Y 6 5 0 6 1 7 72257546 LA 43971 045 quotquot2quot 37 1 94 7 3944 VexV 39 quotJ 0 at la b Given your answer to part a and without doing any additional work1 clearly explain what you know must be true about the pvalue for this sample and why it must be true L 7 1 5 1 c Without referring to the statistical tables clearly explain whether the critical value for the test in a will increase or decrease if the signi cance level decreases to 1 and why will 4 synA X M M M74 76 vegv34 A744 Mow39a U74 SM 4 7 Uet 71 lU 72 aft1 vi rank14 AIL mus7 atyf 44 74quot quot91 lo2 M 7l Vamj Ju d Without performing additional calculations clearly explain how and why the pvalue for the test in a will change if the san39tpie skewness 39o 2 and the sample kurtosis increases to 4 all else constant Z 7 MuJ7 Mal M agK 7 JamL 7 74l76 4 7 41 DbK 3971 JM 4 Amyu WU cm7 vriJ aW r AM SIM 4 noA MM m useu A 7 1 w Ajyiu lt J 0 e Given your work in parts a and b and the 5 signi cance level what is the probability that the random variable is normally distributed and why is it that value lmy3a pro ZWJL 139 W OWV Z Q4LI n A DALEquot 0 1 w e2 M Question 3 25 points total each part is worth 5 points y quot9 y k 39 t39 yV A z 9 Iv Assume that the number of people entering a store within one hour is distributed as a Poisson random variab c with standard deviation equal to 2 Each person entering the store makes a purchase decision independently of the other people who entered the store Each person has a probability of making a purchase equal to 7 The store earns revenues of 125 per person who makes a purchase and has xed costs of 30 per hour a What is the probability that more than 12 minutes elapse between people entering the store 3913quot 3 mrner5 lavr 7 gt 7 7 f4 1 7a 396 quot7 773x7 n7 g L b What is the probability that 5 people enter the store within one hour 77 p quotC r efr orwr m x 6ft quot 3101273 Pr 5 11 o c If 5 people enter the store within one hour what is the probability that 3 people will make purchases H AAA Au 56 Avag u JQ F m j nwg NAM ptp x an 3r 2G 37363 13097 d If 5 people enter the store within one hour what are the store s average pro ts and what is that pro t averaged across x 1 t Mm fa x 35 77yr ILS lx393oz lrff 3Uqf7d Q 71 m m o7 m va 4 1 5 fit en v e Given that the number of people entering the store and the number of people making purchases are not statistically independent what is the joint probability of 5 people entering the store within one hour and 3 people making purchases within that hour yzS VZBJ 3 5 XlM 6 1rd gt93305gt7 397 1075 3 Question 4 25 points total Karripu and Poonai two very athletic cats play an interesting version of basketball cat wins by scoring two points although by their strange scoring methods a game can end in a tie You observe a sample of 8 games ipu39s Poonai s Points Pgints 4 Game 1 l 2 I Game 2 2 0 3 Game 3 0 2 quot 1 M 6 Game 5 2 l I Game 6 l 2 I Game 7 0 2 1 Game 8 l 2 l a 10 pts Test the null hypothesis that the median number of points scored by each cat is equal against the alternative that the cats median points are not equal Use a 15 signi cance level uawc 1 xL 4 IJCo Ho 39 LJ n K f 9 y 05 5 7 007g 39 4gr w 4 NRA 7 M 39 l ll s w mlqr T 95 cl7 cy Ag AukKg 1944 3d3g b 10 pts Test the null hypothesis that the number of points that Karripu scores 0 l or 2 is statistically independent of the number of points that Poonai scores 0 1 or 2 using a 10 level of signi cance 740zl lan v 51 A r ACMg l A Ca IJW9 527 In t lt q 7 7a 53quotquot 1 0 JL Qi dr07v0 3 In3957 roJUi 75 z J 2 I pr Aquot 5 21 r T J L fG 2gt5 37l 7 l 13 g Fgrw I37 1 gt0quot c 5 pts What must the critical values for the test in part a equal 39L 9 y 9 75 a quotfflMJf2r9r32 gt i Cf aneJquot26 1 orsc7o07 p r ltolt 5o ar n gjmK WJ at gazed actx WrM Review g Economics 204 Statistics Dr Martin Spring 2000 You have fty minutes for this review Please allocate your time appropriately If you don t know the answer to one part of a question but you need that answer to work a subsequent part then use an assumed value that will allow you to demonstrate your abilities on that subsequent part Simply write that you are assuming an answer to the part that you couldn39t solve To receive ill credit for an answer you must show your work A correct nal number without an adequate demonstration of the steps leading to that number is not worth full credit You might receive partial credit for an incorrect or an incomplete answer if you show your work or an appropriate diagram Also I can t grade an answer on what you intended to write I can grade an answer only on what you actually write Tum in your formula sheet and the photocopies of the statistical tables with your review Please don t write on the statistical tables Please print your name write the complete honor pledge worth a possible two points of extra credit and sign the honor pledge Printed Name a c 1 Honor Pledge Signature This review is worth 100 points distributed as follows Question 1 25 points h Question 2 25 points Question 3 25 points Question 4 25 points Question 1 25 points total Assume that a lucky student had the opportunity to count the number of pieces of food that Karripu and Poonai the two sweetest cats in the world ate during three meals The student collected the data in the table below Pieces of Food Eaten L Meal Number Poonai Karripu cl 1 6 a l 10 25 75 uquot 6 i z 10 2 5 25 o got 1 I ery 3 A cv J 3 15 4o 1r i3 1 15 Go 1w 54 539 a 10 pts Test the null hypothesis that Karripu eats an average of 10 pieces of food per meal more than Poonai does against the alternative that Karripu s average exceeds 10 pieces more than Poonai s average Use the sample statistic scale and a 10 signi cance level At mlg 5 x to C39 Jo 1 I Gymvb lo 3395 3 J 17 39 Ar WK lt o 39 453 74371 44 mm 4 b 5 pts Calculate the pvalue for the test in part a 7 F6 J 79y l o lt 45quot ftgt1 41 175 2 39 9quot c f1 01r r303 I l l I I A 10 lo 1er quotquot75quot L 5 3quot6 5 10Lr4Lu r OF 7 u 9 z c 5 pts Calculate the power for the test in part a assuming that Karripu really eats an average of 25 pieces of food more than Poonai does quot 5 99 ue h IE 845 9 5 9xd4A 25 r L FA Lek7 lt 320 50 357 W 4 q7r WMj 4 d 5 pts Given your answers to parts b and c what is the probability that Karripu really does average eating more than IO pieces of food more than Poonai does 139 an C q 014 Question 2 25 points total each part is worth 5 points Assume that the time between students entering Summit Co ee is a random variable with an exponential distribution and mean equal to 3 minutes Also assume that you have sampled 100 time intervals a What is the probability that a time interval between students entering Summit Cg 39ee is between 35 and 4 minutes x 352740 Effw 535 LA3 he z tzf Jr f 3l f03 2 3J7gt 092906 b What is the probability that the average time interval between students entering Summit Co ee is between 35 and 4 minutes fora sample of size 100 FgmaFay w g3r 67 44739 SF 3 217 A 392 6 Med Finisherw X Q3 hlco 7 I 0 7 7 c Assume that you have calculated that the con dence interval for the mean for your sample has endpoints 35 and 4 What level of confidence are you using 32 mw L 2y33975 Fa 3quotquot rJlt quot0 quot 73 Blo 3r x 4 r2 F 33 4 73 I 7 76 7 lt 0C 2 I 906 d Clearly explam why the answers to parts a and b are dl erenf 4 Lt 7471 au 740 J 071 Ian746 m e Clearly explain what the answers to parts b and c are different MJUM 77C Sew7017 cr37lr 74 3 d esz 7 7 1 or MC C angry L 7 J rearltde 7 5a leM 37J Question 3 25 points total each part is worth 5 points Assume that you want to test the null hypothesis that the mean value of purchases at a store is 10 and that you have a sample of size 25 with mean equal to 7 and standard deviation equal to 10 You will use a 10 Signi cance level for your hypothesis testmg 4 51quot y A a Test the null hypothesis against the twotailed alternative using the test statistic scale Il l o x 0 A X 700 a A 7 7 2 IO 7 IO f1 2 5 j 39 7 5 r quotV 10f ALprif 7 4 44 4 7 r b Test the null hypothesis against the twotailed alternative using the appropriate con dence interval 5 r quot 40 7 1705 I7u 7 17 F quotgt2 2 721 35 2 oy2L 14ccyquot 64 Mb O r w 4 7070 Q 0 Test the null hypothesis against the appropriate onetailed alternative using the test statistic scale Xv log ll 0 2 O H 39 x 40 7 I 3Jgt pej ccgz 74 54W J5 5 3c J y 39 d Test the null hypothesis against the appropriate onetailed alternative using the appropriate con dence interval 5 U 7 I XL K 1quot v 3 quot7 391 A sdo 17 7 7 l 5 3 5 7 634 9 2 Ajej 54 Amy A 744 adsL 7391 90 I e Regardless of your actual work for parts b and d clearly explain whether or not it is possible to reject the null hypothesis in part b while accepting it in part d g ff c A zUJAdaf l C 4A eZuIJ a39 you In 3 139 7 94 71 AL 74 Aquot 40 XLW 4 7 quot SM ff7L 79 39CJ 1513 AM 29 300 t 7 7l Question 4 25 points total each part is worth 5 points Assume that you have a random variable X that is normally distributed with standard deviation equal to 20 Also assume that have a sample of size 25 with mean equal to 20 that you re using to test the null hypothesis that the mean of X equals 100 against the alternative that the mean doesn t equal lOQ at a 5 signi cance level a Calculate the critical values for the hypothesis test on the sample m test statistic scales 0 ALu X 00 12 7g775 00quot 7 J llZx oo 2IL 0 I7Py glIj 0397P b Without doing any additional calculations clearly explain how and why your answer to part a will change if the signi cance level decreases all else constant 7quot cm Va lJ w 7 sL ach VEb 000 Me A 0 faAJJL lt 39 gt M t d 47 al 74L qLM Do 079 Y 239 J LOAII A7L cur7 71 A quot39 9 0 47 c Without doing any additional calculations clearly explain how and why your answer to part a will change if the sample size decreases all else constant 65490 Ve Kr ca 76644 0173 Sm olt u 0a 71 A CJMUu1 My 790 32de tcaAo e 7 JM cr 474 2 I 0 60 A 7c d Without doing any additional calculations clearly explain how and why your answer to part a will chadgg if the sample mean increases to 22 all else constant 71 mas04w 5444 wear a 7 J cuaL mean cenj J e Clearly why we have critical values for the test in the rst place Why don t we automatically reject the null hypothesis since the sample mean doesn t equal 100 W AM 0 acw 7g fwa Sam417 U424 717 Ar 357 a quot5 mg m 3 Mao Jana me vu y0A5 7543 00 puw 10 fyzvo f 72 AL 444 7b acy frm 2 A olt cf wc can Review 5 Economics 204 Statistics Dr Martin Spring 2000 You have titty minutes for this review Please allocate your time appropriately If you don39t know the answer to one part of a question but you need that answer to work a subsequent part then use an assumed value that will allow you to demonstrate your abilities on that subsequent part Simply write that you are assuming an answer to the part that you couldn t solve To receive full credit for an answer you must show your work A correct nal number without an adequate demonstration of the steps leading to that number is not worth full credit You might receive partial credit for an incorrect or an incomplete answer if you show your work or an appropriate diagram Also I can t grade an answer on what you intended to write I can grade an answer only on what you actually write Turn in your formula sheet and the photocopies of the statistical tables with your review Please don39t write on the statistical tables Please print your name write the complete honor pledge worth a possible two points of extra credit and sign the honor pledge Printed Name 27 Honor Pledge Signature This review is worth 100 points distributed as follows Question 1 40 points Question 2 40 points Question 3 20 points Question 1 40 points total each part is worth 5 points Assume that you are testing the hypothesis that the government raises more money from taxes than is necessary Your faithful research assistant Cyn DePooch has collected data on govenunent expenditures and government tax receipts from 19591997 both measured in billions of 1992 dollars to test your hypothesis She ran the SAS program below and generated the results that follow the program Options linesize72 Data one Infile 39cc1assesEc0204testsS 00 Review 4txt39 expandtabs Input Govt Taxes Number n Label Govt 39Real Government Expenditures Bs39 Taxes 39Real Net Tax Receipts Bs39 Number 390bservation Number39 proc corr var govt taxes proc reg Cynmodel govttaxesldw output outtwo rerror proc plot plot error number3939vrefo run Correlation Analysis Simple Statistics w 14 z Variable N Mean Std Dev Sum GOVT 39 965271795 207136807 87646 TAXES 39 840047722 231685575 32762 Variable Minimum Maximum Label GOVT 617200000 1285000000 Real Government Expenditures 85 TAXES 441394336 1338352604 Real Net Tax Receipts 35 Pearson Correlation Coefficients I Prob gt R under Ho Rho0 N 39 GOVT TAXES GOVT 100000 096703 Heal Government Expenditures 35 00 00001 U 439

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