UIVERSITY OF VICTORIA Mdterm June 6, 08 Solutons Econ 45 Summer A0 08 age AME: STUDET UMBER: V00 Course ame & o. Descrptve Statstcs and robablty Economcs 45 Secton(s) A0 CR: 3067 Instructor: Betty Johnson Duraton: hour 30 mnutes Ths exam has a total of _5_ pages ncludng ths cover page. Students must count the number of pages and report any dscrepancy mmedately to the Invglator. Ths exam s to be answered: In Booklets provded Markng Scheme:. 0 marks. 5 marks 3. 5 marks 4. 5 marks 5. 6 marks 6. 0 marks 7. 0 marks 8. 9 marks 9. 5 marks Materals allowed: on-programmable calculator
Econ 45 Summer A0 08 age MULTILE CHOICE. Choose the one alternatve that best completes the statement or answers the queston. Queston : (0 marks). Inferental statstcs s a process that nvolves all of the followng except: A) estmatng a sample statstc B) estmatng a populaton parameter C) testng a hypothess D) Both A and B ASWER: A. A lne graph that connects ponts that represent the cumulatve percentage of observatons below the upper lmt of each class nterval n a cumulatve frequency dstrbuton s known as a(n): A) ogve B) frequency hstogram C) frequency polygon D) scatter plot ASWER: A 3. In a frequency dstrbuton, what s the number of observatons n a class called? A) Class mdpont B) Class frequency C) Class nterval D) one of the above ASWER: B 4. The total area of bars n a relatve frequency hstogram: A) at least B) at most C) exactly D) any value between 0 and ASWER: C 5. Suppose you are told that the mean sample of numbers s below the medan. What does ths nformaton suggest? A) The dstrbuton s symmetrc. B) The dstrbuton s skewed to the rght or postvely skewed. C) The dstrbuton s skewed to the left or negatvely skewed. D) There s nsuffcent nformaton to determne the shape of the dstrbuton. ASWER: C
Econ 45 Summer A0 08 age 3 6. Whch measures of central locaton are not affected by extremely small or extremely large values data values? A) Arthmetc mean and medan B) Medan and mode C) Mode and arthmetc mean D) Geometrc mean and arthmetc mean ASWER: B 7. What s the relatonshp among the mean, medan and mode n a negatvely skewed dstrbuton? A) They are all equal B) The mean s always the smallest value C) The mean s always the largest value D) The mode s the largest value ASWER: B THE EXT TWO QUESTIOS ARE BASED O THE FOLLOWIG IFORMATIO: A recent survey asked respondents about ther monthly purchases of lottery tckets. The monthly expendtures, n dollars, of ten people who play the lottery are 3, 5,, 0, 8, 35, 3, 0, 0, and 4. 8. What can we say about the shape of the dstrbuton of monthly purchases of lottery tckets? A) Skewed to the left. B) Skewed to the rght. C) Approxmately mound-shaped. D) one of the above. ASWER: C 9. Whch of the followng statements are not true? A) The 75 th percentle s equal to 4. B) The medan s equal to the mode. C) The mean s 9.9. D) The dstrbuton s approxmately symmetrc. Answer D 0. Over the past 0 years, the return on Stock A has averaged 8.4% wth a standard devaton of.%. The return on Stock B has averaged 3.6% wth a standard devaton of 0.9%. Whch of the followng statements s true? A) Stock A has smaller coeffcent of varaton than Stock B. B) Stock B has smaller coeffcent of varaton than Stock A. C) Both stocks exhbt the same coeffcent of varaton. D) Unable to tell wth the gven nformaton. ASWER: C
Econ 45 Summer A0 08 age 4 THE EXT FIVE QUESTIOS ARE BASED O THE FOLLOWIG IFORMATIO: The polce leutenant n charge of the traffc dvson revew the number of traffc ctatons ssued by each of the polce offcers n hs dvson. He fnds that the mean number of ctatons wrtten by each offcer s 3. ctatons per day, wth a standard devaton of 3.. Assume that the dstrbuton of the number of tckets ssued s approxmately mound-shaped.. The coeffcent of varaton for the number of ctatons s: A) 3.36% B) 7.48% C) 6.68 D) Cannot be determned wthout the sample sze. ASWER: A. Suppose that you are also told that the medan for these data was 9.3. Whch of the followng statements may be made about the shape of the dstrbuton? A) It s skewed to the rght. B) It s skewed to the left. C) It s approxmately symmetrc. D) Cannot be determned wthout more nformaton. ASWER: A 3. Whch of the followng statements about the medan s not true? A) It s a measure of central tendency B) It s equal to the second quartle C) It s more affected by extreme values than the mean D) It s equal to the mean n bell-shaped dstrbutons ASWER: C 4. Whch of the followng measures of dsperson are based on devatons from the mean? A) Standard devaton B) Varance C) Range D) Both A and B ASWER: D 5. Expressed n percentles, the nterquartle range s the dfference between the A) 30% and 80% values. B) 45% and 95% values. C) 5% and 75% values. D) 0% and 70% values. ASWER: C
Econ 45 Summer A0 08 age 5 6. Whch measure of central locaton s used to determne an average annual percent ncrease? A) Arthmetc mean B) Weghted mean C) Geometrc mean D) Medan ASWER: C 7. A queston n a market survey asks for a respondent's favorte car model. Whch measure of central locaton should be used to summarze ths queston? A) Arthmetc mean B) Geometrc mean C) Medan D) Mode ASWER: D The accompanyng table shows a prce ndex over 0 tme perods. Suppose the government agency updates ths seres by makng perod 6 the base perod wth a value of 00. erod Index 00.0 3.0 3 9.0 4 4.0 5 5.0 6 3.0 7 33.0 8 44.0 9 47.0 0 5.0 8. What would be the revsed prce ndex for perod 0? A) 86.68 B) 09.9 C). D) 5.7 ASWER: D
Econ 45 Summer A0 08 age 6 9. Whch of the followng statements nvolve descrptve statstcs as opposed to nferental statstcs? A) The Alcohol, Tobacco and Frearms Department reported that Houston had,79 regstered gun dealers n 997. B) Based on a survey of 400 magazne readers, the magazne reports that 45% of ts readers prefer double column artcles. C) The FAA samples 500 traffc controllers n order to estmate the percent retrng due to job stress related llness. D) Based on a sample of 300 professonal tenns players, a tenns magazne reported that 5% of the parents of all professonal tenns players dd not play tenns. ASWER: A 0. The wdth of each bar n a hstogram corresponds to the A) mdpont of the class B) number of observatons n the class C) boundares of the class D) percentage of observatons n the class ASWER: C Queston : SHORT ASWER. (5 Marks) Dscuss brefly the dfference between descrptve and nferental statstcs. ASWER: Descrptve statstcs nclude graphcal and numercal procedures that are used to summarze and process data and to transform data nto nformaton. Inferental statstcs provde the bases for predctons, forecasts, and estmates that are used to transform nformaton nto knowledge. Queston 3: SHORT ASWER. (5 Marks) accompany a measure of central tendency? Why s t necessary for a measure of varaton to ASWER: A measure of central tendency alone does not gve a complete pcture of the data set. The object of summary measures s to vsualze the data set based on these measures. Hence, the measure of central
Econ 45 Summer A0 08 age 7 tendency locates the data set, but a measure of varaton completes the pcture by descrbng the dsperson n the data about the locaton measure. Queston 4: (5 Marks) rove that the populaton varance can be equal to the followng: ( X ) X where = populaton sze X X X Snce X ( ) X Expand X X X Take summaton operator through X X then: = X Queston 5: (6 Marks) The followng table of data relates to the number of caterpllars collected n a month by students n several grade 5 classes: Interval (# of caterpllars) Frequency 0 < x 0 0 < x 40 4 40 < x 60 4
60 < x 80 8 80 < x 00 Compute the () mode, mode=70, or 60 to 80 Econ 45 Summer A0 08 age 8 () arthmetc mean, [0+40+700+60+080]/70=3580/70=5.49 () the medan observaton 35 and 36: 50, or 40 to 60 (v) the populaton varance: =[(69.738)()+447.0(4)+.306(4)+355.590(8)+509.874()]/70 =508.5568/70 =730.4 Sheppard s correcton: 730.4-400/=696.7888 (v) the standard devaton 6.3968 (v) coeffcent of varaton. 5.638% Queston 6: (0 Marks) a) Construct a frequency hstogram to descrbe graphcally the tme (n hours) that students studed for the test. b) Construct a ogve. umber of Hours Frequency.0 but < 3.5 4 3.5 but < 5.0 8 5.0 but < 6.5 5 6.5 but < 8.0 3 Hstogram of study tme Hours 9 8 7 6 5 4 3 0.75 4..5 5.75 7.5 Class mdpont
Econ 45 Summer A0 08 age 9 Cumulatve Frequency 0 Ogve Ogve 0 5 Queston 7 (0 marks) Campus Securty has revewed the number of parkng ctatons ssued per month by each of the seven offcers. The data were:, 9, 0,, 0, 4, and. a) What s the mean number of tckets ssued per day? µ=(6)/7=6.574 b) What s the medan number of ctatons ssued per day? 0 9 0 4 9 = medan ( to < 3.5) (3.5 to < 5) (5 to <6.5) (6.5 to 8) c) What s the mode of the number of tckets ssued per day? d) What s the frst quartle of the number of tckets ssued per day? 5=(*K)/00 =(7*5)/00=.75 locaton: round up to nd observaton:. e) What s the nterquartle of the number of tckets ssued per day? 75=(7*75)/00=5.5 locaton: round up to 6 th observaton: Interquartle range= -=0
Econ 45 Summer A0 08 age 0 Queston 8: (9 marks) (a) Use the followng data to construct the Fsher prce ndex for 06, wth a base value of 00 n 995: (Table your answer wth the two entres.) (3 Marks) Good 995 995 06 06 rce Quantty rce Quantty Eggs 3 4 5 0 bread 5 4 7 995,06 995,06 995,06 L 995,06 L 995,06 L 995,06 5 0 4 7 3 0 7 00 08 60 54 00 00 08 00 8.456404 4 (54) 4 5 34 5 70 00 (4) 50 00 00 70 00 84.786087 9 F 995,06 84.786087 8.456404 3374.76 83.6569 Year Index 995 00 000 83.6569 (b) Generate the Fsher quantty ndex wth a base value of 00 n 995. (3 marks) Q Q Q L F (0)(5) (7)(4) 08 00 00.359 (4)(5) (5)(4) 70 (0)(3) (7)() 4 00 00 3.9304 (4)(3) (5)() 9 3.9304.359 53.0 3.305
(c) Q Econ 45 Summer A0 08 age Determne f the Fsher ndex passes or fals the factor reversal test. (3 Marks) pq t p q 0 0 passes. t 08.60869 9.836569.3305.60869 Queston 9: Evews (5 Marks) Assume the data s a populaton. () What s the standard devaton of STUFF? squroot(3906.933/5)=6.388 () What s the skew of STUFF usng the smple earson measure? [4.6667-37]/6.388=0.644 () How many observatons are n the seres? =5 (v) What s the formula for Sum Sq. Dev.? X (v) What s the range of the data? 57
Formulae Econ 45 Summer A0 08 age Central Locaton: Arthmetc mean x (Grouped data x f x f f x w Weghted arthmetc mean W / w ) Geometrc mean G x Harmonc mean H x Dsperson: opulaton varance x (Mean squared devaton) (Grouped data x f ) Sheppard's correcton c h ( n ) Mean absolute devaton MAD = x Coeffcent of varaton CV = ( / ) 00 Sample varance s x x ercentles: K k /00 Other Measures: Skewness coeffcent skew = ( - medan) / rce Indces: Laspeyres' L = [ p t q 0 ] / [ p 0 q 0 ] aasche's = [ p t q t ] / [ p 0 q t ] Fsher's "deal" F = [ L ] / M.E. rce Index: ME p 0 q 0 q t q0 q t p t M.E. Quantty Index Q ME q t q 0 p 0 p 0 p t p t
Quantty Indces: Laspeyres' Q L = [ q t p 0 ] / [ aasche Q = [ q t p t] / [ Fsher's "deal" Q F = [ Q Q L ] / q 0 p 0 ] q 0 p t ] Econ 45 Summer A0 08 age 3 Tme Reversal test:( * ) t0 Factor Reversal Test: 0 Q0 * p q p q 0 0