AP Statistics 2006 Free-Response Questions Form B

Transcription

1 AP Statstcs 006 Free-Respose Questos Form B The College Board: Coectg Studets to College Success The College Board s a ot-for-proft membershp assocato whose msso s to coect studets to college success ad opportuty. Fouded 1900, the assocato s composed of more tha 5,000 schools, colleges, uverstes, ad other educatoal orgazatos. Each year, the College Board serves seve mllo studets ad ther parets, 3,000 hgh schools, ad 3,500 colleges through major programs ad servces college admssos, gudace, assessmet, facal ad, erollmet, ad teachg ad learg. Amog ts best-kow programs are the SAT, the PSAT/NMSQT, ad the Advaced Placemet Program (AP ). The College Board s commtted to the prcples of ecellece ad equty, ad that commtmet s emboded all of ts programs, servces, actvtes, ad cocers. 006 The College Board. All rghts reserved. College Board, AP Cetral, APCD, Advaced Placemet Program, AP, AP Vertcal Teams, Pre-AP, SAT, ad the acor logo are regstered trademarks of the College Board. Admtted Class Evaluato Servce, CollegeEd, coect to college success, MyRoad, SAT Professoal Developmet, SAT Readess Program, ad Settg the Corerstoes are trademarks owed by the College Board. PSAT/NMSQT s a regstered trademark of the College Board ad Natoal Mert Scholarshp Corporato. All other products ad servces may be trademarks of ther respectve owers. Permsso to use copyrghted College Board materals may be requested ole at: Vst the College Board o the Web: AP Cetral s the offcal ole home for the AP Program: apcetral.collegeboard.com.

2 Formulas beg o page 3. Questos beg o page 6. Tables beg o page 1.

3 Formulas (I) Descrptve Statstcs = s = 1 1 d s p = d d 1 s + s d 1 + d 1 1 y = b + b 0 1 d d y y b = 1 d b = y b 0 1 r 1 = 1 F HG s IF KJ HG y s y y I KJ b 1 = r s y s s b 1 = d y d y 3

4 (II) Probablty P( A B) = P( A) + PB ( ) PA ( B) PAB ( ) = P( A B) PB ( ) E( X ) = µ = p d Var( X) = s = µ p If X has a bomal dstrbuto wth parameters ad p, the: F HG I K J 1 PX ( = k) = k pk ( p) k µ = p s = p( 1 p) µ p = p p( 1 p) s p = If s the mea of a radom sample of sze from a fte populato wth mea µ ad stadard devato s, the: µ = µ s = s 4

5 (III) Iferetal Statstcs Stadardzed test statstc: statstc - parameter stadard devato of statstc Cofdece terval: statstc ± ( crtcal value) ( stadard devato of statstc) Sgle-Sample Statstc Sample Mea Stadard Devato of Statstc σ Sample Proporto p( 1 p) Two-Sample Statstc Dfferece of sample meas Stadard Devato of Statstc σ1 σ + 1 Specal case whe σ σ = σ 1 Dfferece of sample proportos p ( 1 p ) p( 1 p) Ch-square test statstc = a Specal case whe p p 1 p = p b g + 1 observed epected epected f 5

6 STATISTICS SECTION II Part A Questos 1-5 Sped about 65 mutes o ths part of the eam. Percet of Secto II grade 75 Drectos: Show all your work. Idcate clearly the methods you use, because you wll be graded o the correctess of your methods as well as o the accuracy ad completeess of your results ad eplaatos. 1. A large regoal real estate compay keeps records of home sales for each of ts sales agets. Each moth, the compay publshes the sales volume for each aget. Mothly sales volume s defed as the total sales prce of all homes sold by the aget durg a moth. The fgure below dsplays the cumulatve relatve frequecy plot of the most recet mothly sales volume ( hudreds of thousads of dollars) for these agets. (a) I the cotet of ths questo, epla what formato s coveyed by the crcled pot. (b) What proporto of sales agets acheved mothly sales volumes betwee $700,000 ad $800,000? (c) For values betwee 10 ad 11 o the horzotal as, the cumulatve relatve frequecy plot s flat. I the cotet of ths questo, epla what ths meas. (d) A bous s to be gve to 0 percet of the sales agets. Those who acheved the hghest mothly sales volume durg the precedg moth wll receve a bous. What s the mmum mothly sales volume a aget must have acheved to qualfy for the bous? 006 The College Board. All rghts reserved. Vst apcetral.collegeboard.com (for AP professoals) ad (for studets ad parets). 6 GO ON TO THE NEXT PAGE.

7 . A large compay has two shfts a day shft ad a ght shft. Parts produced by the two shfts must meet the same specfcatos. The maager of the compay beleves that there s a dfferece the proportos of parts produced wth specfcatos by the two shfts. To vestgate ths belef, radom samples of parts that were produced o each of these shfts were selected. For the day shft, 188 of ts 00 selected parts met specfcatos. For the ght shft, 180 of ts 00 selected parts met specfcatos. (a) Use a 96 percet cofdece terval to estmate the dfferece the proportos of parts produced wth specfcatos by the two shfts. (b) Based oly o ths cofdece terval, do you thk that the dfferece the proportos of parts produced wth specfcatos by the two shfts s sgfcatly dfferet from 0? Justfy your aswer. 3. Golf balls must meet a set of fve stadards order to be used professoal touramets. Oe of these stadards s dstace traveled. Whe a ball s ht by a mechacal devce, Iro Byro, wth a 10-degree agle of lauch, a backsp of 4 revolutos per secod, ad a ball velocty of 35 feet per secod, the dstace the ball travels may ot eceed 91. yards. Maufacturers wat to develop balls that wll travel as close to the 91. yards as possble wthout eceedg that dstace. A partcular maufacturer has determed that the dstaces traveled for the balls t produces are ormally dstrbuted wth a stadard devato of.8 yards. Ths maufacturer has a ew process that allows t to set the mea dstace the ball wll travel. (a) If the maufacturer sets the mea dstace traveled to be equal to 88 yards, what s the probablty that a ball that s radomly selected for testg wll travel too far? (b) Assume the mea dstace traveled s 88 yards ad that fve balls are depedetly tested. What s the probablty that at least oe of the fve balls wll eceed the mamum dstace of 91. yards? (c) If the maufacturer wats to be 99 percet certa that a radomly selected ball wll ot eceed the mamum dstace of 91. yards, what s the largest mea that ca be used the maufacturg process? 006 The College Board. All rghts reserved. Vst apcetral.collegeboard.com (for AP professoals) ad (for studets ad parets). 7 GO ON TO THE NEXT PAGE.

8 4. The developers of a trag program desged to mprove maual deterty clam that people who complete the 6-week program wll crease ther maual deterty. A radom sample of 1 people erolled the trag program was selected. A measure of each perso s deterty o a scale from 1 (lowest) to 9 (hghest) was recorded just before the start of ad just after the completo of the 6-week program. The data are show the table below. Perso Before Program After Program A B C D E F G H I J K L Total Ca oe coclude that the mea maual deterty for people who have completed the 6-week trag program has sgfcatly creased? Support your cocluso wth approprate statstcal evdece. 006 The College Board. All rghts reserved. Vst apcetral.collegeboard.com (for AP professoals) ad (for studets ad parets). 8 GO ON TO THE NEXT PAGE.

9 5. Whe a tractor pulls a plow through a agrcultural feld, the eergy eeded to pull that plow s called the draft. The draft s affected by evrometal codtos such as sol type, terra, ad mosture. A study was coducted to determe whether a ewly developed htch would be able to reduce draft compared to the stadard htch. (A htch s used to coect the plow to the tractor.) Two large plots of lad were used ths study. It was radomly determed whch plot was to be plowed usg the stadard htch. As the tractor plowed that plot, a measuremet devce o the tractor automatcally recorded the draft at 5 radomly selected pots the plot. After the plot was plowed, the htch was chaged from the stadard oe to the ew oe, a process that takes a substatal amout of tme. The the secod plot was plowed usg the ew htch. Twety-fve measuremets of draft were also recorded at radomly selected pots ths plot. (a) What was the respose varable ths study? Idetfy the treatmets. What were the epermetal uts? (b) Gve that the goal of the study s to determe whether a ewly developed htch reduces draft compared to the stadard htch, was radomzato used properly ths study? Justfy your aswer. (c) Gve that the goal of the study s to determe whether a ewly developed htch reduces draft compared to the stadard htch, was replcato used properly ths study? Justfy your aswer. (d) Plot of lad s a cofoudg varable ths epermet. Epla why. 006 The College Board. All rghts reserved. Vst apcetral.collegeboard.com (for AP professoals) ad (for studets ad parets). 9 GO ON TO THE NEXT PAGE.

10 STATISTICS SECTION II Part B Questo 6 Sped about 5 mutes o ths part of the eam. Percet of Secto II grade 5 Drectos: Show all your work. Idcate clearly the methods you use, because you wll be graded o the correctess of your methods as well as o the accuracy ad completeess of your results ad eplaatos. 6. Sushe Farms wats to kow whether there s a dfferece cosumer preferece for two ew juce products Ctrus Fresh ad Tropcal Taste. I a tal bld taste test, 8 radomly selected cosumers were gve umarked samples of the two juces. The product that each cosumer tasted frst was radomly decded by the flp of a co. After tastg the two juces, each cosumer was asked to choose whch juce he or she preferred, ad the results were recorded. (a) Let p represet the populato proporto of cosumers who prefer Ctrus Fresh. I terms of p, state the hypotheses that Sushe Farms s terested testg. (b) Oe mght cosder usg a oe-proporto z-test to test the hypotheses part (a). Epla why ths would ot be a reasoable procedure for ths sample. (c) Let X represet the umber of cosumers the sample who prefer Ctrus Fresh. Assumg there s o dfferece cosumer preferece, fd the probablty for each possble value of X. Record the -values ad the correspodg probabltes the table below. p( ) 006 The College Board. All rghts reserved. Vst apcetral.collegeboard.com (for AP professoals) ad (for studets ad parets). 10 GO ON TO THE NEXT PAGE.

11 (d) Whe testg the hypotheses part (a), Sushe Farms wll coclude that there s a cosumer preferece f too may or too few dvduals prefer Ctrus Fresh. Based o your probabltes part (c), s t possble for the sgfcace level (probablty of rejectg the ull hypothess whe t s true) for ths test to be eactly 0.05? Justfy your aswer. (e) The preferece data for the 8 radomly selected cosumers are gve the table below. Idvdual Juce Preferece 1 Tropcal Taste Ctrus Fresh 3 Tropcal Taste 4 Tropcal Taste 5 Tropcal Taste 6 Ctrus Fresh 7 Tropcal Taste 8 Tropcal Taste Based o these prefereces ad your prevous work, test the hypotheses part (a). (f) Sushe Farms plas to add oe of these two ew juces Ctrus Fresh or Tropcal Taste to ts producto schedule. A follow-up study wll be coducted to decde whch of the two juces to produce. Make oe recommedato for the follow-up study that would make t better tha the tal study. Provde a statstcal justfcato for your recommedato the cotet of the problem. STOP END OF EXAM 006 The College Board. All rghts reserved. Vst apcetral.collegeboard.com (for AP professoals) ad (for studets ad parets). 11

12 Probablty Table etry for z s the probablty lyg below z. z Table A Stadard ormal probabltes z

13 Probablty Table etry for z s the probablty lyg below z. Table A (Cotued) z z

14 Table etry for p ad C s the pot t* wth probablty p lyg above t ad probablty C lyg betwee t * ad t*. Probablty p t* Table B t dstrbuto crtcal values Tal probablty p df % 60% 70% 80% 90% 95% 96% 98% 99% 99.5% 99.8% 99.9% Cofdece level C 14

15 Table C Table etry for p s the pot ( χ ) wth probablty p lyg above t. χ crtcal values Tal probablty p (χ ) Probablty p df