Section II. Free-Response Questions -46-

Transcription

1 Sectio II Free-Repoe Quetio -46-

2 Formula begi o page 48. Quetio begi o page 51. Table begi o page

3 Formula (I) Decriptive Statitic x =  x i ( ) 2 1 x =  x x - 1 i - p = ( - 1) + ( ) ( - 1) + ( -1) ŷ = b + b x 0 1 ( x - x i )( y - y i ) Â( x - x)  b = 1 2 i b = y - b x 0 1 r = 1 Ê x - x y y i ˆÊ - ˆ i  - 1 Á Á Ë Ë x y b 1 y = r x b 1 = ( y - yˆ ) 2  i i Â( x - x i ) -48-

4 (II) Probability PA (» B) = PA ( ) + PB ( ) - PA ( «B) PAB ( ) = PA ( «B) PB ( ) EX ( ) = μ x =  xp i i ( ) 2 Var( X) = 2 x =  x - μ p i x i If X ha a biomial ditributio with parameter ad p, the: ʈ PX ( = k) = p k (1 p) -k Á Ëk - μ x = p x = p(1 - p) μ p ˆ = p pˆ = p(1 - p) If x i the mea of a radom ample of ize from a ifiite populatio with mea μ ad tadard deviatio, the: μ x = μ x = -49-

5 (III) Iferetial Statitic Stadardized tet tatitic: tatitic - parameter tadard deviatio of tatitic Cofidece iterval: tatitic ± ( critical value) ( tadard deviatio of tatitic) Sigle-Sample Statitic Sample Mea Stadard Deviatio of Statitic Sample Proportio p(1 - p) Two-Sample Statitic Differece of ample mea Stadard Deviatio of Statitic Special cae whe 1 = Differece of ample proportio p1(1 - p1) p2(1 - p2) Special cae whe p1 = p2 p ( 1 - p) Chi-quare tet tatitic = Â ( oberved - expected) 2 expected -50-

6 STATISTICS SECTION II Part A Quetio 1-5 Sped about 65 miute o thi part of the exam. Percet of Sectio II core 75 Directio: Show all your work. Idicate clearly the method you ue, becaue you will be graded o the correcte of your method a well a o the accuracy ad completee of your reult ad explaatio. 1. Caffeie, a chemical foud i may popular beverage, i kow for reducig fatigue. A tudet wated to ivetigate the caffeie cotet i popular beverage, uch a oft drik, eergy drik, tea, ad coffee. The followig data collected by the tudet how the amout of caffeie (i milligram per 12-ouce ervig) for twelve popular beverage (a) Cotruct a appropriate graphical diplay of the amout of caffeie foud i the twelve beverage. (b) Ue the graph i part (a) to write a few etece decribig the ditributio of caffeie cotet for the twelve beverage. (c) A 12-ouce cup of oe popular gourmet coffee cotai over 300 milligram of caffeie. If thi value wa added to the data et of twelve umber above, how would the mea ad media of the data et above compare with the mea ad media of the ew data et with the thirtee umber? Explai how thi compario could be made without performig ay computatio. -51-

7 2. Member of the reearch ad developmet diviio of a bicycle tire maufacturer are ivetigatig tread life of rubber bicycle tire. They have uggeted that a tudy be coducted to determie whether bicycle tire produced uig a ew ythetic rubber compoud have a loger tread life tha the tread life of bicycle tire produced uig the tadard rubber compoud. A reearcher i the diviio uggeted the tudy be deiged i the followig way. Select 60 idetical bicycle ad radomly aig 30 of thoe bicycle to oe group, A, ad the ret to a ecod group, B. All 60 bicycle will be equipped with frot tire produced uig the tadard rubber compoud. However, the bicycle i group A will be equipped with rear tire produced uig the ew ythetic rubber compoud, while the bicycle i group B will be equipped with rear tire produced uig the tadard rubber compoud. A total of 60 bicyclit will be radomly elected from the populatio of tudet at a local uiverity who regularly ride a bicycle. The 60 bicycle will be radomly aiged to the 60 tudet (with a differet bicycle aiged to each tudet), ad the tudet will be aked to ride the bicycle for a ix-moth period. At the ed of the ix-moth period, the reearcher will compare the mea amout of rear tire tread wear for the bicycle i the two group. (a) What type of deig ha bee propoed for the tudy? What i the repoe variable i the deig? (b) Other tha uig a larger ample ize, decribe a better deig for the tudy tha the oe propoed by the reearcher. Explai why your deig i better. (c) For your deig i part (b), idetify a tatitical tet that could be coducted to determie whether tire produced uig the ew compoud have loger tread life tha tire produced uig the tadard compoud. (You do ot have to carry out the tet.) -52-

8 3. A importat method for cotrollig the pread of the H6N2 iflueza (bird flu) viru i chicke i havig a procedure to determie whether chicke are ifected with the viru. It i commo to apply a procedure, called a ELISA tet, to meaure the cocetratio of ati bird flu atibodie i a blood ample take from a chicke. If the ELISA tet reveal a high-eough cocetratio of atibodie, the chicke i aid to tet poitive, ad it i claified a ifected with the viru. Otherwie, the chicke i aid to tet egative, ad it i claified a ot ifected. However, the ELISA tet i a complex procedure that i ot alway accurate. Oe type of mitake, a fale poitive reult, occur whe the ELISA tet give a poitive reult for a chicke that i ot ifected with the viru. A ecod type of mitake, a fale egative reult, occur whe the ELISA tet give a egative reult for a ifected chicke. Coiderig the poibility of fale poitive ad fale egative for tet o idividual chicke, veteriaria have developed the followig procedure for determiig if the H6N2 viru i preet i a large flock of chicke. Radomly elect 10 chicke from the flock. Perform the ELISA tet o a blood ample from each of the 10 chicke. Coclude that the H6N2 viru i preet i the flock if at leat 3 out of the 10 chicke have poitive ELISA tet reult. Suppoe a veteriaria applie the procedure to a flock of 100,000 chicke at a commercial egg productio farm. The ELISA tet i kow to have probability 0.05 of producig a fale poitive reult ad probability 0.10 of producig a fale egative reult for a igle chicke. (a) If o chicke i the flock i ifected with the H6N2 viru, what i the probability that the veteriaria will coclude that the H6N2 viru i ot preet i the flock? Show how you foud your awer. (b) If o chicke i the flock i ifected with the H6N2 viru, what i the probability that the veteriaria will coclude that the H6N2 viru i preet i the flock? Show how you foud your awer. (c) If every chicke i the flock i ifected with the H6N2 viru, what i the probability that the veteriaria will coclude that the H6N2 viru i preet i the flock? Show how you foud your awer. (d) If 20 percet of the chicke i the flock are ifected with the H6N2 viru ad the other 80 percet are ot ifected, what i the probability that the veteriaria will coclude that the H6N2 viru i preet i the flock? Show how you foud your awer. -53-

9 4. The departmet of park ad recreatio of a certai city coduct ummer program for reidet of it ix ditrict. The ummer program iclude operatig ad maitaiig commuity wimmig pool i each of the ditrict a well a offerig port ad recreatioal program for chool-age childre, youg adult, ad older adult. The table below how the proportio of houehold by ditrict out of all houehold that participated i the ummer program, baed o aual data that were collected from imple radom ample each ummer over a 10-year period, edig i the year The proportio are beig ued by the city for plaig purpoe ad for more efficietly targetig the itroductio of future program. Ditrict A B C D E F Proportio of Houehold City leader wat to tet if the proportio that are beig ued by the city are till valid. Data collected by a tatiticia from a imple radom ample thi pat ummer idicated that the followig umber of houehold participated i each ditrict. Ditrict A B C D E F Number of Houehold (a) The tatiticia claim that the data for thi pat ummer provide evidece that the proportio that are beig ued by the city are o loger valid. Give tatitical evidece to jutify the claim. (b) Which oe of the ix ditrict had the greatet chage i participatio ice the year 2000? Ue the iformatio from part (a) to explai your choice. -54-

10 5. Boe mieral deity (BMD) i a meaure of boe tregth. It i defied a the ratio of boe ma to the croectioal area of the boe that i caed, ad it i expreed i uit of gram per quare cetimeter (g/cm 2 ). Recet tudie ugget that peak BMD i wome i achieved betwee age 15 ad 40, ad BMD declie after age 45. Decreaed BMD i aociated with icreaed rik of boe fracture. I a recet tudy, the impact of regular phyical exercie o wome i differig tage of BMD developmet wa examied. A imple radom ample of 59 wome betwee the age of 41 ad 45 ad with o major health problem were erolled i the tudy. The wome were claified ito oe of the two followig group, baed o their level of exercie activity. Sedetary: miimal participatio i phyical exercie i the pat three year (Thi group cotaied 31 wome.) Walker: walk at a aerobic pace at leat 135 miute per week durig the pat three year (Thi group cotaied 28 wome.) (a) The table below how the mea BMD ad correpodig tadard deviatio for each of the two group of wome. Exercie Group Number of Wome Mea BMD Stadard Deviatio Sedetary Walker A t-tet wa coducted to compare the mea BMD level for edetary wome ad walker. The reult of the tet howed a igificat differece at the 0.01 level of igificace. Baed o the reult, ca it be cocluded that covertig edetary wome ito walker would ecearily icreae their BMD level? Explai. -55-

11 (b) There wa ome cocer that wome i the two group may have differet dietary habit that could affect BMD. For example, higher itake of milk or other food ad upplemet that provide additioal calcium to the body could icreae BMD. To examie thi poibility, the reearcher alo aked each woma i the tudy to report o weekly milk coumptio whe he wa age 20 through age 29. The data were the ued to compute a value of calcium obtaied from milk coumptio for each woma i the tudy. BMD level were plotted agait the calcium itake from coumptio of milk for wome i each of the group. The plot are how below, with the leat quare etimate of a regreio lie o each plot. What do the plot idicate about the relatiohip betwee BMD ad calcium itake from milk coumptio from age 20 through age 29? (c) The lie graph i part (b) ugget that edetary wome ted to have lower calcium itake from milk coumptio tha walker do. Aumig that thi i true, decribe the impact, if ay, that it would have o cocluio that ca be reached from t-tet for comparig mea BMD level for the two group, uch a thoe coducted i part (a). -56-

12 STATISTICS SECTION II Part B Quetio 6 Sped about 25 miute o thi part of the exam. Percet of Sectio II core 25 Directio: Show all your work. Idicate clearly the method you ue, becaue you will be graded o the correcte of your method a well a o the accuracy ad completee of your reult ad explaatio. 6. A egieer i developig a polymer material ad i cocered that the mea deity, d, of the material i ot ufficietly cloe to the deired target value of 1.37 kilogram per milliliter (kg/ml). Nie differet ample of the material were prepared. The volume, i milliliter, for the ample were 10, 20, 30, 40, 50, 60, 70, 80 ad 90. The egieer carefully meaured the ma (i kilogram) of each ample. For the rage of volume, the true deity, d, (i kg/ml) of the material ca be etimated by the lope of the leat-quare regreio lie fit to the reultig data. (Recall that deity i defied a ma divided by volume.) Computer output for the regreio aalyi i how below. Ma = (Itercept) + d*volume + Error Coefficiet: Etimate Std. Error t value (Itercept) Volume = (a) Sice the proce the egieer ued to meaure the ma of ample doe ot alway provide the true deity value, the regreio model how above cotai a radom error term. To ue the t-ditributio to perform a tet of hypothee or cotruct a cofidece iterval for the lope of the leat-quare regreio lie, the radom error mut coform to ome model aumptio. Thee iclude the aumptio that the radom error are idepedet of each other. I the cotext of thi experimet, thi mea that the error that the egieer make i meaurig the ma of oe ample ha o ifluece o the error made i determiig the deity of ay other ample. Decribe two other aumptio about the ditributio of the radom error that are eeded to ue the t-ditributio to perform a tet of hypothee or cotruct a cofidece iterval for the lope of the leat quare regreio lie. (b) Aumig that all of the aumptio that you coidered i part (a) are atified, cotruct a 95 percet cofidece iterval for d, the true deity of the material. With repect to the target deity of 1.37 kg/ml, what cocluio ca be reached? (c) I cotext, what doe the itercept i the leat quare regreio lie that i aociated with the computer output above repreet? -57-

13 (d) Ue your awer from part (c) to explai why it might be reaoable to et the itercept equal to 0 ad coider the reultig alterative model Ma = d*volume + Error a a model for the true deity of the material. The egieer fit a regreio model with a itercept of 0 to the data from the ie ample ad obtaied the followig reult for the leat quare etimate of the lope. Ma = d*volume + Error Coefficiet: Etimate Std. Error t value Volume = The egieer the wated to ue the reult to cotruct a 95 percet cofidece iterval for the lope, but could ot decide if it hould be cotructed a ± (2.306)( ) or a ± (2.262)( ). The firt expreio ue the critical value 2.306, the 97.5 percetile of a t-ditributio with 8 degree of freedom, ad the ecod expreio ue the critical value 2.262, the 97.5 percetile of a t-ditributio with 9 degree of freedom. The egieer decided to perform a imulatio tudy to determie the appropriate formula to ue to cotruct a 95 percet cofidece iterval for the lope. (e) To perform the imulatio tudy, the egieer will imulate ample of ma obervatio uig the model Ma = *Volume + Error for the ie volume of material (10, 20, 30, 40, 50, 60, 70, 80, ad 90) ued i the tudy. The tadard deviatio of the radom error i aumed to be Explai how the egieer ca imulate a ample of obervatio for the ie amout uig a computer program that ca geerate radom ample from a tadard ormal ditributio. (f) Give the method of imulatig ample decribed i part (e), explai how the egieer ca determie which of the two expreio provide a appropriate method of cotructig a 95 percet cofidece iterval for the lope of a leat quare regreio lie, uig a model with a itercept of