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1 1. A random sample of 100 people were asked f ther fnances were worse, the same or better than ths tme last year. The sample was splt accordng to ther annual ncome and the results are shown n the table below. Test, at the 5% level of sgnfcance, whether or not the relatve state of ther fnances s ndependent of ther ncome range. State your hypotheses and show your workng clearly. (Total 10 marks). A research worker studyng colour preference and the age of a random sample of 50 chldren obtaned the results shown below. Age n years Red Blue Totals Totals 8 50 Usng a 5% sgnfcance level, carry out a test to decde whether or not there s an assocaton between age and colour preference. State your hypotheses clearly. (Total 11 marks) dexcel Internal Revew 1

2 3. People over the age of 65 are offered an annual flu njecton. A health offcal took a random sample from a lst of patents who were over 65. She recorded ther gender and whether or not the offer of an annual flu njecton was accepted or rejected. The results are summarsed below. Gender Accepted Rejected Male Female Usng a 5% sgnfcance level, test whether or not there s an assocaton between gender and acceptance or rejecton of an annual flu njecton. State your hypotheses clearly. (Total 9 marks) 4. A researcher carred out a survey of three treatments for a frut tree dsease. The contngency table below shows the results of a survey of a random sample of 60 dseased trees. Tree ded wthn 1 year Tree survved for 1 4 years Tree survved beyond 4 years No acton Remove dseased branches Spray wth chemcals Test, at the 5% level of sgnfcance, whether or not there s any assocaton between the treatment of the trees and ther survval. State your hypotheses and concluson clearly. (Total 11 marks) dexcel Internal Revew

3 5. A random sample of 500 adults completed a questonnare on how often they took part n some form of exercse. They gave a response of never, sometmes or regularly. Of those asked, 5% were females of whom 10% never exercsed and 35% exercsed regularly. Of the males, 1.5% never exercsed and 55% sometmes exercsed. Test, at the 5% level of sgnfcance, whether or not there s any assocaton between gender and the amount of exercse. State your hypotheses clearly. (Total 1 marks) 6. A new drug to treat the common cold was used wth a randomly selected group of 100 volunteers. ach was gven the drug and ther health was montored to see f they caught a cold. A randomly selected control group of 100 volunteers was treated wth a dummy pll. The results are shown n the table below. Cold No cold Drug Dummy pll Usng a 5% sgnfcance level, test whether or not the chance of catchng a cold s affected by takng the new drug. State your hypotheses clearly. (Total 11 marks) dexcel Internal Revew 3

4 1. Fnances Worse Same Better Income Under M and above A H 0 : State of fnances and ncome are ndependent (not assocated) H 1 : State of fnances and ncome are not ndependent (assocated) O ( O ) O M A ( O ) O = or = = (awrt 3.55) A1 ν = ( 3 1)( 1) = cv s < so nsuffcent evdence to reject H 0 or not sgnfcant M1 There s no evdence of assocaton between state of fnances and ncome. A1 dexcel Internal Revew 4

5 Note 1 st Row Total Col.Total M1 for some use of. May be mpled by correct Grand Total 1 st A1 for all expected frequences correct for both hypotheses. Must menton state or fnances and ncome at least once Use of relatonshp or correlaton or connecton s B0 nd M1 for at least two correct terms (as n 3 rd or 4 th column) or correct expressons wth ther nd A1 for all correct terms. May be mpled by a correct answer. ( dp or better-allow eg 1.13 ) 3 rd M1 for a correct statement lnkng ther test statstc and ther cv. Must be χ not normal. 4 th A1 for a correct comment n context must menton state or fnances and ncome condone relatonshp or connecton here but not correlaton. No follow through. e.g. There s no evdence of a relatonshp between fnances and ncome [10]. H 0 : No assocaton between age and colour (Independent) H 1 : Assocaton between age and colour (Not ndependent) O (O-) at least one R T CT, 1 G T M1 A1 (O-) M1 A1 3 s.f. or better dexcel Internal Revew 5

6 (O-) awrt = , M1 A1.44 ν = (3 1)( 1) =, χ = ft Insuffcent evdence to reject H 0 No assocaton between age and colour A1ft [11] 3. H 0 : No assocaton between gender and acceptance H 1 : gender and acceptance are assocated Accept Not accept Total Males 170 (180) 110 (100) 80 Females 80 (70) 140 (150) 40 Totals xpected Values M1 A1 O ( O ) ( O ) =.59 (Yates.34) (Condone use of Yates ) M1 A1 ν = 1; (5%) = ; >.59. There s nsuffcent evdence to reject Ho M1 There s no assocaton between a persons gender and ther acceptance A1 (of the offer of a flu jab.) 9 [9] dexcel Internal Revew 6

7 4. No acton Remove dseased branches Spray wth Chemcals Totals Tree ded wthn 1 year 10(7) 5(7) 6(7) 1 Survved 1 4 years 5(7) 9(7) 7(7) 1 Survved > 4 years 5(6) 6(6) 7(6) 18 Totals RT CT GT M1 6 7 A1 3 6 A1 H 0 : Treatment & survval are ndependent (not assocated) H 1 : Treatment & survval are not ndependent (assocated) α = 0.05 = (3 1) (3 1) = 4 CR: χ > ( O ) = Use of ( O ) both ft M1 Any values = A1 awrt 3.48 A1 Snce s NOT n the crtcal regon (e < 9.488) there s nsuffcent evdence to reject H 0. There s no evdence of assocaton between treatment and length of survval. Comparson Concluson M1 A1ft [11] dexcel Internal Revew 7

8 5. Males Females Never Sometmes Regularly Totals M1 convert % to freq A1 (6, 91, 30, 13) A1 (143, 78) H 0 : No assocaton (ndependent) between gender and exercse H 1 : assocaton (not ndependent) between gender and exercse xpected Values Never Sometmes Regularly Totals Males Females M1 A1 at least 3sf α = 0.05 ν = ; CV χ > ; ft ( O ) O OR N = M1 A1 answers n range Not n crtcal regon no evdence of assocaton between A1 ft gender and exercse [1] dexcel Internal Revew 8

9 6. H 0 : Takng drug and catchng a cold are ndependent (not assocated) H 1 : Takng drug and catchng a cold are not ndependent (assocated)(not dtto) Both All totals RT CT = GT M1 A1 A1 Cold NoCold Drug 34(39.5) 66(60.5) 100 Dummy 45(39.5) 55(60.5) O ( O ) ( O ) =.53 (NB wth Yates.09) attempt & add, awrt & 0.5 twce, awrt.53 M1 A1 A1 v = 1, χ 1 (5%) = >.53 1, 3.841, No reason to beleve that the chance of catchng a cold s affected by takng the new drug A1 [11] dexcel Internal Revew 9

10 1. For most canddates ths queston was a good source of marks. Hypotheses were usually correctly phrased n terms of ndependence or assocaton and the calculatons were usually clearly set out although some napproprate roundng sometmes gave an answer of The degrees of freedom and crtcal value caused few problems and most gave a correct concluson n context.. There were some excellent responses wth a large number of correct answers seen. It was unusual not to see hypotheses well stated and the concluson gven correctly n context. 3. Ths queston was answered well and most canddates scored full or almost full marks. The xpected frequences were almost always correct as was the calculaton of the test statstc. Occasonally the hypotheses were the wrong way around and sometmes the concluson was not gven n context. 4. Many canddates produced completely correct solutons to ths queston. Ill-defned hypotheses, poor arthmetc and not gvng the concluson n context were the common errors. 5. Ths queston was very well done wth even very weak canddates managng to gan 11 or 1 marks. Some canddates tred to use percentages rather than frequences and a few dd not seem able to use the χ table correctly. As s often the case hypotheses were sometmes reversed. 6. There was a szeable group of canddates who answered ths queston well as they were able to demonstrate a good grasp of the mathematcal technque requred. The most common errors nvolved accuracy n the workng, careless defntons of the hypotheses and mssng context n the concluson. dexcel Internal Revew 10