AP Statistics Testbank 8

Transcription

1 AP Sttistics Testbnk 8 Multiple-Choice Questions ) Suppose tht you re given sttistics s below: Smple size Smple men Smple Devition Smple Smple The P-vlue ssocited with the hypotheses is closest to = µ, : : µ µ µ ).7 b).3 c).87 d). e).4 ) Suppose tht you re given sttistics s below: Smple size Smple men Smple Devition Smple Smple A 95% confidence intervl for µ is ).9 ±. b).9 ±. c) (.67,.67) d).9 ±.67 e).9 ±.665

2 3) Suppose tht you re given sttistics s below: Smple size Smple men Smple Devition Smple Smple Assuming tht the underlying popultions hve the sme vrinces, the P-vlue ssocited with the hypotheses is closest to ).48 b).89 c).3 d).65 e).935 = µ, : : µ µ < µ 4) Suppose tht we select 3 observtions from ech of two popultions, with the following results: Smple Smple x = 5,87 x = 5, 5 s = 5 s = 75 An estimte for the stndrd devition σ x x is given by ) 3.3 b).4 c) 6.98 d) 77.6 e) 3.49

3 5) Suppose tht we select 3 observtions from ech of two popultions, with the following results: Smple Smple x = 5,87 x = 5, 5 s = 5 s = 75 A test of the hypotheses hs P-vlue : µ =, : µ > ). b).73 c).3 d) e).5 6) Suppose tht we select 3 observtions from popultions nd, where the respective stndrd devitions re denoted σ nd σ. A test of the hypotheses is bsed on the z-sttistic, where : µ =, : µ > ) b) c) z = z = z = x x σ σ x x σ σ x x σ σ d) x x z = σ σ e) z = x x σ σ + 5 5

4 7) Suppose tht we hve tken 5 smples from popultion nd smples from popultion, with the results tht x =.8, s = 3.4, x =.8, s = 4.6. Wht would be the result of testing the hypotheses t the 5% level? = µ, : : µ µ > µ ) Reject becuse P=.47. b) Do not reject becuse P=.3. c) Do not reject becuse P=.47. d) Reject becuse P=.3. e) Do not reject becuse P= ) Suppose tht we hve tken 5 smples from popultion nd smples from popultion, where we know in dvnce tht for these popultions, σ = σ. Assume next tht the result of these smples is tht x =.5, s.9, nd tht x =.8, s. 3. The P-vlue ssocited with the test of hypotheses 4 = 5 = is in the intervl = µ, : : µ µ µ ) P <. 5 b). 5 < P <. c). < P <. 5 d). 5 < P <. e) P >. 9) Suppose tht mtched-pirs design gve rise to the following dt: x x Assume tht µ nd µ re the corresponding popultion mens, nd tht the popultion of differences x x is pproximtely normlly distributed. At 9% confidence intervl for µ is given by ) 3.± b).6 ± c) 3.±. 83 d).6 ±. 83 e) (.5693,.83)

5 ) Suppose tht mtched-pirs design gve rise to the following dt: x x Assume tht µ nd µ re the corresponding popultion mens, nd tht the popultion of differences x x is pproximtely normlly distributed. A test of the hypotheses = µ, : : µ µ > µ t the % level will result in ) Not rejecting becuse P=.56. b) rejecting becuse P=.6. c) not rejecting becuse P=. d) not rejecting becuse P=.5. e) rejecting becuse x > x. ) In simple rndom smple of Toyot cr owners, 83 out of sid they were stisfied with the Toyot front-wheel drive, while in similr survey of Subru owners, 76 out of 8 sid they were stisfied with the Subru front-wheel drive. Wht is 9% confidence intervl for the difference in proportions between Toyot nd Subru cr owners who re stisfied with their drive systems? ). 97 ±. 4 b). 97 ±. 49 c). 97 ±. 8 d). 97 ±. 97 e). 97 ±. 6 ) In simple rndom smple of Toyot cr owners, 83 out of sid they were stisfied with the Toyot front-wheel drive, while in similr survey of Subru owners, 76 out of 8 sid they were stisfied with the Subru front-wheel drive. Assume tht the 9% confidence intervl for the difference ( p p ) in proportions between Toyot ( p ) nd Subru ( p ) cr owners who re stisfied with their drive systems is. 97 ±.8. Now consider the sttements. I. Using this confidence intervl, we my reject : p = p in fvor of : p p t the % level. II. Using this confidence intervl, we my reject : p = p in fvor of : p < p t the 5% level. III. Using this confidence intervl, we my reject : p = p in fvor of : p > p t the 5% level. ) I only b) II only c) III only d) I nd II e) I nd III

6 3) Suppose tht we re to smple from two popultions with unknown proportions p nd p nd tht 95% confidence intervl for p p is to be computed. The most conservtive estimte for the minimum number of smples tken from ech popultion (ssume n = n = n ) in order to obtin mrgin of error of no more thn % is roughly ), smples from ech popultion b) 3, smples from ech popultion c) 4, smples from ech popultion d) 5, smples from ech popultion e) 6, smples from ech popultion 4) In compring life expecttions of two models of refrigertors, the verge yers before complete brekdown of model A refrigertors is compred with tht of 5 model B refrigertors. The 9% confidence intervl is computed to be (6,). Which of the following is the most resonble conclusion? ) The life expectncy of one model is twice tht of nother. b) The men life expectncy of one model is 6, while the men life expectncy of the other model is yers. c) The probbility tht the life expectncies re different is.9. d) The probbility tht the difference in life expectncies is greter thn 6 yers is.9. e) We should be 9% confident tht the difference in life expectncies is somewhere between 6 nd yers. 5) In one experiment to test the effect of lcohol on lrge motor skills, volunteers were rndomly plced into two groups, A nd B. Everyone in group A drnk ounces of lcohol, nd minutes lter everyone in both groups ws timed on mnul dexterity test. The verge completion time for the group A nd B volunteers were 38 nd 3 seconds, respectively. The 9% confidence intervl estimte for the men difference is 7 ± 4 seconds. If µ A nd µ B re the true men completion times, respectively, for people who hve nd hve not drunk ounces of lcohol, how mny of the following sttements re resonble conclusions? I. µ A > µ B with probbility.9. II. There is 9% probbility tht 3 < µ A B <. III. The bove confidence intervl ws clculted by method tht gives correct results (for where µ lies) in 9% of ll cses. ) None b) One c) Two d) Three e) Four A µ B IV. We re 9% confident tht A µ B µ lies between 3 nd seconds.

7 6) A medicl resercher wishes to investigte the effectiveness of exercise versus diet in losing weight. Two groups of 5 overweight dult subjects re used, with subject in ech group mtched to similr subject in the other group on the bsis of number of physiologicl vribles. One of the groups is plced on regulr progrm of vigorous exercise, but with no restriction on diet, nd the other on strict diet, but with no requirement to exercise. The weight losses fter weeks re determined for ech subject nd the difference between mtched pirs of subjects (weight loss of subject in exercise group - weight loss of mtched subject in diet group) is computed. The men of these differences in weight loss is found to be - lbs. with stndrd devition s = 4 lbs. Is this evidence of difference in men weight loss for the two methods? To test this, consider the popultion of differences (weight loss overweight dult would experience fter weeks on the exercise progrm) - (weight loss the sme dult would experience fter weeks on the strict diet). Let µ be the men of this popultion of differences nd ssume their distribution is pproximtely norml. We test the hypotheses : µ = ; : µ using the mtched pirs t test. The P-vlue for this test is ) below.. b) between. nd.5. c) between.5 nd.. d) between. nd.. e) lrger thn.. 7) Scores this yer on the SAT Mthemtics test (SAT-M) for students tking the test for the first time re believed to be normlly distributed with men µ. For students tking the test for the second time, this yer's scores re lso believed to be normlly distributed but with possibly different men µ. The stndrd devitions for first- nd second-time test tkers pper to sty reltively constnt from yer to yer nd we believe these cn be tken to be known, with vlue s = for first-time tkers nd vlue s = 9 for second-time tkers. A rndom smple of the SAT-M scores of students who took the test for the first time this yer ws obtined nd the men of these scores ws x =.5. A rndom smple of the SAT-M scores of 36 students who took the test for the second time this yer ws lso obtined nd the men of these 36 scores ws x = We wish to estimte the difference µ. A 9% confidence intervl for µ is ) 34.6 ± 9.6. b) 34.6 ± c) 34.6 ± d) 34.6 ± e) 34.6 ± 37. 6

8 8) Referring to the bove dt, suppose we wished to determine if those tking the SAT-M for the second time tend to do better thn those tking the test for the first time. To nswer this question, we decide to test the hypotheses =, : µ. : µ > t the 5% significnce level. Bsed on our dt we conclude ) we would not reject the null hypothesis of no difference t the. level. b) we would reject the null hypothesis of no difference t the. level but not t the.5 level. c) we would reject the null hypothesis of no difference t the.5 level but not t the. level. d) we would reject the null hypothesis of no difference t the. level. e) None of the bove. 9) Reserchers compred two groups of competitive rowers: group of skilled rowers nd group of novices. The reserchers mesured the ngulr velocity of ech subject's right knee, which describes the rte t which the knee joint opens s the legs push the body bck on the sliding set. The smple size n, the smple mens, nd the smple stndrd devitions for the two groups re given below. Group n Men Stndrd Devition Skilled Novice The reserchers wished to test the hypotheses : the men knee velocities for skilled nd novice rowers re the sme. : the men knee velocity for skilled rowers is lrger thn for novice rowers. The dt showed no strong outliers or strong skewness, so the reserchers decided to use the two-smple t test. The vlue of the t test sttistic is ).. b).5. c).. d) 4. e) None re true.

9 ) A resercher wished to compre the effect of two stepping heights (low nd high) on hert rte in steperobics workout. A collection of 5 dult volunteers ws rndomly divided into two groups of 5 subjects ech. Group did stndrd step-erobics workout t the low height. The men hert rte t the end of the workout for the subjects in group ws x = 9 bets per minute with stndrd devition s = 9 bets per minute. Group did the sme workout but t the high step height. The men hert rte t the end of the workout for the subjects in group ws x = 95 bets per minute with stndrd devition s = bets per minute. Assume the two groups re independent nd the dt re pproximtely norml. Let µ nd µ represent the men hert rtes we would observe for the entire popultion represented by the volunteers if ll members of this popultion did the workout using the low or high step height, respectively. Suppose the resercher hd wished to test the hypotheses : µ µ µ. = µ, : The P-vlue for the test is (use the conservtive vlue for the degrees of freedom) is ) lrger thn.. b) between. nd.5. c) between.5 nd. d) below. e) none of the bove ) A resercher wished to test the effect of the ddition of extr clcium to yogurt on the "tstiness" of yogurt. A collection of dult volunteers ws rndomly divided into two groups of subjects ech. Group tsted yogurt contining the extr clcium. Group tsted yogurt from the sme btch s group but without the dded clcium. Both groups rted the flvor on scle of to, being "very unplesnt" nd being "very plesnt." The men rting for group ws x = 6.5 with stndrd devition s =.5. The men rting for group ws x = 7. with stndrd devition s =. Assume the two groups re independent. Let µ nd µ represent the men rtings we would observe for the entire popultion represented by the volunteers if ll members of this popultion tsted, respectively, the yogurt with the dded clcium nd the yogurt without it. Assuming two-smple t procedures re sfe to use, 9% confidence intervl for µ is (use the conservtive vlue for the degrees of freedom) ) -.5 ±.5. b) -.5 ±.3. c) -.5 ±.4. d) -.5 ±.5. e) none of the bove

10 Free-Response Questions ) Scores this yer on the SAT Mthemtics test (SAT-M) for students tking the test for the first time re believed to be normlly distributed with men µ. For students tking the test for the second time, this yer's scores re lso believed to be normlly distributed but with possibly different men µ. The stndrd devitions for first- nd second-time test tkers pper to sty reltively constnt from yer to yer nd we believe these cn be tken to be known, with vlue σ = for first-time tkers nd vlue σ = 9 for second-time tkers. A rndom smple of the SAT-M scores of students who took the test for the first time this yer ws obtined nd the men of these scores ws x =.5. A rndom smple of the SAT-M scores of 36 students who took the test for the second time this yer ws lso obtined nd the men of these 36 scores ws x = ) Compute 9% confidence intervl for µ. b) Determine set of hypotheses pproprite to this sitution. c) Test these hypotheses t level α =. 5. 3) Sleep reserchers know tht some people re erly birds (E), preferring to go to bed by p.m. nd rise by 7.m., while others re night owls (N), preferring to go to bed fter p.m. nd rise fter 8.m. A study ws done to compre drem recll for erly birds nd night owls. One hundred people of ech of the two types were selected t rndom nd sked to record their drems for one week. Some of the results re presented below. Number of drems Reclled Proportion Who Reclled Group Men St. Dev. No drems 5 or more drems Erly birds Night owls On the bsis of the bove, we re led to suspect tht night owls hve greter tendency to recll drems s compred with erly birds. ) Set up hypotheses involving mens tht cn be used to test the bove contention. Be sure to define those prmeters you introduce. b) Set up hypotheses involving proportions tht cn be used to test the bove contention. Be sure to define those prmeters you introduce. c) Use the dt bove to test your hypotheses in (). Are the results significnt? d) Use the dt bove to test your hypotheses in (b). Are the results significnt? e) Use the dt bove to construct 95% confidence intervl for the difference of the mens defined in (). f) Use the dt bove to construct 95% confidence intervl for the difference of the proportions defined in (b).

11 4) A mjor university wishes to determine if the verge number of lbortory clsses tht biology mjors enroll in hs chnged over the pst yers. To help in determining whether chnge hs tken plce rndom smples of biology mjors ws tken in ech of two yers (99 nd ), with the results tbulted below: 99 Results Results Number of Lb Clsses Number of Students Number of Lb Clsses Number of Students (Totl) (Totl) x =.83 s =.9 x =.93 s =. 37 ) Stte hypotheses pproprite to this sitution, crefully defining ny prmeters used. b) Test your hypotheses, using α =.. c) Construct confidence intervl for the difference of mens tht could lso be used to test your hypotheses in () t the % level.

12 5) A growing number of employers re trying to hold down the costs tht they py for medicl insurnce for their employees. As prt of this effort, mny medicl insurnce compnies re now requiring clients to use generic-brnd medicines when filling prescriptions. An independent consumer dvoccy group wnted to determine if there ws difference, in milligrms, in the mount of ctive ingredient between certin nme brnd drug nd its generic counterprt. Phrmcies my store drugs under different conditions. Therefore, the consumer group rndomly selected ten different phrmcies in lrge city nd filled two prescriptions t ech of these phrmcies, one of the nme brnd nd the other for the generic brnd of the drug. The consumer group s lbortory then tested rndomly selected pill from ech prescription to determine the mount of ctive ingredient in the pill. The results re given in the following tble. Active Ingredient (in milligrms) Phrmcy Nme brnd Generic brnd Bsed on the bove dt, wht should be the consumer group s lbortory report bout the difference in the ctive ingredient in the two brnds of pills? Give pproprite sttisticl evidence to support your response. (Be sure to stte ny ssumptions you might hve mde in your nlysis.)

13 6) Reserchers wnt to see whether trining increses the cpbility of people to correctly predict outcomes of coin tosses. Ech of twenty people is sked to predict the outcome (heds or tils) of independent tosses of fir coin. After trining, they re retested with new set of tosses. (All 4 sets of tosses re independently generted.) Since the coin is fir, the probbility of correct guess by chnce is.5 on ech toss. The number correct for ech of the people were s follows. Score Before Trining Score After Trining (number correct) (number correct) sum =, sum =,6 To nswer the following questions, you my wish to enter these dt into your clcultor. As check tht you hve entered the dt correctly, the sum of the first column is, nd the sum of the second column is,6. ) Do the dt suggest tht fter trining people cn correctly predict coin toss outcomes better thn the 5 percent expected by chnce guessing lone? Give pproprite sttisticl evidence to support your conclusion. b) Does the sttisticl test tht you completed in prt () provide evidence tht this trining is effective in improving person s bility to predict coin toss outcomes? If yes, justify your nswer. If no, conduct n pproprite nlysis tht would llow you to determine whether or not the trining is effective. c) Would knowing person score before trining be helpful in predicting his or her score fter trining? Justify your nswer.