2. We can be 90% confident that the proportion of people in the city who know how to ski is between and 0.358

Transcription

1 Fial Exam (sample) This particular sample covers sectios iclusive. I. I a particular large Midwest city, a simple radom sample of 400 people reveals that 128 of them kow how to ski. 1. A 99.8% cofidece iterval for the proportio of people i the city who kow how to ski is (0.248, 0.392) 2. We ca be 90% cofidet that the proportio of people i the city who kow how to ski is betwee ad Someoe makes the claim that 36% of the people i the city kow how to ski. 3. Assumig that this claim is true, what is the probability of gettig a sample (of this size) as small as ours (pvalue)? Is there evidece at the 1% level that the proportio of people i the city who ca ski is actually lower tha 0.36? 5. Is there evidece at the 20% level that the proportio who ca ski is actually lower tha 0.36? 6. Is there evidece at the 5% level that the proportio who ca ski is actually lower tha 0.36? 7. Is there evidece at the 6% level that the proportio who ca ski is actually lower tha 0.36? p = = 0.32 p (1 p ) 0.32( ) p ± z = 0.32 ± z 400 where z = (99.8%), 1.645(90%) p = proportio of people i the city who ca ski H 0 : p = 0.36 H a : p < p Z = = = 1.67 p (1 p) 0.36 (1 0.36) 400 p value = is ot less tha 1% but is less tha 20%, 5%, ad 6%.

2 II. We study the umber of cars to pass a certai itersectio betwee 11am ad 12pm each weekday. We pick 50 weekdays at radom, ad the average cout is 83 cars. Based o aalysis of other itersectios, the stadard deviatio is 10 cars. 8. We ca be 99.5% cofidet that the mea umber of cars of all weekdays is betwee ad A 60% cofidece iterval for the mea umber of cars of all weekdays is (81.811, ) Jack claims that, o average, 86 cars pass by durig these hours. 10. Assumig that Jack is correct, what is the probability of gettig a sample average (from a sample of this size) as low as ours (p-value)? Assumig that Jack is correct, what is the probability of gettig a sample average (from a sample of this size) as differet from 86 as ours (p-value)? Is there evidece at the 1% level that the average for this itersectio is really lower tha claimed? 13. Is there evidece at the 0.1% level that the average for this itersectio is really lower tha claimed? 14. Is there evidece at the 5% level that the average for this itersectio is really lower tha claimed? 15. Is there evidece at the 20% level that the average for this itersectio is really lower tha claimed? 16. Is there evidece at the 20% level that the average for this itersectio is ot as claimed? 17. Is there evidece at the 15% level that the average for this itersectio is ot as claimed? 18. Is there evidece at the 5% level that the average for this itersectio is ot as claimed? 19. Is there evidece at the 1% level that the average for this itersectio is ot as claimed? x ± z σ = 83 ± z where z = (99.5%), 0.841(60%) H 0 : μ = 86 H a : μ < 86 x μ Z = ( σ = ) ( 10 ) = (tail) Compare to levels (12 15). H 0 : μ = 86 H a : μ = (tails) Compare to levels (16 19).

3 III. Scietists are studyig the effectiveess of St. Joh s wort i treatig medical depressio. A sample of 98 patiets receive the St. Joh s wort extract, ad 14 soo show remissio. A sample of 102 receive a placebo (o drug), ad 5 soo show remissio. 20. What is the pooled sample proportio? Assumig that St. Joh s wort has o effect, what is the probability that for two samples of these sizes, the wort group s proportio of remissios would be as much higher tha the other group s as it is (p-value)? Assumig that St. Joh s wort has o effect, what is the probability that for two samples of these sizes, the two samples proportio of remissios would be as differet as they are (p-value)? Is there evidece at the 5% level that St. Joh s wort has a positive effect? 24. Is there evidece at the 1% level that St. Joh s wort has a positive effect? 25. Is there evidece at the 0.1% level that St. Joh s wort has a positive effect? 26. Is there evidece that St. Joh s wort has ay effect (positive or egative) at the 5% level? 27. Is there evidece that St. Joh s wort has ay effect (positive or egative) at the 15% level? 28. Is there evidece that St. Joh s wort has ay effect (positive or egative) at the 25% level? p 1 = proportio of all wort patiets who show remissio p 2 = proportio of all placebo patiets who show remissio p 1 = = p 2 = = H 0 : p 1 = p 2 H a : p 1 > p 2 pooled p = = z = p 1 p 2 = p (1 p ) ( ) z = (tail) Compare to levels (23 25). H 0 : p 1 = p 2 H a : p 1 p = (tails) Compare to levels (26 28).

4 IV. I early November of 1922, a expeditio discovered the tomb of Kig Tutakhame. 36 people etered the tomb ad thus were exposed to Kig Tut s Curse. 25 people stayed outside ad thus were ot exposed to the curse. The mea age at death (lifespa) of those exposed was 71 years. The mea lifespa of those ot exposed was 75 years. Assume that the stadard deviatio for the lifespas is i ay case 9 years ad assume that the results vary ormally i ay case. Let 1 be the ideal lifespa for those ot exposed, ad let 2 be the ideal lifespa for those exposed. 29. Assumig that there is o differece i these two ideal lifespas (i.e., that the curse has o effect), what is the probability that for two samples of these sizes, the average lifespa of the cursed group would be as much lower as it is (p-value)? Is there evidece of Kig Tut s Curse at the 5% level? 31. Is there evidece of Kig Tut s Curse at the 10% level? 32. Is there evidece of Kig Tut s Curse at the 15% level? 33. Is there evidece of Kig Tut s Curse at the 1% level? σ 1 = 9 σ 2 = 9 x 1 = 75 x 2 = 71 1 = 25 2 = 36 H 0 : μ 1 = μ 2 H a : μ 1 < μ 2 z = x 1 x 2 (σ 1 )2 1 + (σ 2 )2 2 = z = (tail) Compare to levels (30 33).

5 V. All respectable dragos make marks o their cave walls to represet the umber of kights they ve eate. After the extictio of dragos, a archaeologist (some call him Tom) examies drago caves, makig a list of 12 dragos lifespas X i years (by carbo datig their remais) ad umber of kights eate Y. From this list we compute: 34. Compute SS(X): x = 2598 x 2 = y = 258 y 2 = 9384 xy = Compute SS(Y): Compute SS(XY): What is the correlatio betwee X ad Y? What is the liear regressio lie of Y o X: y = x What is the predicted umber of kights eate for a drago who lived 100 years? 49 kights SS(x) = (x i ) 2 ( x i) 2 SS(y) = (y i ) 2 ( y i) 2 SS(xy) = (x i y i ) ( x i)( y i ) r = = = = = 3837 = (2598)(258) = y = mx + b where m = r SS(y) SS(x) = = ad b = y m x = ( ) = y = x = y = (100) = 49

6 VI. A simple radom sample of 20 icidets reveals that the average time it takes the Hulk to revert to Bruce Baer is 18 miutes with a stadard deviatio of 3 miutes. 40. Fid a 95% cofidece iterval for the Hulk s average time to revert: (16.60, 19.40) Baer complais that it takes 20 miutes o average before the Hulk reverts. 41. The test statistic (positive versio) for this hypothesis test has a value of Do we have evidece at the 25% level that he is overestimatig his average time? 43. Do we have evidece at the 10% level that he is overestimatig his average time? 44. Do we have evidece at the 5% level that he is overestimatig his average time? 45. Do we have evidece at the 1% level that he is overestimatig his average time? oe tail % we re x = 18 s = 3 = 20 df = 20 1 = df here x ± t s = 18 ± t 3 20 where t = (for 95%) H 0 : μ = 20 H a : μ < 20 x μ t = ( s ) = ( 3 ) = Our p-value is betwee 0.05% ad 0.5%, so it s smaller tha 25%, 10%, 5%, 1%.

7 VII. Batma is iterested i whether he experieces a temporary lost of agility followig his recovery from exposure to the Scarecrow s fear gas, so he uses a scored agility test before ad after such ecouters. A sample of 13 icidets reveals a average score differece (before after) of +2 poits with a stadard deviatio of 3.65 poits. 46. The test statistic (positive versio) for this hypothesis test has a value of Do have evidece at the 25% level that the fear gas temporarily lowers his agility? 48. Do have evidece at the 10% level that the fear gas temporarily lowers his agility? 49. Do have evidece at the 5% level that the fear gas temporarily lowers his agility? 50. Do have evidece at the 1% level that the fear gas temporarily lowers his agility? o d = 2 s d = 3.65 = 13 df = 13 1 = 12 t = d 0 ( s d ) = 2 ( ) = oe tail 0.025% we re 0.05 df here Our p-value is betwee 2.5% ad 5%, so it s smaller tha 25%, 10%, 5%, but ot smaller tha 1%.