AP Statistics Review Ch. 8

Transcription

1 AP Statistics Review Ch. 8 Name 1. Each figure below displays the samplig distributio of a statistic used to estimate a parameter. The true value of the populatio parameter is marked o each samplig distributio. Which is the samplig distributio of a statistic that is a ubiased estimator of the parameter of iterest? Statistic I True Value Statistic II True Value Statistic III True Value A. Statistic I oly B. Statistic II oly C. Statistic III oly D. Both Statistic I ad Statistic II are ubiased E. Noe of the three statistics are ubiased.

2 2. Four differet statistics are beig cosidered for estimatig a populatio characteristic. The samplig distributios of the four statistics are show here. Which statistic is most likely to result i a estimate that is close to the true value of the populatio characteristic? True Value of Populatio Characteristic Statistic II Statistic I Statistic III Statistic IV A. Statistic I B. Statistic II C. Statistic III D. Statistic IV E. Statistics III ad IV are equally likely to result i a estimate that is close to the true value. 3. Give a choice of ubiased statistics, the best statistic would be the oe with a samplig distributio that A. is most closely approximated by a ormal curve. B. has the smallest variace. C. has the largest stadard error. D. is most early symmetric. E. is most closely approximated by a t distributio. 4. To estimate the proportio of faculty at a state uiversity who ow a home, a radom sample of faculty is selected. For which of the followig combiatios of ad ˆp would it be appropriate to use the cofidece iterval pˆ(1 pˆ) pˆ (z critical value)? A. = 20 ad ˆp = 0.40 B. = 40 ad ˆp = 0.20 C. = 100 ad ˆp = 0.05 D. = 150 ad ˆp = 0.45 E. = 200 ad ˆp = 0.02

3 5. Suppose that a radom sample of 100 high school classrooms i the state of Califoria is selected ad a 95% cofidece iterval for the proportio that has iteret access is (0.62, 0.78). Which of the followig is a correct iterpretatio of the 95% cofidece level? A. The method used to costruct the iterval will produce a iterval that icludes the value of the populatio proportio about 95% of the time i repeated samplig. B. We are 95% cofidet that the sample proportio is betwee 0.62 ad C. There is a 95% chace that the true proportio of high school classrooms i Califoria that have iteret access is betwee 0.62 ad D. We are 95% cofidet that the true proportio of high school classrooms i Califoria that have iteret access is betwee 0.62 ad E. Noe of the above is a correct iterpretatio of the cofidece level. 6. Which of the followig must be true of a sample i order for it to be appropriate to use a z cofidece iterval to estimate the populatio proportio? A. The sample is a radom sample from the populatio of iterest. B. pˆ 10 ad (1 pˆ ) 10 C. The populatio distributio is ormal. D. All of the above are required assumptios to use the z cofidece iterval to estimate the populatio proportio. E. Oly (a) ad (b) are required assumptios to use the z cofidece iterval to estimate the populatio proportio. 7. Which of the followig would ted to decrease the width of a cofidece iterval? I. Icreasig the sample size II. Usig a higher cofidece level III. Usig a lower cofidece level A. I oly B. II oly C. III oly D. I ad II oly E. I ad III oly

4 8. Each idividual i a radom sample of 60 college studets was asked how much moey he/she spet o food i a typical moth. The data was the used to costruct a 99% cofidece iterval for the true mea amout spet o food for all studets at the college i a typical moth. The cofidece iterval was (240, 380). Which of the followig is closest to the 95% cofidece iterval costructed usig the same sample data? A. (240, 380) B. (250, 370) C. (260, 360) D. (270, 350) E. (280, 340) 9. Each idividual i a radom sample of 50 iteret users was asked how may miutes he/she speds olie i a typical day. The data was the used to costruct a 99% cofidece iterval for the mea umber of miutes spet olie i a typical day for all iteret users. The cofidece iterval was (80, 200) miutes per day. Which of the followig is a correct iterpretatio of the cofidece iterval? A. There is a 99% chace that the mea umber of miutes spet olie i a typical day of all iteret users is betwee 80 ad 200. B. We are 99% cofidet that the sample mea is betwee 80 ad 200 miutes. C. We are 99% cofidet that for all iteret users the mea umber of miutes spet olie i a typical day is betwee 80 ad 200. D. 99% of all iteret users will sped betwee 80 ad 200 miutes olie i a typical day. E. 99% of the people i the sample spet betwee 80 ad 200 miutes olie i a typical day. 10. Two-hudred visitors to a atioal park were selected at radom ad each was asked whether they iteded to stay more tha 2 ights i the park o that visit. Assumig that the sample size is large eough, which of the followig cofidece itervals should be used to estimate the proportio of all visitors to this atioal park who will stay more tha 2 ights o their visit? A. x (t critical value) s B. pˆ (t critical value) s C. pˆ ( zcritical value) s D. x (z critical value) E. pˆ z critical value pˆ(1 pˆ)

5 11. Two-hudred visitors to a atioal park were selected at radom ad each was asked how far they had traveled to get to the park o that visit. Which of the followig cofidece itervals should be used to estimate the mea umber of miles traveled by all visitors to the park? A. x (t critical value) s B. (t critical value) s C. ( z critical value) D. x (z critical value) E. pˆ z critical value pˆ(1 pˆ) 12. Suppose you take a simple radom sample from a populatio kow to be ormally distributed, but the value of is ukow. Your sample size is 10. Which formula below should be used to fid the 90% cofidece iterval for the mea? s A. x B. x s C. x D. x E. x

6 13. A large sample ( = 250) was used to compute A. 50% cofidece iterval. B. 68% cofidece iterval. C. 84% cofidece iterval. D. 90% cofidece iterval. E. 95% cofidece iterval. x s. This iterval is equivalet to a 14. Which of the followig is certai to reduce the width of a cofidece iterval? A. larger sample size ad higher cofidece level B. larger sample size ad lower cofidece level C. smaller sample size ad higher cofidece level D. smaller sample size ad lower cofidece level E. Noe of the above. 15. The critical value for a 99% cofidece iterval for p = A B C D E What is the differece betwee a t-iterval ad z-iterval? A. z-itervals are used if you kow the populatio s stadard deviatio ad t-itervals are used whe you do ot kow the populatio s stadard deviatio. B. t-itervals are used if you kow the populatio s stadard deviatio ad z-itervals are used whe you do ot kow the populatio s stadard deviatio. C. t-itervals are used whe the sample size is much larger tha i z-itervals D. z-itervals are used whe the data are ot ormally distributed E. t-itervals use the sample size where z-itervals use -1 degrees of freedom 17. Assumig that the data came from a radom sample, what other coditios must be met i order to perform a z-iterval o a sigle proportio? A. The sample size is greater tha 30 or the data are roughly ormal. B. The populatio of iterest is greater tha the sample size. C. The populatio is greater tha 10 times the sample size. D. The probability of success times sample size is greater tha 10 as well as the probability of failure times the sample size is greater tha 10. E. Both C ad D must be true for a proportio iterval to be costructed.

7 18. A researcher wats to fid a reasoable iterval of the mea time it takes to complete a order over the phoe at a call ceter for a large retail compay. He has selected a sample of 18 calls ad recorded the time to completio o the phoe to be 4.27 miutes with a stadard deviatio of 0.78 miutes. What is the appropriate iterval if he wats to use a 98% cofidece level? æ A ± ö æ ç B ± ö æ ç C ± ö ç è 18 ø è 17 ø è 18 ø æ D ± ö æ ç E ± ö ç è 18 ø è 17 ø 19. What size would a sample eed to be i order for it to be withi 3 percetage poits o a 95% cofidece iterval whe the proportio of successes is 36%? A. 984 B. 983 C. 256 D. 255 E Which of the followig would ot decrease the width of a cofidece iterval? I. icreasig the sample size II. icreasig the stadard deviatio III. decreasig the cofidece iterval A. I oly B. II oly C. III oly D. I ad II E. II ad III 21. I a study with a ew atibiotic for childre, a radom sample of 64 showed that 52 of the childre had positive results withi 12 hours. Fid a 95% cofidece iterval for the proportio of childre who will experiece positive results withi 12 hours o this atibiotic. A B C D E The average umber of mior medical eeds at ay Army hospital i Irag over a 49-day period was foud to be 11.2 with a stadard deviatio of 3.6 per day. With what degree of cofidece ca we say that the mior medical eeds at the Army hospital per day is betwee 10.3 ad 12.1? A. 80% B. 85% C. 90% D. 95% E. 98%

8 23. What assumptios are ecessary for a 95% t-iterval with a sample size of 9 to be valid? I. The sample was selected radomly from the populatio of iterest. II. The populatio stadard deviatio is kow III. The sample ca be show to be approximately ormal with o skewess. A. I oly B. III oly C. I ad II oly D. I ad III oly E. I, II ad III 24. A 98% cofidece iterval for a populatio mea is foud to be 127 ± 18. Which of the followig is a correct iterpretatio of this level? A. There is 98% probability that the true mea is cotaied i this iterval. B. There is 98% probability that the sample mea is cotaied i this iterval. C. There is 98% probability that aother iterval will give a true mea of 127. D. If 100 samples of the same size are take agai, about 98 of them will cotai the true mea value. E. If 100 samples of the same size are take agai, about 98 of them will cotai the true sample value. 25. Whe the sample size is icreased what effect does this have o the size of the cofidece iterval? A. It makes it wider. Depedig o how much the sample is icreased by, this will make the iterval wider. B. It will make the iterval arrower. The larger the sample gets, the smaller the width of the iterval. C. It will have o effect o the width because this would oly chage based o the level of cofidece. D. It will double the width of the iterval sice you are ow multiplyig by ad additioal size. E. It will cut the iterval i half sice you are ow dividig by a additioal size. 26. You have bee asked to compute a 96% cofidece iterval for a populatio mea. If the populatio stadard deviatio is kow to be 7 ad the sample size is 40, what is the value of z* that will be used to perform this calculatio? A B C D E A commuity wats to see if there is eough support to hold a tow childre s festival. Forty-two percet of the 150 residets said they would participate i the evet. What would be the approximate stadard error for this sample proportio? A B C D E

9 28. A Fortue 500 compay wats to desig a study to estimate the proportio of employees who would prefer a pay icrease to a icrease i retiremet beefits. The compay statisticia wats to geerate a 95% cofidece iterval with a margi of error beig 0.04 or smaller. What is the miimum sample size eeded for this study? A. 600 employees B. 601 employees C. 307 employees D. 156 employees E. 157 employees 29. A cosumer group just published a study statig that 72% of Americas believe corporatios are ot cocered about public safety, with a 90% cofidece level ad a margi of error of 2%. What does this mea? A. If the poll were coducted agai, there is a 90% probability that Americas who believe corporatios are ot cocered about safety are withi a margi of error of 2%. B. There is a 90% chace that the proportio of Americas who believe corporatios are t cocered with public safety is betwee 70% ad 74%. C. From 90% of all Americas, about 70% to 74% believe corporatios are ot cocered with public safety. D. They are 90% cofidet that the true percetage of Americas who believe corporatios are ot cocered with public safety is betwee 70% ad 74%. E. Niety out of 100 Americas who believe corporatios are ot cocered with public safety is 70% ad 74% i repeated samplig. Free Respose 30. A large Midwester forest plated by the CCC at the ed of the depressio is ow too thick for safe tree growth ad may sectios are begiig to die. The state forestry commissio has hired a lumber compay to begi cuttig dow eough trees to allow the healthier trees to cotiue to grow. A sample of 204 trees foud that 28% of the trees must be removed to allow further growth. a. What coditios eed to be checked i order to calculate a cofidece iterval? b. If the forestry commissio wats to estimate the proportio of the total forest populatio that will eed to be removed, calculate a 95% cofidece iterval to predict the amout of cuttig that will occur. c. Iterpret this iterval ad explai why this is reasoable.

10 31. A AP coordiator i a urba school district madates that AP studets take a practice exam, called a Mock Exam, i the sprig to help idetify studet potetial scores o the upcomig exam i May. A radom sample of AP studets Mock Exam was take ad the scores recorded. The results for 28 of the studets were foud to be: AP Mock Exam Scores a. Costruct a 90% cofidece iterval for the mea score o the AP Mock Exams give i this district. Justify your aswer. b. Explai why the aswer is reasoable c. The district has foud i previous years that their studets ted to score a passig score o the atioal exam (that is 3-5) if the Mock scores are at least a 70. Ca the AP coordiator feel cofidet i this year s results? Explai how you arrived at this coclusio. 32. A humae society wated to estimate with 95 percet cofidece the proportio of households i its couty that ow at least oe dog. (a) Iterpret the 95 cofidece level i this cotext. The humae society selected a radom sample of households i its couty ad used the sample to estimate the proportio of all households that ow at least oe dog. The coditios for calculatig a 95 percet cofidece iterval for the proportio of households i this couty that ow at least oe dog were checked ad verified, ad the resultig cofidece iterval was 0.417± (b) A atioal pet products associatio claimed that 39 percet of all America households owed at least oe dog. Does the humae society s iterval estimate provide evidece that the proportio of dog owers i its couty is differet from the claimed atioal proportio? Explai. (c) How may households were selected i the humae society s sample? Show how you obtaied your aswer.

11 Aswer Key: 1. C 16. A 2. C 17. E 3. B 18. D 4. D 19. A 5. A 20. B 6. E 21. A 7. E 22. C 8. B 23. D 9. C 24. D 10. E 25. B 11. A 26. E 12. C 27. D 13. B 28. B 14. B 29. D 15. B 30. a. Sice this is a proportios case, the followig four coditios must be met to ru this iterval. We eed to assume the sample came from a simple radom sample of trees. 204(0.28) = , this is ok ad 204(0.72) = , this is ok 10(204) = 2040, as log as the forest had more tha 2040 trees, the you ca proceed with a cofidece iterval. b. Based o a 95% cofidece level, the proportio of trees from the forest eedig to be removed would reasoably be withi (0.218, 0.341) c. I am 95% cofidet that the true proportio of trees eedig to be removed from the forest is betwee 21.8% ad 34.1%. I am cofidet i this because the method used will capture the true proportio of trees eedig to be removed 95 out of 100 times i repeated samplig. 31. a. Sice this is meas data ad we do ot kow the populatios stadard deviatio, a t-iterval eeds to be used. The justificatio is foud i checkig the coditios eeded for a t-iterval. The problem stated it was a radom sample so we will assume this is sufficiet. Sice the sample size is slightly uder 30, we will still check for the data beig roughly ormal. Either a boxplot or ormal probability plot would suffice. I both cases there is o reaso to doubt the sample data are roughly ormally distributed so we ca proceed to the calculatios. The iterval would be (69, 76) b. This is a reasoable solutio because the method used for geeratig a t-iterval will capture the true test average 90 out of 100 times i repeated samplig. c. Sice the iterval is from 69 to 76, I am 90% cofidet that is the true mea AP Mock score is cotaied i this iterval. Further, sice the district is lookig for studets to score 70 ad 70 is cotaied i the iterval foud; it is a reasoable coclusio that the district coordiator ca be cofidet i the studets results this year.

12 32. a. The 95 percet cofidece level meas that if oe were to repeatedly take radom samples of the same size from the populatio ad costruct a 95 percet cofidece iterval from each sample, the i the log ru 95 percet of those itervals would succeed i capturig the actual value of the populatio proportio of households i the couty that ow at least oe dog. b. No. The 95 percet cofidece iterval ± is the iterval (0.298, 0.536). This iterval icludes the value 0.39 as a plausible value for the populatio proportio of households i the couty that ow at least oe dog. Therefore, the cofidece iterval does ot provide evidece that the proportio of dog owers i this couty is differet from the claimed atioal proportio. c. The sample proportio is 0.417, ad the margi of error is Determiig the sample size ( 0.417) ( ) requires solvig the equatio =1.96 for. Thus, = (0.417)( )» 65.95, so the humae society must have selected 66 households for its sample.