STA 2023 Practice Questions Exam 2 Chapter 7 sec 9.2. Case parameter estimator standard error Estimate of standard error


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1 STA 2023 Practice Questios Exam 2 Chapter 7 sec 9.2 Formulas Give o the test: Case parameter estimator stadard error Estimate of stadard error Samplig Distributio oe mea x s t (1) oe p ( 1 p) CI: prop. p pˆ ST: pˆ(1 pˆ) p 1 p ) ( 0 0 z Memorize these Formulas: Geeral Format for Cofidece Iterval: estimator +/ (t or z) est. stadard error Geeral Format of Test Statistic: estimator # from Ho ( t or z) estimate of stderr Determiig sample size for estimatig proportios ad meas. Practice Problems o ext page.
2 1. You take a radom sample from some populatio ad form a 96% cofidece iterval for the populatio mea, Which quatity is guarateed to be i the iterval you form? a) 0 b) c) x d) Suppose you coduct a sigificace test for the populatio proportio ad your pvalue is Give a 0.10 level of sigificace, which of the followig should be your coclusio? a) accept H O b) accept H A c) Fail to reject H A d) Fail to reject H O e) Reject H o 3. Decreasig the sample size, while holdig the cofidece level the same, will do what to the legth of your cofidece iterval? a) make it bigger b) make it smaller c) it will stay the same d) caot be determied from the give iformatio 4. Decreasig the cofidece level, while holdig the sample size the same, will do what to the legth of your cofidece iterval? a) make it bigger b) make it smaller c) it will stay the same d) caot be determied from the give iformatio 5. If you icrease the sample size ad cofidece level at the same time, what will happe to the legth of your cofidece iterval? a) make it bigger b) make it smaller c) it will stay the same d) caot be determied from the give iformatio 6. Which of the followig is a property of the Samplig Distributio of x? a) if you icrease your sample size, x will always get closer to the populatio mea. b) the stadard deviatio of the sample mea is the same as the stadard deviatio from the origial populatio c) the mea of the samplig distributio of x is the populatio mea. d) x always has a Normal distributio. 7. Which of the followig is true about pvalues? a) a pvalue must be betwee 0 ad 1. b) if a pvalue is greater tha.01 you will ever reject H O. c) pvalues have a N(0,1) distributio d) Noe of the above are true.
3 8. Suppose that we wated to estimate the true average umber of eggs a quee bee lays with 95% cofidece. The margi of error we are willig to accept is 0.5. Suppose we also kow that s is about What sample size should we use? a) 1536 b) 1537 c) 2653 d) What should be the value of z used i a 93% cofidece iterval? a) 2.70 b) 1.40 c) 1.81 d) What are the possible values of xbar for all samples of a give from this populatio? To aswer this questio, we would eed to look at the: a) test statistic b) zscores of several statistics c) stadard ormal distributio d) samplig distributio e) probability distributio of x 11. Why do we use iferetial statistics? a) to help explai the outcomes of radom pheomea b) to make iformed predictios about parameters we do t kow c) to describe samples that are ormal ad large eough (>30) d) to geerate samples of radom data for a more reliable aalysis 12. A 95% cofidece iterval for the mea umber of televisios per America household is (1.15, 4.20). For each of the followig statemets about the above cofidece iterval, choose true or false. a) The probability that is betwee 1.15 ad 4.20 is.95. b) We are 95% cofidet that the true mea umber of televisios per America household is betwee 1.15 ad c) 95% of all samples should have xbars betwee 1.15 ad d) 95% of all America households have betwee 1.15 ad 4.20 televisios. e) Of 100 itervals calculated the same way (95%), we expect 95 of them to capture the populatio mea. f) Of 100 itervals calculated the same way (95%), we expect 100 of them to capture the sample mea. 13. Whe doig a sigificace test, a studet gets a pvalue of This meas that: I. Assumig Ho were true, this sample s results were a ulikely evet. II % of samples should give results which fall i this iterval. III. We reject Ho at ay reasoable alpha level. a) II oly b) III oly c) I ad III d) I, II, ad III
4 14. Parameters ad statistics a) Are both used to make ifereces about x b) Describe the populatio ad the sample, respectively. c) Describe the sample ad the populatio, respectively. d) Describe the same group of idividuals. 15. A waiter believes that his tips from various customers have a slightly right skewed distributio with a mea of 10 dollars ad a stadard deviatio of 2.50 dollars. What is the probability that the average of 35 customers will be more tha 13 dollars? a) almost 1 b) almost zero c) d) Questios A certai brad of jelly beas are made so that each package cotais about the same umber of beas. The fillig procedure is ot perfect, however. The packages are filled with a average of 375 jelly beas, but the umber goig ito each bag is ormally distributed with a stadard deviatio of 8. Yesterday, Jae wet to the store ad purchased four of these packages i preparatio for a Sprig party. Jae was curious, ad she couted the umber of jelly beas i these packages  her four bags cotaied a average of 382 jelly beas. 16. I the above sceario, which of the followig is a parameter? a) The average umber of jelly beas i Jae s packages, which is 382. b) The average umber of jelly beas i Jae s packages, which is ukow. c) The average umber of jelly beas i all packages made, which is 375. d) The average umber of jelly beas i all packages made, which is ukow. 17. If you wet to the store ad purchased six bags of this brad of jelly beas, what is the probability that the average umber of jelly beas i your bags is less tha 373? a).2709 b).3085 c).4013 d) Why ca we use the Z table to compute the probability i the previous questio? a) because p>15 ad (1p) > 15 b) because is large i this problem c) because the distributio of jelly beas is Normal d) because the average is large 19. Accordig to the cetral limit theorem, what is the stadard deviatio of the samplig distributio of the sample mea? a) The stadard deviatio of the populatio b) The stadard deviatio of the sample c) The stadard deviatio of the populatio divided by the square root of the sample size d) The stadard deviatio of the sample divided by the square root of the sample size
5 Questios Researchers are cocered about the impact of studets workig while they are erolled i classes, ad they d like to kow if studets work too much ad therefore are spedig less time o their classes tha they should be. First, the researchers eed to fid out, o average, how may hours a week studets are workig. They kow from previous studies that the stadard deviatio of this variable is about 5 hours. 20. A survey of 200 studets provides a sample mea of 7.10 hours worked. What is a 95% cofidece iterval based o this sample? a) (6.10, 8.10) b) (6.41, 7.79) c) (6.57, 7.63) d) (7.10, 8.48) 21. Suppose that this cofidece iterval was (6.82, 7.38). Which of these is a valid iterpretatio of this cofidece iterval? a) There is a 95% probability that the true average umber of hours worked by all UF studets is betwee 6.82 ad 7.38 hours. b) There is a 95% probability that a radomly selected studet worked betwee 6.82 ad 7.38 hours. c) We are 95% cofidet that the average umber of hours worked by studets i our sample is betwee 6.82 ad 7.38 hours. d) We are 95% cofidet that the average umber of hours worked by all UF studets is betwee 6.82 ad 7.38 hours. 22. We have 95% cofidece i our iterval, istead of 100%, because we eed to accout for the fact that: a) the sample may ot be truly radom. b) we have a sample, ad ot the whole populatio. c) the distributio of hours worked may be skewed d) all of the above 23. The researchers are ot satisfied with their cofidece iterval ad wat to do aother study to fid a shorter cofidece iterval. What should they chage to esure they fid a shorter cofidece iterval? a) They should icrease their cofidece level ad icrease their sample size. b) They should icrease their cofidece level but decrease their sample size. c) They should decrease their cofidece level but icrease their sample size. d) They should decrease their cofidece level ad decrease their sample size. 24. Suppose our pvalue is.044. What will our coclusio be at alpha levels of.10,.05, ad.01? a) We will reject H o at alpha=.10, but ot at alpha=.05 b) We will reject H o at alpha=.10 or.05, but ot at alpha=.01 c) We will reject H o at alpha=.10,.05, or.01 d) We will ot reject H o at alpha=.10,.05, or.01
6 25. For each of the followig situatios, ca we use the Z table to compute probabilities (T/F): a. Weights of adults are approximately Normally distributed with mea 150 lbs ad stdev 25 lbs. We wat to kow the probability that a radomly selected perso weights more tha 200 pouds. b. Weights of adults are approximately Normally distributed with mea 150 lbs ad stdev 25 lbs. We wat to kow the probability that the average weight of 10 radomly selected people is more tha 200 pouds. c. Weights of adults are approximately Normally distributed with mea 150 lbs ad stdev 25 lbs. We wat to kow the probability that the average weight of 50 radomly selected people is more tha 200 pouds. d. Salaries at a large corporatio have mea of $40,000 ad stdev of $20,000. We wat to kow the probability that a radomly selected employee makes more tha $50,000. e. Salaries at a large corporatio have mea of $40,000 ad stdev of $20,000. We wat to kow the probability that the average of te radomly selected employees is more tha $50,000. f. Salaries at a large corporatio have mea of $40,000 ad stdev of $20,000. We wat to kow the probability that the average of fifty radomly selected employees is more tha $50,000. g. A club has 50 members, 10 of which thik the presidet should be deposed. What is the probability that, if we select 20 members at radom, 18% or more i our sample thik the presidet should be deposed? h. A club has 5000 members, 1000 of which thik the presidet should be deposed. What is the probability that, if we select 91 members at radom, 18% or more i our sample thik the presidet should be deposed? Questios Recet studies have show that 20% of Americas fit the medical defiitio of obese. A radom sample of 100 Americas is selected ad the umber of obese i the sample is determied. 26. What is the samplig distributio of the sample proportio? a) ˆp ~ N(10, 0.2) b) ˆp ~N(2, 1.27) c) ˆp ~N(0.2, 0.04) d) Ca ot be determied 27. What is the probability that the sample proportio is greater tha 0.24? a) b) c) d) 1.0
7 28. A auto isurace compay has 32,000 cliets, ad 5% of their cliets submitted a claim i the past year. We will take a sample 3,200 cliets, ad determie how may of them have submitted a claim i the past year. What is the samplig distributio of ˆp? a) ˆp ~ N(3200, 0.2) b) ˆp ~N(160, 152) c) ˆp ~N(0.05, ) d) Ca ot be determied Questios Suppose 20 doors come to a blood drive. Assume that the blood doors are ot related i ay way, so that we ca cosider them idepedet. The probability that the door has typeo blood is 0.06, which is costat from door to door. Let X = the umber of doors that have typeo blood. 29. For a sample of 100 doors, what is the samplig distributio of the sample proportio? a) ˆp ~ Biomial (100, 0.06) b) ˆp ~Normal (0.06, ) c) ˆp ~Normal(6, 2.37) d) Ca ot be determied 30. For a sample of 300 doors, what is the samplig distributio of the sample proportio? a) ˆp ~ Biomial (200, 0.06) b) ˆp ~Normal (12, 3.359) c) ˆp ~Normal(0.06, ) d) Ca ot be determied 31. For the sample of 300 doors, what is the probability that the sample proportio is greater tha 0.10? a) b) c) d) The executives at Sadbachia, Ic. havig recetly solved their widget crises, have aother major problem with oe of their products. May cities are sedig complaits that their mahole covers are defective ad people are fallig ito the sewers. Sadbachia, Ic. is pretty sure that oly 4% of their mahole covers are defective, but they would like to do a study to cofirm this umber. They are hopig to costruct a 95% cofidece iterval to get withi 0.01 of the true proportio of defective mahole covers. How may mahole covers eed to be tested? a) 8 b) 1476 c) 9604 d) 9605
8 33. The workers at Sadbachia, Ic. took a radom sample of 800 mahole covers ad foud that 40 of them were defective. What is the 95% CI for p, the true proportio of defective mahole covers, based o this sample? a) (37.26, 42.74) b) (.035,.065) c) (.047,.053) d) (.015,.085) 34. Matchig Researchers are desigig a study to determie whether the age of the victim is a factor i reported scams. The researchers are testig to see if more tha half of all reported scams victimize the elderly. They radomly sample 350 reported scams over the past 10 years from the Better Busiess Bureau archives, ad ote that, for 287of them, the victim is over 65 years old. Match the followig symbols with the correct umber o the right: p a) 0.50 phat b) 65 p 0 c) 287 x d) 350 e) f) g) ukow 35. Whe are pvalues egative? a) whe the test statistic is egative. b) whe the sample statistic is smaller tha the proposed value of the parameter c) whe the cofidece iterval icludes oly egative values d) whe we fail to reject the ull hypothesis e) ever 36. Let x 1, x 2,, x 50 be idepedet observatios from a distributio X which is ot ormal. Suppose it is kow that the mea of this distributio is 48 ad the stadard deviatio is 5. What ca we say about the sample mea xbar? a) xbar = 48 b) xbar is distributed approximately ormal with mea 48 ad stadard error 5 c) xbar is distributed approximately ormal with mea 48 ad stadard error 5/ (50) d) xbar caot be approximated with the ormal distributio sice X is ot ormal 37. Suppose that the probability that a UF basketball player, makes a free throw is p = Now suppose that he shoots 100 free throws over the course of a basketball seaso (sample of 100 idepedet free throws). Fid the approximate probability that he makes less tha 65% of his free throws durig the course of the seaso. a) b) c) d) 0.591
9 38. Suppose the average weight for adult males (age 18 or older) i Alachua Couty is 190 lbs with a stadard deviatio of 20. Now suppose we take a radom sample of 143 adult males (age 18 or older) i Alachua Couty. What is the probability that the average weight of our 143 subjects is bigger tha 193 lbs? a) b) c) d) Caot say from the iformatio provided 39. Refer to questio 38, but this time suppose we take a radom sample of 16 males from Alachua Couty. What is the probability that the average weight of our 16 subjects is bigger tha 193 lbs? a) b) c) d) Caot say from the iformatio provided 40. Suppose the probability that Barry Bods, a famous baseball player, gets a hit i a give at bat is p = 0.3. If Barry has 400 at bats i a sigle seaso (sample of 400 idepedet at bats), what is the mea ad stadard error of the samplig distributio phat (the sample proportio of hits per at bat)? a) mea = 0.3, stadard error = b) mea = 0.3, stadard error = c) mea = 0.7, stadard error = d) mea = 0.7, stadard error = Suppose the pvalue for a test is.02. Which of the followig is true? a) We will ot reject H 0 at alpha =.05 b) We will reject H 0 at alpha =.01 c) We will reject H 0 at alpha = 0.05 d) We will reject H 0 at alpha equals 0.01, 0.05, ad 0.10 e) Noe of the above is true. 42. A radom sample of married people were asked "Would you remarry your spouse if you were give the opportuity for a secod time?"; Of the 150 people surveyed, 127 of them said that they would do so. Fid a 95% cofidece iterval for the proportio of married people who would remarry their spouse. a) ± b) ± c) ± d) ± e) ± 0.113
10 43. You would like to estimate the proportio of "regular users of vitamis" i a large populatio. I order to fid a cofidece iterval for the proportio, a) we must assume that we have a radom sample from a ormal populatio b) we must assume that we have a radom sample from a biomial populatio where p> 15 ad (1p)> 15 c) we must assume that the populatio is ormal (but we do ot require a radom sample because of the Cetral Limit Theorem). d) we do ot eed to assume that the populatio is ormal or that the sample is radom (because of the Cetral Limit Theorem). e) We do ot eed to assume aythig. 44. A survey was coducted to get a estimate of the proportio of smokers amog the graduate studets. Report says 38% of them are smokers. Chatterjee doubts the result ad thiks that the actual proportio is much less tha this. Choose the correct choice of ull ad alterative hypothesis Chatterjee wats to test. a) Ho: p=.38 versus Ha: p.38. b) Ho: p=.38 versus Ha: p >.38. c) Ho: p=.38 versus Ha: p<.38. d) Noe of the above. 45. A political poll of Americas was coducted to ivestigate their opiios o gu cotrol. Each perso was asked if they were i favor or gu cotrol or ot i favor of gu cotrol  o respodets were removed from the results. The survey foud that 25% of people cotacted were ot i favor of gu cotrol laws. These results were accurate to withi 3 percetage poits, with 95% cofidece. Which of the followig is NOT correct? a) The 95% cofidece iterval is approximately from (22% to 28%). b) We are 95% cofidece that the true proportio of people ot i favor of gu cotrol is withi 3 percetage poits of 25%. c) I approximately 95% of polls o this issue, the cofidece iterval will iclude the sample proportio. d) If aother poll of similar size were take, the percetage of people IN FAVOR of gu cotrol would likely rage from 72% to 78%. 46. Suppose we are iterested i fidig a 95% cofidece iterval for the proportio p of UF udergraduate studets who are from the state of Florida. We take a radom sample of 20 studets, ad we fid that 17 of them are from Florida. Which of the followig is the smallsample cofidece iterval for p, usig 95% cofidece? a) (.694, 1.000) b) (.629,.954) c) (.850,.930) d) (.688, 1.000)
11 47. Which of the followig statemets about smallsample ad largesample cofidece itervals for proportios are true? I. The largesample cofidece iterval formula for proportios is valid if p 15 ad (1p) 15. II. Largesample cofidece itervals always cotai the true parameter value, whereas smallsample cofidece itervals may ot. III. We form smallsample cofidece itervals by usig the largesample formula after addig 4 successes ad 4 failures. a) I ad III oly b) II oly c) I oly d) I, II, ad III Questios 4850: Suppose we are iterested i fidig a 95% cofidece iterval for the mea SAT Verbal score of studets at a certai high school. Five studets are sampled, ad their SAT Verbal scores are 560, 500, 470, 660, ad What is the stadard error of the sample mea? a) b) c) d) What is the 95% cofidece iterval for the populatio mea? a) (462.3, 669.7) b) (469.9, 662.1) c) (486.3, 645.7) d) (492.8, 639.2) 50. The method used to calculate the cofidece iterval i the previous questio assumes which oe of the followig? a) The sample mea equals the populatio mea. b) The sample stadard deviatio does ot deped o the sample draw. c) The populatio has a approximately ormal distributio. d) The degrees of freedom df A sample of size 45 is draw from a slightly skewed distributio. What is the approximate shape of the samplig distributio? a) Skewed Distributio b) Biomial Distributio c) Normal Distributio d) Uiform Distributio
12 Questios We kow that 65% of all Americas prefer chocolate over vailla ice cream. Suppose that 1000 people were radomly selected. 52. The stadard error of the sample proportio is a) b) c) d) The Samplig Distributio of the sample proportio is a) Biomial ( 1000, 0.65) b) Normal( 0.65, ) c) Normal(10000,0.65) d) Noe of the above 54. What is the probability that our sample will have more tha 70% of people prefer chocolate ice cream? a) b) c) 0.70 d) oe of the above 55. We are doig a experimet where we record the umber of heads whe we get whe we flip a ubiased coi may times. For what sample sizes below would the samplig distributio of the sample proportio be approximately ormally distributed? a) 5 b) 28 c) 50 d) All of the above e) Noe of the above 56. For a test with the ull hypothesis Ho: p = 0.5 vs. the alterative Ha: p > 0.5, the ull hypothesis was ot rejected at level alpha=.05. Das wats to perform the same test at level alpha=.025. What will be his coclusio? a) Reject H 0. b) Fail to Reject H 0. c) No coclusio ca be made. d) Reject Ha. 57. The ull hypothesis Ho: p=.5 agaist the alterative Ha: p>.5 was rejected at level alpha=0.01. Nate wats to kow what the test will result at level alpha=0.10. What will be his coclusio? e) Reject H 0. f) Fail to Reject H 0. g) No coclusio ca be made. h) Reject Ha.
13 58. A ull hypothesis was rejected at level alpha=0.10.what will be the result of the test at level alpha=0.05? a) Reject Ho b) Fail to Reject Ho c) No coclusio ca be made d) Reject Ha Questios Commercial fisherme workig i certai parts of the Atlatic Ocea sometimes fid their efforts beig hidered by the presece of whales. Ideally, they would like to scare away the whales without frighteig the fish. Oe of the strategies beig experimeted with is to trasmit uderwater the souds of a killer whale. O the 52 occasios that that techique has bee tried, it worked 24 times (that is, the whales left the area immediately). Experiece has show, though, that 40% of all whales sighted ear fishig boats leave o their ow accord, ayway, probably just to get away from the oise of the boat. 59. What would a reasoable hypothesis test be: a) Ho: p=0.4 versus Ha: p = 0.46 b) Ho: p=0.46 versus Ha: p > 0.46 c) Ho: p=0.46 versus Ha: p 0.46 d) Ho: p=0.4 versus Ha: p > Suppose you wat to test Ho: p=0.4 versus Ha: p > 0.40 at the 0.05 level of sigificace. What would your coclusio be? a) Reject Ho. b) Accept Ho. c) Accept Ha. d) Fail to reject Ho. 61. Which of the followig are the assumptios that must be satisfied i order to be able to coduct a sigificace test for p? I The data is obtaied from a radom sample II The variable is categorical III The variable is quatitative IV The populatio size is large V The populatio is ormally distributed VI The sample size is sufficietly large VII The samplig distributio of pˆ is approximately ormal a) I, IV, ad VII b) I, II, ad VII c) I, III, ad VI d) I, IV, V ad VI
14 ANSWERS 1. C. The formula for the cofidece iterval for a populatio mea is: x t s, which was based o the sample Mea. So, x is guarateed to be i the iterval you form. 2. D. Use the rule : pvalue <alpha, reject H 0. The Pvalue is greater tha the sigificace level (=.10), so we ca coclude the data do ot provide sufficiet evidece to reject the ull hypothesis (H 0 ). Fail to reject H A. The formula for cofidece iterval is: x t s where (t s ) is the margi of error. Other thigs beig equal, the margi of error of a cofidece iterval icreases as the sample size decreases. So, whe the sample size decreases, the legth of the cofidece iterval will become bigger. 4. B. Similar to the previous questio. Other thigs beig equal, the margi of error of a cofidece iterval decreases as the cofidece level (t score) decreases. So, the legth of the cofidece iterval will become smaller whe the cofidece level decreases. 5. D. From the results of the previous two questios, we kow that whe the sample size icreases, the cofidece iterval will be smaller. However, it will become bigger as the cofidece level icreases. Therefore, we caot coclude how the cofidece iterval will be i this questio, sice we do t have eough iformatio to determie whether the chage i sample size or the cofidece level is more ifluetial here. 6. C. The samplig distributio of a statistic is the distributio of values take by the statistic i all possible samples of the same size from the same populatio. The mea of the samplig distributio of x is the populatio mea. 7. A. Sice the Pvalue is a probability, so it must be betwee 0 ad B. For 95% cofidece, z = For a margi of error of 0.5, we have 2 2 z* s = m = 1.96*10 = So, the sample size should be (Always 0.5 roud up to the ext higher whole umber whe fidig ). 9. C. Take the 93% ad chage it to a decimal (0.93). Take = 0.07 (this is the area i the tails). Divide this umber i half (0.035.) Look i the middle of the table for the etry This correspods to  z= Thus, z=1.81. I short look up (10.93)/2=.035 i the middle of the table ad z is the absolute value of the z score.
15 10. D. The samplig distributio of x is the distributio of values take by x i all possible samples of the same size from the same populatio. 11. B. Because we ifer coclusios about the populatio from data o selected Idividuals (all sample). 12. a. F. I a very large umber of samples, 95% of the cofidece itervals would cotai the populatio mea. If the edpoits of the CI are give, use the term cofidece, ot probability. b. T. The defiitio of cofidece iterval. We are 95% cofidece that the ukow lies betwee (1.15, 4.20). c. F. The ceter of each iterval is at x, ad therefore varies from sample to sample. So, whe 100 itervals calculated the same way, we ca expect 100 of them to capture their ow sample mea. Not oly 95% of them. d. F. This setece states that idividuals (all America households) is i that iterval. This is wrog. CI made statemets about ot idividuals. e. T. I a very large umber of samples, 95% of the cofidece itervals would cotai the populatio mea. f. T. The ceter of each iterval is at x, ad therefore varies from sample to sample. So, whe 100 itervals calculated the same way, we ca expect 100 of them to capture the sample mea. 13. C. Use the rule : pvalue <alpha, reject H 0. Our usual alpha levels are.10,.05, ad.01. We reject H 0 at all these levels, so III is true. II is ot true because there is ot a iterval i HT. I is true because the defiitio of the pvalue is the probability that you would see a result this extreme if the ull were true. This pvalue is so low that the probability of gettig a sample like this if H 0 were true is ulikely. 14. B. A parameter is a umber that describes the populatio. A statistic is a umber that describes the sample. 15. B This problem is a questio about the samplig distributio of the sample meas. The amout of moey eared i tips is a quatitative variable. The sample mea has a Normal distributio with mea equal to 10 ad stadard error equal to 2.5. Draw the picture z 7.09 The probability greater tha 7.09 is a very small umber almost zero
16 16. C. A parameter is a umber that describes the populatio. So here, the parameter should be the average umber of jelly beas i all packages made, which is A. z= =.61. Look the table A, the probability of beig less tha .61 is / C. Sice the umber of jelly beas follows the ormal distributio, we ca use the z table. 19. C. Accordig the cetral limit theory, whe is large, the samplig distributio of the sample mea x is approximately ormal. That is, x ~,. s 20. B. The formula for the cofidece iterval for a populatio mea is: x t.however, is large, so we ca use the z istead of the t. x z s. x =7.1. For 95% cofidece, z = So the cofidece iterval is * 200 =7.1.69=(6.41, 7.79) 21. D. The defiitio of cofidece iterval. We are 95% cofidet that the ukow populatio mea work hours lies betwee 6.82 ad A is wrog because it was the term probability whe the umbers are give. B is wrog because it talks about idividuals rather tha the populatio mea. C is wrog because of it estimates the average i our sample. A CI estimates the average i our populatio. 22. B. The estimate ( x i this case) is our guess for the value of the ukow parameter ( ). So, we eed to calculate the margi of error shows how accurate we believe our guess is, based o the variability of the estimate. That s why we have 95% cofidece i our iterval, istead of 100%. 23. C. From the coclusio of questio 4 ad 5, we kow that the cofidece iterval will become arrower whe the size icreases ad the cofidet level decreases. 24. B. We will reject the ull whe the pvalue is smaller tha the sigificace level. The pvalue of this test is 0.044, which is smaller tha the levels at.10,.05, but larger tha.01. So we reject the H 0 whe =0.10 ad.05, but fail to reject the ull whe = a. T. Sice the populatio has a ormal distributio, we ca use the Normal table for the probability that oe perso is more tha 200 lbs. b. T. Sice the populatio has a ormal distributio, the samplig distributio of x is ormal. So, we ca use the z table. c. T. Sice the populatio has a ormal distributio, the samplig distributio of x is also ormal. So, we ca use the z table.
17 d. F. The distributio is ot Normal because the 68,95,99.7% rule does ot apply. The sample size is quite small (1), so the CLT does ot apply. So, we ca t use the z table. e. F. The sample size is quite small (10), So the CLT does ot apply. So, we ca t use the z table. f. T. Accordig to the CLT, whe we draw a SRS of size from ay populatio with mea ad fiite stadard deviatio. Whe is large, the samplig distributio of the sample mea x is approximately ormal. x ~N(, ). Here, the sample size is large, so we ca apply the CLT. Therefore, we ca use the Z table to fid the probability. p ( 1 p) g. F. pˆ ~N( p, ) whe values of, p satisfyig p 15 ad (1p) 15. However, p 10 =20* =4<15, therefore, you ca t use Normal table here to fid this probability. NOTE 50 we leared i class that you ca make CI for this data if you add 2 successes ad 2 failures. The trick oly works for CI for p ot for sigificace tests, or fidig this type of probability, or doig problems about meas. p ( 1 p) h. T. pˆ ~N( p, ) whe values of, p satisfyig p 15 ad (1p) p =91* =18.2. (1p)=91*(4000/5000)=72.8. So we ca use the z table here C pˆ ~N( p, p ( 1 p) ) whe values of, p satisfyig p 15 ad (1p) 15. p =0.2*100=20> 15 ad (1p)=100*0.8 = So, pˆ ~N( 0.2, 0.04 ) 0.2(0.8) 100 ) pˆ ~N( 0.2, 27. B Use the samplig distributio of the sample proportio that you used above ad the z score. z 1.0 Look up 1.00 i the table is listed i the table. This is the 0.04 proportio less tha, we wat the proportio greater tha so we take = C pˆ ~N( p, p ( 1 p) ) whe values of, p satisfyig p 15 ad (1p) 15. p =0.05*3200=160 > 15 ad (1p)=3200*0.95= So, pˆ ~N( 0.05, pˆ ~N( 0.05, ) 0.05(0.95) 3200 )
18 p ( 1 p) 29. D pˆ ~N( p, ) whe values of, p satisfyig p 15 ad (1p) 15. p =0.06*100=6 < 15 So, the distributio of the sample proportio is ukow. p ( 1 p) 30. C pˆ ~N( p, ) whe values of, p satisfyig p 15 ad (1p) 15. p =0.06*300=18 > 15 ad (1p)=300*0.94= So, pˆ ~N( 0.06, 0.06(0.94) 300 ) pˆ ~N( 0.06, ). 31. A Use the samplig distributio of the sample proportio that you used i problem 52 ad the zscore. z 2.90 Look up 2.90 i the table. P(z< 2.90) = We wat the probability that our sample proportio is greater tha 0.10 so we take = ( z) p(1 p) (1.96) (.04)(.96) 32. B. = = 2 2 m.01 So should be 1476 = B. ˆp == X =40/800=0.05 estimate of the stadard error of ˆp = pˆ(1 pˆ) = 0.05*(0.95) 800 From the z table, we fid the value of z to be = So the CI is ˆp z*se pˆ =.05 (1.960)(.0077)= (.035,.065) 34. G. The populatio proportio p is ukow, ad that s why we wat to estimate it by the sample proportio. E. Phat is the sample proportio. pˆ =X/=287/350=.82 A. Sice the researchers are testig to determie if more tha half of all reported scams victimize the elderly, the p 0 should be 0.5. C. From the sample size, we record the cout X of success ( here ifers the victim over 65 years old). The X should be 287. D. The total sample size is 350.
19 35. E. Sice the Pvalue is a probability, so it must be betwee 0 ad C Note that we are takig 50 idepedet observatios from the distributio. This is a large eough sample size to use the Cetral Limit Theorem for xbar. Hece xbar is distributed approximately ormal with mea 48 ad stadard error 5/ (50) 37. A Here, we are samplig 100 idepedet observatios from a populatio with proportio p = Therefore, the samplig distributio of the sample proportio has mea 0.75 ad stadard error ((0.75*0.25)/100) = Now, sice p = 75 > 15 ad (1p) = 25 > 15, we ca coclude that the samplig distributio of the sample proportio is approximate ormally distributed with mea 0.75 ad stadard error Now, we ca compute the zscore ad get z = ( ) / = The, we use a ztable to fid P(Z < 2.31) = B Note that we are takig a radom sample of 143 adult males. Sice this sample size is larger tha 30, we ca use the Cetral Limit Theorem ad coclude that average weight is approximately ormally distributed eve though we do ot kow the distributio of the weight of adult males i Alachua Couty. So the average weight is distributed ormally with mea 190 ad stadard error 20/ (143) = Now, we ca compute the zscore ad get z = ( ) / = Fially, we ow use the ztable to fid P(Z > 1.79) = D We are ot give the distributio of the populatio we are samplig from ad our sample size is oly 16 (<30). Hece, we caot give a approximate probability here. 40. B Here, the umber of hits is a biomial radom variable with = 400 ad probability of a hit p = 0.3. Therefore, the mea of the samplig distributio of the proportio of hits has mea 0.3 ad stadard error C Rule: pvalue < alpha Reject Ho. At alpha = 0.10, reject Ho. At alpha = 0.05, reject Ho. At alpha = 0.01, fail to reject Ho. 42. D pˆ z pˆ(1 pˆ) *(1.8467) B For a cofidece iterval for the populatio proportio, we must assume that the data comes from a radom sample, that we have categorical data, p> 15 ad (1p)> C The size of the populatio does ot affect how accurate the results are. The size of the sample affects how accurate a predictio we ca make.
20 45. C is the icorrect statemet. The cofidece iterval is suppose to estimate the populatio proportio ot the sample proportio. A is just givig the cofidece iterval that is o.k. B is talkig about estimatig the populatio proportio with the cofidece iterval that is correct. D is estimatig the complemet of ot i favor of gu cotrol i favor of gu cotrol. 46. B p ˆ = 17 > 15, but (1 pˆ ) = 3 < 15. Therefore, we ca compute the cofidece iterval usig the large sample formula if we add 2 successes ad 2 failures. The 17 2 pˆ pˆ 1 pˆ se (.7917)(.2083) ad the resultig 95% cofidece iterval is p ˆ 1.96( se) (.629,.954). 47. C The oly correct statemet is the first oe The largesample cofidece iterval formula for proportios is valid if p 15 ad (1p) 15. The large sample cofidece iterval oly cotai the true value a certai percetage of the time. A 95% CI will cotai the value 95% of the time. You add 2 successes ad 2 failures. 48. B First, we use a calculator to fid that the sample stadard deviatio s = The se s A The 95% cofidece iterval for the populatio mea is x t. 025 se. I this particular problem, we have x se Usig df = 1 = 4, we look up (i a table) that t.025 = The our cofidece iterval is x t se (462.3, 669.7) C Assumptios for the cofidece iterval for the mea are as follows: data is quatitative, radom sample, data comes from a ormal distributio. Oly statemet (c) is true.
21 51. C Accordig to the Cetral Limit Theorem for large the samplig distributio of sample mea is Normal. 52. B The stadard error is (0.65 * 0.35)/ 1000= B pˆ ~N( p, p ( 1 p) ) whe values of, p satisfyig p 15 ad (1p) 15., p satisfyig 1000* ad 1000* So, pˆ ~N( 0.65, pˆ ~N( 0.65, ) 0.65(1 0.65) 1000 ) 54. B The Z value for this ( )/ = 3.32 Now P( Z> 3.32) = 1 P(Z 3.32) = C p = 0.5 = the probability of gettig heads whe you flip a ubiased(fair) coi You eed to have p> 15 ad p> 15. This happes whe = 50. (50*.5=25 ad 50*(1.5) = 25) 56. B The hypothesis was ot rejected at level alpha=.05.so p value was higher tha 0.05 ad so higher tha as well. So the test will agai fail to reject the ull hypothesis at level = A The hypothesis was rejected at level=0.01.so, p value was less tha 0.01 ad so less tha 0.10 as well. Hece the test will agai reject the ull hypothesis at level= C The hypothesis was rejected at level=0.10.so p value was less tha 0.10.But that might be more tha 0.05 or might be less tha 0.05 which we do t kow from above iformatio. Therefore we do t kow what will happe for the test at level= D They wat to show that more whales tur away tha usual with the extra souds emitted. p(1 p) 0.4(1 0.4) 60. D Solutio: phat is 24/52= se pˆ p Thus, z se The probability shaded greater tha is ( ) = pvalue = pvalue is ot less tha alpha So, we fail to reject Ho. 61. B The assumptios of the hypothesis test for a proportio are the data must be categorical, data must come from a radom sample, p> 15 ad (1p)> 15.
1. C. The formula for the confidence interval for a population mean is: x t, which was
s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : pvalue
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