PRACTICE EXAMINATION 1

Practice Examination 1

Section IQuestions l-40

Spend 90 minutes on this part of the exam.

Directions: The questions or incomplete statements that follow are each followed by five sug-gested answers or completions. Choose the response that best answers the question oicompletesthe statement.

1. Following is a histogram of home sale prices(in thousands of dollars) in one communitv:

Which of the following statements are tfue?I. The median price was $125,000.

II. More homes sold for between $100,000and $125,000 than for over $125,000.

m. $10,000 is a reasonable estimate of thestandard deviation in selling prices.

(A) I only(B) II only(C) III only(D) AX are true.(E) None is true.

2. .Which of the following can stemplots show?I. Symmetry

II. GapsIIL ClustersIV. Outliers(A) I, [I, and III(B) I, II, and IV(C) I, III, and IV(D) II, III, and IV(E) I, il, III, and IV

3. The average yearly snowfall in a city is 55 .inches. What is the standard deviation if l|Voof the years have snowfalls above 60 inches?Assume yearly snowfalls are normallydistributed.(A) 4.83(B) 5.18(c) 6.04(D) 8.93(E) The standard deviation cannot be

computed from the information given.

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.?RACTICE EXAMINATIONS

4. To determine the average cost of running fora congressional seat, a simple random sampleof 50 politicians is chosen and the politicians'records examined. The cost figures show amean of $125,000 with a standard deviationof $32,0CI. Which of the following is the bestinterpretation of a 90Vo confidence intervalestimate for the average cost of running foroffice?(A) 9AVo of politicians running for a

congressional seat spend between$117,500 and $132,500.

(B) 90% of politicians running for acongressional seat spend a mean dollaramount that is between $117.500 and$132,500. !

(C) r$fe arc 90Vo confident that politiciansrunning for a congressional seat spendbetween $117,500 and $132,500.

(D) We arc 90c/o confident that politiciansrunning for a congressional seat spend amean dollar amount between $117,500and $132,500.

(E) We arc 9OVo confident that in the chosensample, the mean dollar amount spentrunning for a congressional seat is :,between $117,500 and $132,500. . i

5. In one study on the effect that eating meatproducts has on weight level, an SRS of 500subjects who admitted to eating meat at leastonce a day had their weights compared withthose of an independent SRS of 500 peoplewho claimed to be vegetarians. In a secondstudy, an SRS of 500 subjects were served atleast one mqat meal per day for 6 months,while an independent SRS of 500 others werechosen to receive a strictly vegetarian diet for6 months, with weights compared after 6months.(A) The first study is a controlled

experiment, while the second is anobservational study.

(B) The first study is an observational srudy,while the second is a controlledexperiment.

(C) Both studies are controlled experiments.(D) Both studies are observational studies.(E) Each study is part controlled experiment

and part observational study.

6. A plumbing contractor obtains 6AVo of herboiler circulators from a company whosedefect rate is 0.005, and the rest from acompany whose defect rate is 0.010. Whatproportion of the circulators can be expectedto be defective? If a circulator is defective,what is the probability that it came from thefirst company?(A) .0070, .429(B) .0070, .600(c) .0075, .500(D) .0075, .600(E) .0150,.571

7. A kidney dialysis center periodically checks asample of its equipment and performs a majorrecalibration if readings are sufficiently offtarget. Similarly, a fabric factory periodicallychecks the sizes of towels coming off anassembly line and halts production ifmeasurements are sufficiently off target. Inboth situations, we have the null hypothesisthat the equipment is performingsatisfactorily. For each situation, which isthe more serious concern, a Type I or$pe II error?(A) Dialysis center; lYpe I error, towel

manufacturer: Type I error($) Dialysis center: Type I error, towel

manufacturer: Type II error(C) Dialysis center: Type II error, towel

manufacturer: Type I error(D) Dialysis center: Type II error, towel

manufacturer: Tlpe II error(E) This is impossible to answer without

making an expected value judgmentbetween human life and accurate towelsizes.

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In the following table, what value for nresults in a table showing perfectindependence?

(A)(B)(c)(D)(E)

9. During the years 1886 through 2000 therewere an average of 8.7 tropical cyclones peryear, of which an average of 5.1 becamehurricanes. Assuming that the probability ofany cyclone becoming a hurricane is.independent of what happens to any othercyclone, if there are five cyclones in one year,what is the probability that ar least threebecome hurricanes?(A) .313(B) .34s(c) .s86(D) .558(E) .686

10. Given the probabilities P(A) = .3 and p(B) =.2, what is the probability of the union p(A UB) if A and B are mutually exclusive? If Aand B are independent? If B is a subset of A?(A) .44, .5, .2(B) .44, .5, .3(c) .5, .44,.2(D) .5, .44,.3(E) 0, .5, .3

PRACTICE EXAMINATI,ON I I

11. A nursery owner claims that a recent droughtstunted the growth of 22Vo of all herevergreens. A botanist tests this clairn byexamining an SRS of 1100 evergreens andfinds that 268 trees in the sample show signsof stunted growth. With Hs: p = .22 and Ho:p * .22, what is the value of the test statistic?

.244 - .22z=-:'-!Tmb@

_ _. .24 - .22-_:

{1100(.22Xr - .22)

.244 -.22(u)z=-:\v, / . -w1 uoo

(A)

30507075100

(B)

12.

(D) z- 2

_ .244 _ .22, ,E,r i=26

A medical research team tests fo, tu-orreduction in a sample of patients using thre.edifferent dosages of an experimental canc€rdrug. Which of the following is tnre?(A) There are three explanatory variables

and one response variable.(B) There is one explanatory variable with

three levels of response.(C) Tumor reduction is the only explanatory

variable, but there are three r€sponsevariables corresponding to the differentdosages.

(D) There are three levels of a singleexplanatory variable.

(E) Each explanatory level has an associaEpdlevel ofresponse.

40 60

50 n

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370 PRACTICEEXAMINATIONS

13. The graph below shows cumulativeproportion plotted against age for apopulation.

The median is approximately what age?(A) 17(B) 30(c) 3s(D) 40(E) 45

14. Mr. Bee's statistics class had a standarddeviation of Ll.2 on a standardized test, whileMr. Emfs class had a standard deviation of 5.0,,on the same test. Which of the following is .the most reasonable conclusion concerningthe two classes'performance on the test?(A) Mr. Bee's class is less heterogeneous

than Mr. Em's.(B) Mr. Em's class is more homogeneous

than Mr. Bee's.(C) Mr. Bee's class performed twice as well

as Mr. Em's..(D) Mr. Em's class did not do as well as Mr.

Bee's.(E) Mr. Bee's class had the higher mean, but

this may not be statistically significant.

15. A college admissions officer is interested incomparing the SAT math scores of highschool applicants who have and have nottaken AP Statistics. She randomly pulls thefiles of five applicants who took AP Statisticsand five applicants who did not, and proceedsto run a t-test to compare the mean SAT mathscores of the two groups. Which of thefollowing is a necessary assumption?(A) The population variances from each

group are known.(B) The population variances from each

group are unknown.(C) The population variances from the two

groups are equal.(D) The population of SAT scores from each

group is normally distributed.(E) The samples must be independent simple

random samples, and for each sample npand n(l -p) must both be at least 10.

15. The appraised values ofhouses in a city havea mean of $125,000 with a standard deviationof $23,000. Because of a new teachers'contract, the school district needs an extrallVo in funds compared to the previous year.To raise this additional money, the cityinstructs the assessment office to raise allappraised house values by $5,000. What willbe the new mean and standard deviation ofthe appraised values of houses in the city?(A) Mean: $130,000, standard deviation:

$23,000(B) Mean: $130,000, standard deviation:

$28,000(C) Mean: $142,500, standard deviation:

$2s,300(D) Mean: $142,500, standard deviation:

$30,300(E) Mean: $143,000, standard deviation:

$30,800

c 1.0oE& o.aIo.o 0.6

'#o5 o.rEtu o.z

30 40 50 60 70

Age

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Score:

Number ofpeople:

17. Apsychologist hypothesizes that scor€s on anaptitude t€st are normally distributed with amean of 70 and a standard deviation of 10. Ina random sample of size 100, scores aredistributed as in the table below. What is thea-

,i statistic for a goodness_of_fit test?

PRACTICE EXAMb{ATION 1 371

19. Define a new measurement as the differencebetween the 60th and 40th percentile scores ina population. This measurem€nt will giveinformation concerning(A) central tendency.(B) variability.(C) symmenry.(D) skewness.(E) clustering.

20. Suppose X and y are random variables withE(n - 37, va{X) = 5, E(n = 62, andvar(I4= 12. What are the expected value andvariance of the random variable X + y?(A) E(X + y) - 99, var(X + I) = g.5(B) E(X+ n - 99,var(X+ f) = t3(C) E(X t Y) = 99, var(X + y) = 17(D) E(X + y) - 49.5, var(X + y) = 17(E) There is insufficient informatiori to

answer this question.21. Which of the following can affect the value of

. the correlation r?(A) A change in measurement units(B) A change in which variable is cailed x

and which is called v(C) Adding the same colstant to all values

of the .r-variable(D) All of the above can affect the r value.(E) None of the above can affect the r value.

22. The American Medical Association (AMA)wishes to determine the percentage ofobstetricians who are considerinfbaving ttreprofession because of the rapidly increasingnumber of lawsuits against obstetricians. Howlarge a sample should be taken to find theanswer to within l3%o at the 95Vo confidencelevel?(A) 6(B) 33(c) s34(D) 7s2(E) 1068

below 60 6G70 70-80 above 80

l0 40 35 l5

, ̂ ., (10 - 16)2 @0

.?4)2 * Qs - z+)2 -,_

(rs - ro)2l.'' -lo-- t -40 35

r 15

, r , 0o-ro)2 +(40-34)2 ,es_34)2, 05_16)2\-, t6 - ---- * -J;- + -G--

,o, ( ro-zs)2

*(+o-zs)2 _(3s_2s)2 .0s_2rz- 25 -- 25-=--T---T

p; !9=3'),. + @o -?s)2 + Qs - 2s)2 * os - 2r210 40

--T-* ' 15

@) (r0:?i l2

+@0-2s)2 +es-2s)2 _es_2$216 34 ---- l fr-;-

18. A talk show host recently reported that inresponse to his on_air question, g2clo of themore than 2500 e-mail messages receivedthrough his publicized address supported thedeath penalty for anyone convicted of sellingdrugs to children. What does this show?(A) The survey is meaningless because of

voluntary response bias.(B) No meaningful conclusion is possible

without knowing something more aboutthe characteristics of his listeners.

(C) The survey would have been moremeaningful if he had picked a randomsample of the 2500 listeners whoresponded.

(D) Th,g survey would have been morEmeaningful if he had used a controlgroup.

(E) This was a legitimate sample, randomlydrawn from his listeners, and ofsufficient size to be able to conclude thatmost of his listeners support the deathpenalty for such a crime.

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372 PRACTTCEEXAMINATIONS

23. Following are parts of the probabilitydistributions for the random variables Xand I.

IfX and Iare independent and thejointprobability P(X = 2, Y = 3) = .03, what isP(X = 4\?(A) .10(B) .ls(c) .20(D) .2s(E) .30

24. Which of the following sets has the smallest. standard deviation? Which has the largest?

r, L,2,3,4,5,6,7I I I . 1, 1, ' . ] , ,4,7,7,7

III I . 1, 4, 4, 4, 4,4,7(A) I, n(B) II, m ;"(c) trL I(D) tr,I(E) trI, n

25. Aresearcher plans a study to examine long-ferm confidence in the U.S. economy amongthe adult population. She obtains a simplerandom sample of 30 adults as they leave aWall Street office.building one weekdayafternoon. All but two of the adults agree toparticipate in the survey. Which of thefollowing are true statements?

I. Proper use ofchance as evidenced bythe simple random sample makes this awell-designed survey.

II. The high response rate makes this awell-designed survey.

III. Selection bias makes this a poorlydesigned survey.

(A) I only(B) II only(C) III only(D) I and tr(E) None of the statements is true.

Which among the following would result inthe narrowest confidence interval?(A) Small sample size and 957o confidence(B) Small sample size and 99Vo confidence(C) Large sample size and 957a confidence(D) Large sample size and 99%o confidence(E) This cannot be answered,wifhout

knowing an appropriate standarddeviation.

Consider the following three scatterplots:

What is the relationship among r; 12, drrd 13,the correlations associated with the first,second, and third scatterplots, respectively?(A) rr I r21 13(B) rr <ry<rz(C) rz<ry<rr(D) rs 1\ 112(E) ry <12<rr

28. There are two games involving flipping acoin. In the first game you win a prize if youcan throw between 45Vo and 557o heads: Inthe second game you win if you can throwmore than 807o heads. For each game wouldyou rather flip the coin 30 times or 300times?(A) 30 times for each game(B) 300 times for each game(C) 30 times for the first game and 300

times for the second(D) 300times for the first game and 30

times for the second(E) The outcomes of the games do not

depend on the number of flips.

26.

27.

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ii;

j

. l i

I

29. City planners are trying to decide amongvarious parking plan options ranging frommore on-str€et spaces to multilevel facilitiesto spread-out small lots. Before making adecision, they wish to test the downtownmerchants'claim that shoppers park for anaverage of only 47 minutes in the downtownarea. The planners have decided to tabulateparking durations for 2ZS shoppers and to

. reject the merchants' claim if tlie samplemean exceeds 50 minutes. If the merchants'claim is wrong and the true mean is 5lminutes, what is the probability that therandom sample will lead to a mistaken failureto reject the merchants' claim? Assume thatthe standard deviation in parking durations is27 minutes.

/ <n_47\(A) Pl z<+l

\ 271422s)

8\ dz> s0-47)\ 2714225)

{c) p(2.#)\ 2714225)

@'t p(z> so - 5l )\ 2714225 )

n {z> sl-47)\ - 271422s )

Consider the following parallel boxplotsindicating the starting satraries (in thousandsof dollars) for blue collar and white collarworkers at a particular production plant:

15" 20 25 30 35 40 15 50 55

PRACTICE EXAMR.IATION I 373

Which of the following are true statements?L The rang€s are the same.

II. The interquartile ranges are the same.trL Because of symmeffry, their medians are

the same.(A) I only(B) tr only(C) I and tr(D) I and III(E) II and III

31. The mean Law School Aptitude Test (LSAT)score for applicants to a particular law schoolis 650 with a standard deviation of 45.Suppose that only applicants with scoresabove 700 are considered. What percentage ofthe applicants considered have scores below740? (Assume tlre scores are normallydistributed.)(A) l3.3Vo(B) l7.r%o(C) 82.9Vo(D) 86.7,Vo(E) 97.7Vo

32. Lf all the other variables remain constanr,which of the following will increase thepower of a hypothesis test?

I. Increasing the sample size.II. Increasing the significance level;

III. Increasing the probability of a Tlpe IIerTor.

,(A) I only(B) II only(C) III only

' (D) I and II(E) AII are rrue.

f,tfti*,e:rl

*.

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374 PRACTICEEXAMINATIONS

33. A researcher planning a survey of schoolprincipals in a particular state has lists of theschool principals employed in each of the 125school districts. The procedure is to obtain arandom sample of principals from each of thedisnicts rather than grouping all the liststogether and obtaining a sample from theentire group. Which of the followingstatements about the resulting stratifiedsample are true?

I. It is not a simple random sample.II. It is easier and less costly to obtain than

a simple random sample.III. It gives comparative information that a

simple random sample wouldn't give.(A) I only(B) I and II(C) I and trI(D) II and III(E) I, II, and III

34. A simple random sample is defined'by(A) the method of selection.(B) examination of the outcome.(C) both of the above.(D) how representative the sample is of the

population.(E) the size of the sample versus the size of "

the population.

35. Ch'anging from a 90Vo confidence intervalestimate for a population proportionto a99%oconfidence interval estimate, with all otherthings being equal,(A) increases the interval sizeby 9Vo.(B) decreases the interval sizeby 9%.(C) increases theinterval sizeby 57Vo.(D) decreases the interval sizeby 57Vo.(E) This question cannot be answered

without knowing the sample size.

36. Which of the following are true statements?I. If there is sufficient evidence to reject a

null hypothesis at the 107o level, thenthere is sufficient evidence to reject it atthe 57o level.

II. Whether to use a one- or a two-sidedtest is typically decided after"the data aregathered.

III. If a hypothesis test is conducted at thelVo level, there is a LVo chance ofrejecting the null hypothesis.

(A) I only(B) II only(C) III only(D) I, II, and III(E) None are true.

37. A population is normally distributed withmean 25. Consider all samples of size 10. The

x-zJvanaDle

-g{10

(A) has a normal distribution.(B) has a /-distribution with df = 10.(C) has a r-distribution with df = 9.(D) has neither a normal distribution or a t-

distribution.(E) has either a normal or a r-distribution

depending on the characteristics of the" population standard deviation.

38. A conelation of .6 indicates that thepercentage of variation in y that is explainedby the variation in.r is how many times thepercentage indicated by a correlation of .3?(A) 2(B) 3(c) 4(D) 6(E) There is insufficient information to

answer this question.

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39,As reported on CNN, in a May 1999 nationalp,oll43Vo ofhigh school students expressedfear about going to school. Which of thefollowing best describes what is meant by thepoll having a margin of error of SVo?(A) It is likely that the rrue proportion of

high school students afraid to go toschool is between 38Vo and4g%o.

(B) Five percent of the students refused toparticipate in the poll.

(C) Between 38Vo and 48Vo of thosesurveyed expressed fear about going toschool.

(D) There is a .05 probability that the 43Voresult is in error.

(E) If similar size polls were repeatedlytaken, they would be wrong about 5Vo ofthe time.

PRA'CTTCE EXAMINATION 1 375

40. In a high school of 1650 sfudents, l3}havepersonal investments in the stock market. Toestimate the total stock investment by studentsin this school, two plans are proposed. plan Iwould sample 30 students at random, find aconfidence interval estimate of their averageinvestment, and then multiply both ends ofthis interval by 1650 to get an intervalestimate of the total invesffnent. plan tr wouldsample 30 students at random from among the132 who have investments in the market, finda confidence interval estimate of their averag€investment, and then multipty both ends ofthis interval by 132 to get an interval estimateof the total investment. Which is the betterplan for estimating the total stock marketinvestment by students in this school?(A) Plan I(B) Ptan II

" (C) Both plans use random sarnples and so

will produce equivalent results.(D) Neither plan will give an accurate

estimate.(E) The resulting data must be seen to

evaluate which is the better plan.

,

STOP

376 PRACTICEEXAMINATTONS

Secfion ll

Pqrt AQuestions 1-5

Spend about 65 minutes on this part of the exam.Percentage of Section II grade-75

You must show all work and indicate the methods you use. You will be graded on the correct-ness of your methods and on the accuracy of your results and explanations.

1. An AP Statistics student takes a random survey of 100 high school teachers and classifies departmentaffiliation versus music preference. The results are as follows:

Humanities/Soc. Sci.

Math/?hysical Sci.

Fine Arts

Classical Country Jazz

15 ) 20

t0 15 10

l5 ) 5

(a) What is the probability that a teacher sdlected at random from this sample is a fine arts teacherwho prefers country music?

(b) What is the probability that a teacher selected at"random from this sample is a math/physicalscience teacher given that he/she prefers jazz?

(c) What is the probability that a teacher selected at random from this sample prefers classical musicgiven that he/she is a humanities/social science teacher?

(d) Is there evidence of a relationship between departrnent affiliation and music preference? Explain.Ahigh school principal would like to know whether or not a new integrated math curriculum is workingbetter than the traditional style of separation of topics. He notes that of the six math teachers, three areusing the new approach and three the traditional. He asks them to administer a common final exam andplans to compare the average student scores from each of the two groups. If one group does signifi-cantly better than the other, he'll ask all of the teachers to adopt the indicated teaching mode.

(a) Comment on the principal's scheme.(b) Design an appropriate experiment for the principal to use.(c) How would you modify the experiment if you are concerned that a confounding variable is the

abitity of a student as measured by the student's scoring above or below 475 ona national exam?

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PRACTICE EXAMINATION 1 377

3' A printing firm takes on new jobs on a first come, first serv€d basis. Typic ally,:2lVoof the orders aresmall, taking 3 hours to finish, 3ovo of the orders take 8 hours to finish, 35Zo tak€ 12 hours, una rcf,take ?-0 hours.(a) what is the average number of hours the jobs take to finish?(b) Suppose in any given week that when the total number of hours reaches or exceeds 40, no further

orders will be taken. Explain how you would use simulation with a random number table toestimate the distribution of the number of new jobs the firm takes on each week and the disributionof the number of job-hours the firm accepts each week.

(c) Run three trials of your simulation using the random number table below. Give your results,explaining your procedure in action.

84177 06757 17613 15582 51506 81435 41050 g2o3r 06449 05059s9884 31180 s3115 84469 9,1868 57967 05811 84514 TSOrr 1300663395 55041 15866 06589 l3rr9 7r}2o 85940 91932'06488 74987

4' In a study of migraine headache sufferers, each of 500 subjects flipped a coin to determine who wasgiven either a placebo or an experimental pain medication- After one hour each was asked to jud.gerelief on a scale of 1 t'o l0 with "no reliefi = I and "complete relief" = 10. The results appear in thefollowing frequency table:

Pain reliefFre{uency of

placebo takersFrequency of experimental' medication takers

1 39 l8

2 68 2r3 50 204 22 335 2l 49

6 23 407 16 318 9 t39 7 6

l0 9 5

n =264i= 3.76

n=236i= 4.92

(a) Use a graphical display to compare the effectiveness of the placebo versus the experirnental med-ication in treating migraine headaches. Write a few sentences, based on your graph, to comparethe results of the subjects receiving placebo versus the subjects receiving the medication.

(b) Does there appear to be significantly greater pain relief from taking the medication as opposed tothe placebo? Give a statistical justification for your answer.

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378 PRACTICE EXAMINATIONS

5. A guidance counselor believes that eating a good breakfast leads to higher grades. She interviews 20students taking the health course that she also happens to teach and records grade point averages andaverage calories consumed for breakfast. Some computer graphical output follows;

I

ta

'a lI

a lI

taI

II

Ta!

Residuals Versus CaloriesResponse is GPA

f ; to

2

t l500 1000

Calories

I

r500

(a)(b)(c)

Histogram of the ResidualsResponse is GPA

-1.00 -0.75 -o.50 -0.25 0.00 0.25 0.50 0.7s 1.00 1.25 1.s0Residual

t

Is this an observational study or a controlled experiment? Explain.Comment on the design of the study.The counselor plans to do a test of significance for evidence of an association between GPA andbrealdast calories. Comment on whether the necessary assumptions are met for inference on slopefor a least squares, regression line.

t l

I

a

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PRACTICE EXAMINATION I 379

Section ll

Porf B

auestion 6Spend about 25 minutes on this part of the exam.

Percentage of Section tr grade-2

5. For a final project, three AP Statistics students plan a study of right-hand versus left-hand grip strengthsof male high school students in a large city. From each of the city's five high schools, ttrey pict a simplerandom sample.of 40 male students and measure right-hand and left-hand grip strengths using a grip-ping device that gives values on a 1 to l0 scale. In their sample, right-hanJgrip srengths *"

"pp-i-imately normally distributed with a mean of 7.3 and a'standard deviation oi1.g, wtrite teft-trand gripsftengths are approximately normally distributed with a mean of 6.1 and a standard deviation of 1.2.There ate 58 students whose left-hand grip strength is greater than that of the right-hand, while l4Zstudents show higher grip strength in ttri rignt nanO.(a) Explain why the students did or did not obtain a simple random sample of all male high school

students in the city. Explain why your answer would or would not "L*g"

if you knew that attfive high schools had the same number of male students.

(b) Calculate and interpret a gOVo confidence interval estimate for the proportion of male high schoolstudents in the city for which the right-hand grip strength is greatir ttran ttrat of the leftl

(c) Suppose that two male students are choseny't random. Whaf is the approximate probability thatthe right-hand grip strength of the first student is greatep thari the tet-trana grip strength of tfr"second?

(d) Based on your answers to parts (b) and (c), is there evidence that left-hand and right-hand gripstrengths of male students in the city are independent? Explain.

STOP