AP Exam Practice 2018 Page 1 of 59 Exploring Data

APExamPractice:ExploringData

NAME_____________________________________________1.

(a) Write a few sentences comparing the distributions of the yearly salaries at the two corporations. (b) Suppose both corporations offered you a job for $36,000 a year as an entry-level accountant.

(i) Based on the boxplots, give one reason why you might choose to accept the job at corporation A. (ii) Based on the boxplots, give one reason why you might choose to accept the job at corporation B.

2015#1

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2.Independentrandomsamplesof500householdsweretakenfromalargemetropolitanareaintheUnitedStatesfortheyears1950and2000.Histogramsofhouseholdsize(numberofpeopleinahousehold)fortheyearsareshownbelow.

(a) Compare the distributions of household size in the metropolitan area for the years 1950 and 2000. (b) A researcher wants to use these data to construct a confidence interval to estimate the change in mean

household size in the metropolitan area from the year 1950 to the year 2000. State the conditions for using a two-sample t-procedure, and explain whether the conditions for inference are met.

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3.(b)Foreachregion,describehowtheyearlyfrequencieschangedoverthetimeperiodfrom1997to2010.

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2015#1:YearlySalaries

Student“F:”

Student“E:”

F:Nocomparisonsofcenterorspread,only“$”forcontext.INCOMPLETEE:Nomentionof“potentialtoearnmoremoney”PARTIAL

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2012#3:HouseholdSizes

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2013#6ab(ExampleB)

(b)

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2013#6ab(ExampleC)

(b)

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APExamPractice:BivariateData

(2016#6)

1.

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2. (a)Basedonthescatterplot,describetherelationshipbetweenarmspanandheightforthesampleof12seniors.Letxrepresentheight,ininches,andletyrepresentarmspan,ininches.Twoscatterplotsofthesamedataareshownbelow.Graph1showsthedatawiththeleastsquaresregressionline andgraph2showsthedatawiththeliney=x.

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(b) The criteria described in the table below can be used to classify people into one of three body shape categories:

square, tall rectangle, or short rectangle.

(i) For which graph, 1 or 2, is the line helpful in classifying a student’s body shape as square, tall rectangle, or short rectangle? Explain.

(c) Using the best model for prediction, calculate the predicted arm span for a senior with height 61 inches.

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3.

(b) Interpret the value of r2 .

(c) Comment on the appropriateness of this linear regression for modeling the relationship between the transformed variables.

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ANSWERS to AP Exam Practice: Bivariate Data 2016 #6: Academic Semesters

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Score: E P I Reason:

Score: E P I Reason:

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Score: E P I Reason:

ab: P (b lacks nondeterministic lang.)

cd: E e:P

(retracting old is incorrect

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APExamPractice:Sampling 1.Atacertainuniversity,studentswholiveinthedormitorieseatatacommondininghall.Recently,somestudentshavebeencomplainingaboutthequalityofthefoodservedthere.Thedininghallmanagerdecidedtodoasurveytoestimatetheproportionofstudentslivinginthedormitorieswhothinkthatthequalityofthefoodshouldbeimproved.Oneevening,themanageraskedthefirst100studentsenteringthedininghalltoanswerthefollowingquestion.

Manystudentsbelievethatthefoodinthedininghallneedsimprovement.Doyouthinkthatthequalityoffoodservedhereneedsimprovement,eventhoughthatwouldincreasethecostofthemealplan?

_____Yes_____No_____Noopinion

(a)Inthissetting,explainhowbiasmayhavebeenintroducedbasedonthewaythisconveniencesamplewasselectedandsuggesthowthesamplecouldhavebeenselecteddifferentlytoavoidthatbias.

(b)Inthissetting,explainhowbiasmayhavebeenintroducedbasedonthewaythequestion

waswordedandsuggesthowitcouldhavebeenwordeddifferentlytoavoidthatbias.

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2.

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(c)Whichofthetwosamplingmethods,Method1orMethod2,willresultinlessvariabilityinthe

diametersofthe200tortillasinthesampleonagivenday?Explain.

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3. An apartment building has nine floors and each floor has four apartments. The building owner wants to install new carpeting in eight apartments to see how well it wears before she decides whether to replace the carpet in the entire building.

The figure below shows the floors of apartments in the building with their apartment numbers. Only the nine apartments indicated with an asterisk (*) have children in the apartment.

(a) For convenience, the apartment building owner wants to use a cluster sampling method, in which the floors are clusters, to select the eight apartments. Describe a process for randomly selecting eight different apartments using this method. (b) An alternative sampling method would be to select a stratified random sample of eight apartments, where the strata are apartments with children and apartments with no children. A stratified random sample of size eight might include two randomly selected apartments with children and six randomly selected apartments with no children. In the context of this situation, give one statistical advantage of selecting such a stratified sample as opposed to a cluster sample of eight apartments using the floors as clusters.

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ANSWERS to APExamPractice:Sampling 1.2004B#2--DiningHall

a)Studentswhoarriveearlymayhavedifferentopinionsaboutfoodqualitythanotherstudents(late-diners),thusproducingabiasedsample.Thisbiascanbeavoidedbytakingasimplerandomsampleofalldormresidents.

b)Thephraseaboutthedininghallfoodneedingimprovementmayleadmorestudentstorespondthatthefoodreallydoesneedimprovement.Also,thephraseindicatingincreasedcostmayalsoinfluencestudentstoanswerinsincerely.

Betterwording:“Doyoufeelthequalityoffoodneedsimprovement?” 2.2015#6(abconly) Inpart(a)theresponsesaysnoANDeitherarguesthatthesamplewillonlybeselectedfrom

oneproductionlineandnottheentirepopulation Inpart(b)theresponsechoosesMethod1ANDreferstoarelevantcharacteristicofthe

histogram(shape,center,orvariability).Inpart(c)theresponsechoosesMethod2ANDeitherjustifiesbystatingthatthesamplecomesfromonlyoneproductionline.

3.2011#3--SamplingApartments

a) Usearandomnumbergeneratortoselecttwodifferentrandomintegersfrom1to9.Thetwonumberscorrespondtothetwofloorsthatwillbesampled,foratotalofeightapartments.

b) Usingtheclustersamplingmethoddescribedabove,itwouldbepossibletoselectNOapartmentswithchildrenif,say,floors3and6werepicked.Sinceit’sreasonabletobelievethatcarpetsofapartmentswithchildrenwillhavemorewearthanapartmentswithoutchildren,aclustersamplemaynotprovidetheownerwithagoodrepresentativesampleofcarpetwearinthisbuilding.

Astratifiedsampleasdescribedwillallowtheownertoseehowwellthecarpetwearsinbothapartmentswithandwithoutchildren.

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2011#3:Apartments(StudentSamples)Student#1,Part(a):

Part(b):

Student#2:Part(a):

Part(b):

1a:Nocluster,nomethod(I);b:(E)2a:No“how”(P);b:weak,but(E)

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7GreatDesignProblems(fromtheAPStatisticsExam) NAME___________________________________1.AgriculturalexpertsaretryingtodevelopabirddeterrenttoreducecostlydamagetocropsintheUnitedStates.Anexperimentistobeconductedusinggarlicoiltostudyitseffectivenessasanontoxic,environmentallysafebirdrepellant.TheexperimentwilluseEuropeanstarlings,abirdspeciesthatcausesconsiderabledamageannuallytothecorncropintheUnitedStates.Foodgranulesmadefromcornaretobeinfusedwithgarlicoilineachoffiveconcentrationsofgarlic—0percent,2percent,10percent,25percent,and50percent.Theresearcherswilldeterminetheadversereactionofthebirdstotherepellantbymeasuringthenumberoffoodgranulesconsumedduringatwo-hourperiodfollowingovernightfooddeprivation.Therearefortybirdsavailablefortheexperiment,andtheresearcherswilluseeightbirdsforeachconcentrationofgarlic.Eachbirdwillbekeptinaseparatecageandprovidedwiththesamenumberoffoodgranules.

(a)Fortheexperiment,identify

i.thetreatmentsii.theexperimentalunitsiii.theresponsethatwillbemeasured

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2.Asdogsage,diminishedjointandhiphealthmayleadtojointpainandthusreduceadog’sactivitylevel.Suchareductioninactivitycanleadtootherhealthconcernssuchasweightgainandlethargyduetolackofexercise.Astudyistobeconductedtoseewhichoftwodietarysupplements,glucosamineorchondroitin,ismoreeffectiveinpromotingjointandhiphealthandreducingtheonsetofcanineosteoarthritis.Researcherswillrandomlyselectatotalof300dogsfromtendifferentlargeveterinarypracticesaroundthecountry.Allofthedogsaremorethan6yearsold,andtheirownershavegivenconsenttoparticipateinthestudy.Changesinjointandhiphealthwillbeevaluatedafter6monthsoftreatment.(a)Whatwouldbeanadvantagetoaddingacontrolgroupinthedesignofthisstudy?(b)Assumingacontrolgroupisaddedtotheothertwogroupsinthestudy,explainhowyou

wouldassignthe300dogstothesethreegroupsforacompletelyrandomizeddesign.(c)Ratherthanusingacompletelyrandomizeddesign,onegroupofresearchersproposes

blockingonclinics,andanothergroupofresearchersproposesblockingonbreedofdog.Howwouldyoudecidewhichoneofthesetwovariablestouseasablockingvariable?

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3. Studentsaredesigninganexperimenttocomparetheproductivityoftwovarietiesofdwarffruittrees.Thesitefortheexperimentisafieldthatisborderedbyadenselyforestedareaonthewestside.Thefieldhasbeendividedintoeightplotsofapproximatelythesamearea.Thestudentshavedecidedthatthetestplotsshouldbeblocked.Fourtrees,eachoftwovarieties,willbeassignedatrandomtothefourplotswithineachblock,withonetreeplantedineachplot.

Thetwoblockingschemesshownbelowareunderconsideration.Foreachscheme,oneblockisidentifiedbythewhiteregionandtheotherblockindicatedbythegreyregioninthefigures.

a. Whichoftheblockingschemes,AorB,isbetterforthisexperiment?Explain. b. Eventhoughthestudentshavedecidedtoblock,theymustrandomlyassignthevarietiesof

thetreestotheplotswithineachblock.Whatisthepurposeofthisrandomizationinthecontextoftheexperiment?

Forest

BlockingSchemeA

Forest

BlockingSchemeB

Key Block1Block2

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4.Thedentistsinadentalclinicwouldliketodetermineifthereisadifferencebetweenthenumberofnewcavitiesinpeoplewhoeatanappleadayandinpeoplewhoeatlessthanoneappleaweek.Theyaregoingtoconductastudywith50peopleineachgroup.Fiftyclinicspatientswhoreportthattheyroutinelyeatanappleadayand50clinicpatientswhoreportthattheyeatlessthanoneappleaweekwillbeidentified.Thedentistswillexaminethepatientsandtheirrecordstodeterminethenumberofnewcavitiesthepatientshavehadoverthepasttwoyears.Theywillthencomparethenumberofnewcavitiesinthetwogroups.

a.Whyisthisanobservationalstudyandnotanexperiment?b.Explaintheconceptofconfoundinginthecontextofthisstudy.Includeanexampleofapossibleconfoundingvariable.

c.Ifthemeannumberofnewcavitiesforthosewhoateanappleadaywasstatisticallysignificantlysmallerthanthemeannumberofnewcavitiesforthosewhoatelessthanoneappleaweek,couldoneconcludethatthelowernumberofnewcavitiescanbeattributedtoeatinganappleaday?Explain.

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5.Abiologistisinterestedinstudyingtheeffectofgrowth-enhancingnutrientsandiffererntsalinity(salt)levelsinwateronthegrowthofshrimps.Thebiologisthasorderedalargeshipmentofyoungtigershrimpsfromasupplyhouseforuseinthestudy.Theexperimentistobeconductedinalaboratorywhere10tigershrimpsareplacedrandomlyintoeachof12similartanksinacontrolledenvironment.Thebiologistisplanningtouse3differentgrowth-enhancingnutrients(A,B,andC)andtwodifferentsalinitylevels(lowandhigh).

(a)Listthetreatmentsthatthebiologistplanstouseinthisexperiment.(b)Usingthetreatmentslistedinpart(a),describeacompletelyrandomizeddesignthatwill

allowthebiologisttocomparetheshrimps’growthafter3weeks.(c)Giveonestatisticaladvantagetohavingonlytigershrimpsintheexperiment.Explainwhy

thisisanadvantage.(d)Giveonestatisticaldisadvantagetohavingonlytigershrimpsintheexperiment.Explainwhy

thisisadisadvantage.

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6.Amanufacturerofbootsplanstoconductanexperimenttocompareanewmethodofwaterproofingtothecurrentmethod.Theappearanceofthebootsisnotchangedbyeithermethod.Thecompanyrecruits100volunteersinSeattle,whereitrainsfrequently,towearthebootsastheynormallywouldfor6months.Attheendofthe6months,thebootswillbereturnedtothecompanytobeevaluatedforwaterdamage.

(a)Describeadesignforthisexperimentthatusesthe100volunteers.Includeafewsentences

onhowitwouldbeimplemented. (b)Couldyourdesignbedoubleblind?Explain

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7.

a)Whatwastheresponsevariableinthisstudy?

Identifythetreatments.

Whatweretheexperimentalunits?

b)Giventhatthegoalofthestudyistodeterminewhetheranewlydevelopedhitchreducesdraft

comparedtothestandardhitch,wasrandomizationusedproperlyinthisstudy?Justifyyouranswer.

c)Giventhatthegoalofthestudyistodeterminewhetheranewlydevelopedhitchreducesdraft

comparedtothestandardhitch,wasreplicationusedproperlyinthisstudy?Justifyyouranswer.

d)Plotoflandisaconfoundingvariableinthisexperiment.Explainwhy.

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SOLUTIONSto7GreatDesignProblems

1.BirdsandGarlicGranules(2010#1a)

2.Dogs’HipHealth(2007#2)

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PEP=2(Nocontextin(a);nocomparisonin(c))EEE=4(Complete)

EEP=3(“eleminateanyvariables”)

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3.FruitTrees(2001#4)

4.DentistsandApples(1999#3)

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5.Shrimp(2006#5)

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6.Boots(2002#2)

7.2006B,#5

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AP Exam Practice: Probability

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3. A shopping mall has three automated teller machines (ATMs). Because the machines receive heavy use, they sometimes stop working and need to be repaired. Let the random variable X represent the number of ATMs that are working when the mall opens on a randomly selected day. The table shows the probability distribution of X.

(a)WhatistheprobabilitythatatleastoneATMisworkingwhenthemallopens?(b)WhatistheexpectedvalueofthenumberofATMsthatareworkingwhenthemallopens?(c)WhatistheprobabilitythatallthreeATMsareworkingwhenthemallopens,giventhatat

leastoneATMisworking?

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3. There are four runners on the New High School team. The team is planning to participate in a race in which each runner runs a mile. The team time is the sum of the individual times for the four runners. Assume that the individual times for the four runners are all independent of each other. The individual times, in minutes, of the runners in similar races are approximately normally distributed with the following means and standard deviations.

a) Runner 3 thinks he can run a mile in less than 4.2 minutes in the next race. Is that likely to happen? Explain.

b) The distribution of possible team times is approximately normal. Find the mean and standard

deviation of this distribution. c) Suppose the team’s best time to date is 18.4 minutes. What is the probability that the team will

beat its own best time in the next race?

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3. Each full carton of Grade A eggs consists of 1 randomly selected empty cardboard container and 12 randomly selected eggs. The weights of such full cartons are approximately normally distributed with a mean of 840 grams and a standard deviation of 7.9 grams. (a) What is the probability that a randomly selected full carton of Grade A eggs will weigh more than

850 grams? (b) The weights of the empty cardboard containers have a mean of 20 grams and a standard deviation

of 1.7 grams. It is reasonable to assume independence between the weights of the empty cardboard containers and the weights of the eggs. It is also reasonable to assume independence among the weights of the 12 eggs that are randomly selected for a full carton.

Let the random variable X be the weight of a single randomly selected Grade A egg. i) What is the mean of X? ii) What is the standard deviation of X ?

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ANSWERS: AP Exam Practice: Probability 2016 #4: Model Rockets

2015 #3: ATM’s

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2002 #3 (Four Runners)

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2013 #3: Grade A Eggs

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AP Exam Practice: Sampling Distributions 2.Alocalradiostationplays40rock-and-rollsongsduringeach4-hourshow.Theprogramdirectoratthestationneedstoknowthetotalamountofairtimeforthe40songssothattimecanalsobeprogrammedduringtheshowfornewsandadvertisements.Thedistributionofthelengthsofrock-and-rollsongs,inminutes,isroughlysymmetricwithameanlengthof3.9minutesandastandarddeviationof1.1minutes.

(a)Describethesamplingdistributionofthesamplemeansonglengthsforrandomsamplesof40rock-

and-rollsongs.(b)Iftheprogrammanagerschedules80minutesofnewsandadvertisementsforthe4-hour(240-

minute)show,only160minutesareavailableformusic.Approximatelywhatistheprobabilitythatthetotalamountoftimeneededtoplay40randomlyselectedrock-and-rollsongsexceedstheavailableairtime?

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3.TrainscarrybauxiteorefromamineinCanadatoanaluminumprocessingplantinnorthernNewYorkstateinhoppercars.Fillingequipmentisusedtoloadoreintothehoppercars.Whenfunctioningproperly,theactualweightsoforeloadedintoeachcarbythefillingequipmentatthemineareapproximatelynormallydistributedwithameanof70tonsandastandarddeviationof0.9ton.Ifthemeanisgreaterthan70tons,theloadingmechanismisoverfilling.

(a)Ifthefillingequipmentisfunctioningproperly,whatistheprobabilitythattheweightofthe

oreinarandomlyselectedcarwillbe70.7tonsormore?Showyourwork.(b)Supposethattheweightoforeinarandomlyselectedcaris70.7tons.Wouldthatfactmake

yoususpectthattheloadingmechanismisoverfillingthecars?Justifyyouranswer.(c)Ifthefillingequipmentisfunctioningproperly,whatistheprobabilitythatarandomsample

of10carswillhaveameanoreof70.7tonsormore?Showyourwork.(d)Basedonyouranswerinpart(c),ifarandomsampleof10carshadameanoreweightof

70.7tons,wouldyoususpectthattheloadingmechanismwasoverfillingthecars?Justifyyouranswer.

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2. Four different statistics have been proposed as estimators of a population parameter. To investigate the behavior of these estimators, 500 random samples are selected from a known population and each statistic is calculated for each sample. The true value of the population parameter is 75. The graphs below show the distribution of values for each statistic.

(a) Which of the statistics appear to be unbiased estimators of the population parameter? How can you tell? (b) Which of statistics A or B would be a better estimator of the population parameter? Explain your choice.

(c) Which of statistics C or D would be a better estimator of the population parameter?

Explain your choice.

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ANSWERS: AP Statistics Exam Practice: Sampling Distributions 2010#2(RadioStation)

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2004B#3(TrainsCarryingBauxite)

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2009#2(TireManufacturer)

2008B#2(FourHistograms) a)A,CandDareunbiasedsincetheirmeansareall≈75.b)Asinceitisanunbiasedestimator(itsmeanis≈75).StatisticBisbiased(itsmeanis≈85)

c)Cisbetter.BothCandDareunbiased(centered≈75),butChaslessvariabilitythanD.

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AP Exam Practice: Inference 2006B #2

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2005 #4

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2004 #6:

Deeper question:

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2006 #4

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2008 #5

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2012 #5: Windmills

(a) Use the computer output above to determine the equation of the least squares regression line. Identify all variables used in the equation. (b) How much more electricity would the windmill be expected to produce on a day when the wind velocity is 25 mph than on a day when the wind velocity is 15 mph? Show how you arrived at your answer. (c) What proportion of the variation in electricity production is explained by its linear relationship with wind velocity? (d) Is there statistically convincing evidence that electricity production by the windmill is related to wind velocity? Explain.

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2015 #2 Restaurant Discounts 2. To increase business, the owner of a restaurant is running a promotion in which a customer’s bill can be

randomly selected to receive a discount. When a customer’s bill is printed, a program in the cash register randomly determines whether the customer will receive a discount on the bill. The program was written to generate a discount with a probability of 0.2, that is, giving 20 percent of the bills a discount in the long run. However, the owner is concerned that the program has a mistake that results in the program not generating the intended long-run proportion of 0.2.

The owner selected a random sample of bills and found that only 15 percent of them received discounts. A confidence interval for p, the proportion of bills that will receive a discount in the long run, is 0.15 ± 0.06. All conditions for inference were met.

(a) Consider the confidence interval 0.15 ± 0.06.

(i) Does the confidence interval provide convincing statistical evidence that the program is not working as intended? Justify your answer.

(ii) Does the confidence interval provide convincing statistical evidence that the program generates the

discount with a probability of 0.2 ? Justify your answer.

A second random sample of bills was taken that was four times the size of the original sample. In the second sample 15 percent of the bills received the discount.

(b) Determine the value of the margin of error based on the second sample of bills that would be used to compute an interval for p with the same confidence level as that of the original interval. (c) Based on the margin of error in part (b) that was obtained from the second sample, what do you conclude

about whether the program is working as intended? Justify your answer.