Master Harvard Business School Business Analytics with this 2025 study guide. Covers data analysis, statistical modeling, regression, optimization, forecasting, and data-driven decision-making for MBA students.

Business Analytics HBS with correct
answers 2025


A/Bvtestv-vcorrect-answersv-
Anvexperimentvthatvcomparesvthevvaluevofvavspecifiedvdependentvvariablev(suchvasvthevlikelihoodv
thatvavwebvsitevvisitorvpurchasesvanvitem)vacrossvtwovdifferentvgroupsv(usuallyvavcontrolvgroupvand
vavtreatmentvgroup).vThevmembersvofveachvgroupvmustvbevrandomlyvselectedvtovensurevthatvthevo


nlyvdifferencevbetweenvthevgroupsvisvthev"manipulated"vindependentvvariablev(forvexample,vthevsi
zevofvthevfontvonvtwovotherwise-
identicalvwebvsites).vAnvA/Bvtestvisvavhypothesisvtestvthatvtestsvwhethervthevmeansvofvthevdepende
ntvvariablevarevthevsamevacrossvthevtwovgroups.v(AnvA/Bvtestvcanvalsovbevusedvtovtestvwhethervano
thervparameter,vsuchvavstandardvdeviation,visvthevsamevacrossvtwovgroups.)



adjustedvR-squaredv-vcorrect-answersv-
Avmeasurevofvthevexplanatoryvpowervofvavregressionvanalysis.vAdjustedvR-squaredvisvequalvtovR-
squaredvmultipliedvbyvanvadjustmentvfactorvthatvdecreasesvslightlyvasveachvindependentvvariablevi
svaddedvtovavregressionvmodel.vUnlikevR-
squared,vwhichvcanvnevervdecreasevwhenvavnewvindependentvvariablevisvaddedvtovavregressionvm
odel,vAdjustedvR-
squaredvdropsvwhenvanvindependentvvariablevisvaddedvthatvdoesvnotvimprovevthevmodel'svtruevex
planatoryvpower.vAdjustedvR2vshouldvalwaysvbevusedvwhenvcomparingvthevexplanatoryvpowervofvr
egressionvmodelsvthatvhavevdifferentvnumbersvofvindependentvvariables.

,alternativevhypothesisv-vcorrect-answersv-
Anvalternativevhypothesisvisvthevtheoryvorvclaimvwevarevtryingvtovsubstantiate,vandvisvstatedvasvthev
oppositevofvavnullvhypothesis.vWhenvourvdatavallowvusvtovnullifyvthevnullvhypothesis,vwevsubstantia
tevthevalternativevhypothesis.



asymmetricvdistributionv-vcorrect-answersv-
Avprobabilityvdistributionvthatvisvnotvsymmetricvaroundvthevmean.



averagev-vcorrect-answersv-
Thevmostvcommonvstatisticvusedvtovdescribevthevcentervofvthevvaluesvinvavdatavset.vThevmeanvisvals
ovknownvasvthevaverage.vForvavdistributionvthatvhasvdiscretevvalues,vthevmeanvisvequalvtovsumvofvth
evvaluesvofvallvthevdatavpointsvinvthevset,vdividedvbyvthevnumbervofvdatavpoints.



basevcasev-vcorrect-answersv-
ThevcategoryvofvavcategoricalvvariablevforvwhichvavdummyvvariablevisvNOTvincludedvinvavregressionv
model.vAvregressionvmodelvwithvavcategoricalvvariablevthatvhasvnvcategoriesvshouldvhavevn-
1vdummyvvariables.vThevcoefficientsvofvthevdummyvvariablesvincludedvinvthevregressionvmodelvarev
interpretedvinvrelationvtovthevbasevcase.vThevanalystvcanvselectvanyvcategoryvtovbevexcludedvfromvt
hevregressionvmodel;vhowever,vdifferentvbasevcasesvleadvtovdifferentvinterpretationsvofvthevdumm
yvvariables'vcoefficients.vForvexample,vsupposevwevarevtryingvtovdeterminevthevaveragevdifferencev
invheightvbetweenvmenvandvwomenvinvavsample,vandvsupposevthatvonvaveragevmenvarev5vinchesvta
llervthanvwomenvinvthevsample.vIfvwevusevFemalevasvthevbasevcasevthenvthevcoefficientvforvthevdum
myvvariablevforvMalevwouldvbev+5.vIfvwevusevMalevasvthevbasevcase,vthevcoefficientvforvthevdummyv
variablevforvFemalevwouldvbev-5.



biasv-vcorrect-answersv-Thevtendencyvofvavmeasurementvprocessvtovover-vorvunder-
estimatevthevvaluevofvavpopulationvparameter.vAlthoughvavsamplevstatisticvwillvalmostvalwaysvdiffe
rvfromvthevpopulationvparameter,vforvanvunbiasedvsample,vthevdifferencevwillvbevrandom.vInvcontr
ast,vforvavbiasedvsample,vthevstatisticvwillvdiffervinvavsystematicvwayv(e.g.,vtendvtovbevtoovhigh).vSom
evcommonvreasonsvforvbiasvincludevnon-randomvsamplingvmethodsvandvnon-
neutralvquestionvphrasing.

, biasedvsamplev-vcorrect-answersv-
Avsamplevthatvisvnotvrepresentativevofvthevpopulationvfromvwhichvitvisvcollected.vSamplingvpractice
svthatvcanvintroducevbiasvincludevpoorlyvphrasedvsurveyvquestionsvandvnon-randomvsampling.



bimodalvdistributionv-vcorrect-answersv-Avmulti-
modalvdistributionvwithvtwovclearlyvdiscernablevpeaks.vThevtwovpeaksvmayvbevofvthevsamevheightv(
thatvis,vhavevequalvfrequency),vorvonevmayvbevthevtruevmodevwhilevthevothervhasvavveryvhighv(butv
notvthevhighest)vfrequency.



binv-vcorrect-answersv-
Avrangevofvvaluesvusedvtovcategorizevdata.vInvavhistogram,vobservationsvarevdividedvintovavsetvofvno
n-
overlappingvbins,veachvcorrespondingvtovavrangevofvvalues.vThevbinsvarevconstructedvtovensurevtha
tvthevsetvofvbinsvcontainsvallvobservationsvinvthevdatavset.vThevheightvofvthevbarvcorrespondingvtovav
binvisvequalvtovthevnumbervofvobservationsvinvthevdatavsetvthatvfallvwithinvthatvbin'svrange.vTypicall
y,vallvbinsvinvavgivenvhistogramvarevthevsamevwidthv(i.e.,vthevdifferencevbetweenvthevlargestvvalueva
ndvthevsmallestvvaluevisvthevsamevforveachvbin).vInvanvExcelvhistogram,veachvbinvisvlabeledvbyvthevva
luevofvthevuppervboundaryvofvthevbin'svrange.vForvexample,vinvavhistogramvwithvthreevbinsv(eachvof
vwidthv1),vlabeledv1,v2,vandv3,vthevbinvlabeledv2vcontainsvallvobservationsvgreatervthanv1vandvlessvth


anvorvequalvtov2.vSeevhistogram.



binomialvdistributionv-vcorrect-answersv-
Avdistributionvofvthevpossiblevsuccessfulvoutcomesvinvavgivenvnumbervofvtrials,vwherevtherevarevonl
yvtwovpossiblevoutcomesvforveachvtrial,vandveachvtrialvhasvthevsamevprobabilityvofvsuccessv(e.g.,vfli
ppingvavcoin).vForvexample,vthevbinomialvdistributionvforvthevnumbervofv"heads"vthatvresultvfromvfl
ippingvavcoinv50vtimesvspecifiesvthevprobabilityvforveachvpossiblevoutcome,vfromvobservingv0v"hea
ds"vtovobservingv50v"heads".vThevbinomialvdistributionvisvusedvtovcreatevconfidencevintervalsvforvp
roportions.



CentralvLimitvTheoremv-vcorrect-answersv-
Avtheoremvstatingvthatvifvwevtakevsufficientlyvlargevrandomly-
selectedvsamplesvfromvavpopulation,vthevmeansvofvthesevsamplesvwillvbevnormallyvdistributedvreg
ardlessvofvthevshapevofvthevunderlyingvpopulation.v(Technically,vthevunderlyingvpopulationvmustvh
avevavfinitevvariance.)