PROBABILITY & STATISTICS #5
QUESTIONS WITH CORRECT ANSWERS
Sam vvran vvan vvexperiment vvto vvtest vvoptimum vvpower vvand vvtime vvsettings vvfor
vvmicrowave vvpopcorn. vvHis vvgoal vvwas vvto vvdeliver vvpopcorn vvwith vvfewer vvthan vv8%
vvof vvthe vvkernels vvleft vvunpopped, vvon vvaverage. vvHe vvdetermined vvthat vvpower vv9
vvat vv4 vvminutes vvwas vvthe vvbest vvcombination. vvTo vvbe vvsure vvthat vvthe vvmethod
vvwas vvsuccessful, vvhe vvpopped vv8 vvmore vvbags vvof vvpopcorn vv(selected vvat
vvrandom) vvat vvthis vvsetting. vvAll vvwere vvof vvhigh vvquality, vvwith vvthe vvpercentages vvof
vvunpopped vvkernels vvshown vvbelow.
3.7, vv10.7, vv4.6, vv5.8, vv6.1, vv9.9, vv12.3, vv4. vv- vvAnswer vvHo: vvu vv= vv8
Ha: vvu vv<8
Claimed vvHypothesis vvMean, vvH0: vv8
Sample vvMean, vvx: vv7.25
Standard vvdeviation, vvσ:3.2293
Sample vvSize, vvn: vv8
t= vv-0.657
p-value vv= vv0.2661
Yes, vvthere vvis vvenough vvevidence vvsuggest vvthat vvless vvthan vv11% vvof vvthe vvkernels
vvare vvleft vvunpopped vvwhen vvthe vvspecified vvpower vvand vvtime vvsettings vvare vvused
Using vvt-tables, vvsoftware, vvor vva vvcalculator, vvestimate vvthe vvcritical vvvalue vvof vvt
vvfor vva vv99% vvconfidence vvinterval vvwith vvdf=24.Round vvto vvthree vvdecimal vvplaces
vvas vvneeded. vv- vvAnswer vv2.797
Using vvthe vvt-tables, vvsoftware, vvor vva vvcalculator, vvestimate vvthe vvcritical vvvalue vvof
vvt vvfor vvthe vvgiven vvconfidence vvinterval vvand vvdegrees vvof vvfreedom.80%
vvconfidence vvinterval vvwith vvdf vv= vv11 vv- vvAnswer vv1.363
The vvdistribution vvof vvscores vvon vva vvtest vvfor vva vvparticular vvclass vvis vvskewed vvto
vvthe vvright. vvThe vvprofessor vvwants vvto vvpredict vvthe vvmaximum vvscore vvand
vvunderstand vvthe vvdistribution vvof vvthe vvsample vvmaximum. vvShe vvsimulates vvthe
vvdistribution vvof vvthe vvmaximum vvof vvthe vvtest vvfor vv36 vvdifferent vvtests vv(with vvn vv=
vv5). vvThe vvhistogram vvto vvthe vvright vvshows vva vvsimulated vvsampling vvdistribution vvof
vvthe vvsample vvmaximum vvfrom vvthese vvtests. vv
a) vvWould vva vvNormal vvmodel vvbe vva vvuseful vvmodel vvfor vvthis vvsampling
vvdistribution? vvExplain
b) vvThe vvmean vvof vvthis vvdistribution vvis vv46.9 vvand vvthe vvSD vvis vv3.5. vvWould vvyou
vvexpect vvabout vv68% vvof vvthe vvsamples vvto vvhave vvtheir vvmaximums vvwithin vv3.5 vvof
, vv46.9? vvWhy vvor vvwhy vvnot? vv- vvAnswer vvA. vvNo. vvThe vvsampling vvdistribution vvof
vvthe vvmaximum vvis vvskewed vvto vvthe vvright, vvso vva vvNormal vvmodel vvwould vvnot vvbe
vvuseful vvfor vvthis vvsampling vvdistribution.
B. vvNo. vvThe vv68-95-99.7 vvRule vvis vvbased vvon vvthe vvNormal vvdistribution.
A vvwaiter vvbelieves vvthe vvdistribution vvof vvhis vvtips vvhas vva vvmodel vvthat vvis vvslightly
vvskewed vvto vvthe vvright, vvwith vva vvmean vvof vv$9.60 vvand vva vvstandard vvdeviation vvof
vv$5.40. vv
a) vvExplain vvwhy vvyou vvcannot vvdetermine vvthe vvprobability vvthat vva vvgiven vvparty
vvwill vvtip vvhim vvat vvleast vv$20. vvChoose vvthe vvcorrect vvanswer vvbelow.
b) vvCan vvyou vvestimate vvthe vvprobability vvthat vvthe vvnext vv4 vvparties vvwill vvtip vvan
vvaverage vvof vvat vvleast vv$15? vvExplain.
c) vvIs vvit vvlikely vvthat vvhis vv10 vvparties vvtoday vvwill vvtip vvan vvaverage vvof vvat vvleast
vv$15? vvExplain. vv- vvAnswer vvA. vvThis vvdistribution vvis vvskewed, vvmeaning vvit vvis
vvnon-symmetric vvand vvdoes vvnot vvmeet vvthe vvconditions vvfor vvthe vvNormal vvmodel,
vvso vvthe vvprobabilities vvof vvvalues vvwithin vvthis vvdistribution vvcannot vvbe
vvdetermined.
B. vvNo. vvA vvsample vvof vv4 vvparties vvis vvprobably vvnot vvlarge vvenough vvfor vvthe vvCLT
vvto vvallow vvthe vvuse vvof vva vvNormal vvmodel vvto vvestimate vvthe vvdistribution vvof
vvaverages.
C. vvWhile vva vvsample vvof vv10 vvparties vvmay vvnot vvbe vvlarge vvenough vvto vvuse vva
vvNormal vvmodel, vvit vvis vvlikely vvthat vvthe vvsample vvof vvaverages vvis vvstarting vvto
vvapproach vva vvNormal vvdistribution. vvThe vvstandard vvdeviation vvof vvthis vvdistribution
vvis vvabout vv$1.71, vvmeaning vvthat vvan vvaverage vvtip vvof vv$15 vvis vvmore vvthan vv3
vvstandard vvdeviations vvabove vvthe vvmean. vvEven vvif vvthe vvdistribution vvis vvslightly
vvskewed, vvthis vvis vvstill vvunlikely.
A vvstudy vvmeasured vvthe vvwaist vvsize vvof vv975 vvmen, vvfinding vva vvmean vvof vv36.24
vvinches vvand vva vvstandard vvdeviation vvof vv4.09 vvinches. vvA vvhistogram vvof vvthese
vvmeasurements vvis vvshown vvto vvthe vvright.
a) vvDescribe vvthe vvhistogram vvof vvthe vvwaist vvsizes.
b) vvTo vvexplore vvvariation vvof vvthe vvmean vvfrom vvsample vvto vvsample, vvthey
vvsimulated vvby vvdrawing vvmany vvsamples vvof vvsize vv2, vv5, vv10, vvand vv20 vvwith
vvreplacement, vvfrom vvthe vv975 vvmeasurements. vvThe vvhistograms vvfor vveach
vvsimulation vvare vvshown vvin vvthe vvaccompanying vvtable. vvExplain vvhow vvthese
vvhistograms vvdemonstrate vvwhat vvthe vvCentral vvLimit vvTheorem vvsays vvabout vvthe
vvsampling vvdistribution vvmodel vvfor vvsample vvmeans. vv- vvAnswer vvA. vvThe
vvhistogram vvis vvskewed vvto vvthe vvright.
B. vvThese vvsimulations vvappear vvto vvdemonstrate vvwhat vvthe vvCentral vvLimit
vvTheorem vvsays vvabout vvthe vvsampling vvdistribution vvmodel vvfor vvsample vvmeans.
vvAll vvof vvthe vvhistograms vvare vvcentered vvnear vv36 vvinches. vvAs vvn vvgets vvlarger,
vvthe vvhistograms vvapproach vvthe vvNormal vvshape, vv
and vvthe vvvariability vvin vvthe vvsample vvmeans vvdecreases. vv
The vvhistograms vvare vvfairly vvNormal vvby vvthe vvtime vvthe vvsample vvreaches vvsize vv5.
QUESTIONS WITH CORRECT ANSWERS
Sam vvran vvan vvexperiment vvto vvtest vvoptimum vvpower vvand vvtime vvsettings vvfor
vvmicrowave vvpopcorn. vvHis vvgoal vvwas vvto vvdeliver vvpopcorn vvwith vvfewer vvthan vv8%
vvof vvthe vvkernels vvleft vvunpopped, vvon vvaverage. vvHe vvdetermined vvthat vvpower vv9
vvat vv4 vvminutes vvwas vvthe vvbest vvcombination. vvTo vvbe vvsure vvthat vvthe vvmethod
vvwas vvsuccessful, vvhe vvpopped vv8 vvmore vvbags vvof vvpopcorn vv(selected vvat
vvrandom) vvat vvthis vvsetting. vvAll vvwere vvof vvhigh vvquality, vvwith vvthe vvpercentages vvof
vvunpopped vvkernels vvshown vvbelow.
3.7, vv10.7, vv4.6, vv5.8, vv6.1, vv9.9, vv12.3, vv4. vv- vvAnswer vvHo: vvu vv= vv8
Ha: vvu vv<8
Claimed vvHypothesis vvMean, vvH0: vv8
Sample vvMean, vvx: vv7.25
Standard vvdeviation, vvσ:3.2293
Sample vvSize, vvn: vv8
t= vv-0.657
p-value vv= vv0.2661
Yes, vvthere vvis vvenough vvevidence vvsuggest vvthat vvless vvthan vv11% vvof vvthe vvkernels
vvare vvleft vvunpopped vvwhen vvthe vvspecified vvpower vvand vvtime vvsettings vvare vvused
Using vvt-tables, vvsoftware, vvor vva vvcalculator, vvestimate vvthe vvcritical vvvalue vvof vvt
vvfor vva vv99% vvconfidence vvinterval vvwith vvdf=24.Round vvto vvthree vvdecimal vvplaces
vvas vvneeded. vv- vvAnswer vv2.797
Using vvthe vvt-tables, vvsoftware, vvor vva vvcalculator, vvestimate vvthe vvcritical vvvalue vvof
vvt vvfor vvthe vvgiven vvconfidence vvinterval vvand vvdegrees vvof vvfreedom.80%
vvconfidence vvinterval vvwith vvdf vv= vv11 vv- vvAnswer vv1.363
The vvdistribution vvof vvscores vvon vva vvtest vvfor vva vvparticular vvclass vvis vvskewed vvto
vvthe vvright. vvThe vvprofessor vvwants vvto vvpredict vvthe vvmaximum vvscore vvand
vvunderstand vvthe vvdistribution vvof vvthe vvsample vvmaximum. vvShe vvsimulates vvthe
vvdistribution vvof vvthe vvmaximum vvof vvthe vvtest vvfor vv36 vvdifferent vvtests vv(with vvn vv=
vv5). vvThe vvhistogram vvto vvthe vvright vvshows vva vvsimulated vvsampling vvdistribution vvof
vvthe vvsample vvmaximum vvfrom vvthese vvtests. vv
a) vvWould vva vvNormal vvmodel vvbe vva vvuseful vvmodel vvfor vvthis vvsampling
vvdistribution? vvExplain
b) vvThe vvmean vvof vvthis vvdistribution vvis vv46.9 vvand vvthe vvSD vvis vv3.5. vvWould vvyou
vvexpect vvabout vv68% vvof vvthe vvsamples vvto vvhave vvtheir vvmaximums vvwithin vv3.5 vvof
, vv46.9? vvWhy vvor vvwhy vvnot? vv- vvAnswer vvA. vvNo. vvThe vvsampling vvdistribution vvof
vvthe vvmaximum vvis vvskewed vvto vvthe vvright, vvso vva vvNormal vvmodel vvwould vvnot vvbe
vvuseful vvfor vvthis vvsampling vvdistribution.
B. vvNo. vvThe vv68-95-99.7 vvRule vvis vvbased vvon vvthe vvNormal vvdistribution.
A vvwaiter vvbelieves vvthe vvdistribution vvof vvhis vvtips vvhas vva vvmodel vvthat vvis vvslightly
vvskewed vvto vvthe vvright, vvwith vva vvmean vvof vv$9.60 vvand vva vvstandard vvdeviation vvof
vv$5.40. vv
a) vvExplain vvwhy vvyou vvcannot vvdetermine vvthe vvprobability vvthat vva vvgiven vvparty
vvwill vvtip vvhim vvat vvleast vv$20. vvChoose vvthe vvcorrect vvanswer vvbelow.
b) vvCan vvyou vvestimate vvthe vvprobability vvthat vvthe vvnext vv4 vvparties vvwill vvtip vvan
vvaverage vvof vvat vvleast vv$15? vvExplain.
c) vvIs vvit vvlikely vvthat vvhis vv10 vvparties vvtoday vvwill vvtip vvan vvaverage vvof vvat vvleast
vv$15? vvExplain. vv- vvAnswer vvA. vvThis vvdistribution vvis vvskewed, vvmeaning vvit vvis
vvnon-symmetric vvand vvdoes vvnot vvmeet vvthe vvconditions vvfor vvthe vvNormal vvmodel,
vvso vvthe vvprobabilities vvof vvvalues vvwithin vvthis vvdistribution vvcannot vvbe
vvdetermined.
B. vvNo. vvA vvsample vvof vv4 vvparties vvis vvprobably vvnot vvlarge vvenough vvfor vvthe vvCLT
vvto vvallow vvthe vvuse vvof vva vvNormal vvmodel vvto vvestimate vvthe vvdistribution vvof
vvaverages.
C. vvWhile vva vvsample vvof vv10 vvparties vvmay vvnot vvbe vvlarge vvenough vvto vvuse vva
vvNormal vvmodel, vvit vvis vvlikely vvthat vvthe vvsample vvof vvaverages vvis vvstarting vvto
vvapproach vva vvNormal vvdistribution. vvThe vvstandard vvdeviation vvof vvthis vvdistribution
vvis vvabout vv$1.71, vvmeaning vvthat vvan vvaverage vvtip vvof vv$15 vvis vvmore vvthan vv3
vvstandard vvdeviations vvabove vvthe vvmean. vvEven vvif vvthe vvdistribution vvis vvslightly
vvskewed, vvthis vvis vvstill vvunlikely.
A vvstudy vvmeasured vvthe vvwaist vvsize vvof vv975 vvmen, vvfinding vva vvmean vvof vv36.24
vvinches vvand vva vvstandard vvdeviation vvof vv4.09 vvinches. vvA vvhistogram vvof vvthese
vvmeasurements vvis vvshown vvto vvthe vvright.
a) vvDescribe vvthe vvhistogram vvof vvthe vvwaist vvsizes.
b) vvTo vvexplore vvvariation vvof vvthe vvmean vvfrom vvsample vvto vvsample, vvthey
vvsimulated vvby vvdrawing vvmany vvsamples vvof vvsize vv2, vv5, vv10, vvand vv20 vvwith
vvreplacement, vvfrom vvthe vv975 vvmeasurements. vvThe vvhistograms vvfor vveach
vvsimulation vvare vvshown vvin vvthe vvaccompanying vvtable. vvExplain vvhow vvthese
vvhistograms vvdemonstrate vvwhat vvthe vvCentral vvLimit vvTheorem vvsays vvabout vvthe
vvsampling vvdistribution vvmodel vvfor vvsample vvmeans. vv- vvAnswer vvA. vvThe
vvhistogram vvis vvskewed vvto vvthe vvright.
B. vvThese vvsimulations vvappear vvto vvdemonstrate vvwhat vvthe vvCentral vvLimit
vvTheorem vvsays vvabout vvthe vvsampling vvdistribution vvmodel vvfor vvsample vvmeans.
vvAll vvof vvthe vvhistograms vvare vvcentered vvnear vv36 vvinches. vvAs vvn vvgets vvlarger,
vvthe vvhistograms vvapproach vvthe vvNormal vvshape, vv
and vvthe vvvariability vvin vvthe vvsample vvmeans vvdecreases. vv
The vvhistograms vvare vvfairly vvNormal vvby vvthe vvtime vvthe vvsample vvreaches vvsize vv5.
