Example8:
An IntelligenceQuotientor IQ is a scorederivedfrom differentstandardizedtestsattemptingto measure
the levelof intelligenceof an adulthumanbeing.The averagescoreof the test is and the standard
deviationis
1.Whatis the percentageof the populationthat has a scorebetween and?
2.Whatpercentageof the populationhas a scoreabove?
Solution:
- Usingthe normalprobabilitydensityfunction,
and substituting and
The percentageof the populationthat has a scorebetween and is
Again,the integralof doesnot havean elementaryanti-derivativeand thereforecannotbe evaluated.
Usingthe programingfeatureof a scientificcalculatoror a mathematicalcomputersoftware,we get
That is,
Whichsays that of the populationhas an IQ scorebetween and
- To measurethe probabilitythat a personselectedrandomlywill havean IQ scoreabove140,
This integralis evenmoredifficult to integratesinceit is an improperintegral.To avoidthe messywork,we
can arguethat sinceit is extremelyrare to meetsomeonewith an IQ scoreof over we can approximate
the integralfrom to Then