A school nurse examined a population of 1,000 children in an attempt to detect h

Question

A school nurse examined a population of 1,000 children in an attempt to detect hearing impairment. The prevalence of a hearing impairment in this population is known to be 20%. The sensitivity of the examination is 50% and its specificity is 80%. All children labeled as "positive" (i.e., suspected of having a hearing impairment) by the school nurse are sent for further examination by an audiologist. The sensitivity of the audiologist's examination is 98% and its specificity is 90%.

What is the negative predictive value (NPV) of the audiologist’s exam?

Explanation / Answer

probabilty of hearing impairment=P(H)=0.2

probability of +ve result when actually hearing impairment =P(+ve|H)=0.5

probability of -ve result when actually not hearing impairment =P(-ve|NH)=0.8

probabilty of +ve result=P(+ve)=P(H)*P(+ve|H)+P(NH)*P(+ve|NH) =0.2*0.5+0.8*0.2=0.26

hence probabilty of Hearing inpairment given test is +ve =P(H|+ve)=0.2*0.5/0.26=0.3846

and probabilty of not Hearing inpairment given test is +ve=P(NH|+ve)=0.8*0.2/0.26=0.6154

thereofre probabilty of -ve result =0.3846*0.02+0.6154*0.9=0.5614

hence P(NH|-ve)=0.6154*0.9/0.5614=0.9865

hence option D

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