2.30 points total) A medical diagnostic test with 97% accuracy sounds impressive

Question

2.30 points total) A medical diagnostic test with 97% accuracy sounds impressive, but in fact, implementation of such a test can have many issues, particularly for a rare disease. You may find it helpful to consult Problem 2.25 of the OpenIntro Statistics text (note that answers to odd number problems are provided in the appendix of the text) Consider the following scenario. A medical test has been developed to detect a rare, but serious, disease for which the only treatment is difficult, with possible life threatening side effects. The disease itself occurs in only 1% of the population. If the disease is present then the test will return a positive result 97% of the time, (a quantity also referred to as sensitivity or the true positive rate). If no disease is present, then the test will return a negative result 97% of the time (a quantity also referred to as specificity or the true negative rate). In general, the sensitivity and specificity of an analytical method need not be the same, but here we are taking both to be equal to 97%, which in this case is also the accuracy of the test

Explanation / Answer

Ans:

Given that:

P(disease)=0.01

P(not desease)=1-0.01=0.99

P(positive/desaese)=0.97

P(negative/desease)=0.03

P(negative/not desease)=0.97

P(positive/not desease)=0.03

b)

P(desease/positive)=P(positive/desease)*P(desease)/P(positive)

Now,

P(positive)=P(positive/desease)*P(desease)+P(positive/not desease)*P(not desease)

=0.97*0.01+0.03*0.99

=0.0394

P(desease/positive)=(0.97*0.01)/0.0394

=0.2462

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