Question

3. A diagnostic test has 95% sensitivity (the probability a person with the condition tests positive = 0.95) and 95% specificity (the probability a person without the condition tests negative = 0.95). In a population of people given the test, 1% of the people have the condition (probability a person has the condition = 0.01). (a) What proportion of the people will test positive? (b) Given a person has tested positive, what is the probability he/she has the condition?

Answer 1

Answer)

Given sensitivity = 0.95

And specificity = 0.95

Sensitivity = (true positive)/(true positive + false negative)

Specificity = (true negative)/(true negative + false positive)

It is given that 1% of the people have the condition

Lets say we have a sample size of 1000

Now 1% of 1000 have the condition

= 10

Now considering the sensitivity

Sensitivity = (true positive)/(true positive + false negative)

Now 10 has the condition

So, true positive + false negative must be equal to 10

Given sensitivity = 0.95

So, 0.95 = (true positive)/(10)

True positive = 9.5

So, false negative = 10-9.5 = 0.5

Now considering the specificity

Specificity = (true negative)/(true negative + false positive)

True negative + false positive = 1000-10 = 990

So, true negative = 990*0.95 = 940.5

And false positive = 49.5

A)

False positive = 49.5

True positive = 9.5

Total = 59

So, required proportion = 59/1000 = 0.059

B)

Total positive results = 59

And out of them 9.5 are true

So, required probability is 9.5/59 = 0.16101694915