The Rapid Test is used to determine whether someone has HIV in the low risk segm

Question

The Rapid Test is used to determine whether someone has HIV in the low risk segment of the population. The test results in either a "+", indicating that the patient appears to have HIV (H) or "-", indicating that the patient appears to NOT have HIV (H^c). Clinical studies over the last 5 years have estimated the probability of a false positive to be 0.02 [ P(+/H^c) = 0.02 ] and false negative to be 0.06 [P(-/H) = 0.06]. The low risk segment of the population is known to have a 3% probability of having HIV. A randomly selected person is given the test, resulting in a "+" test results. Calculate the probability that this patient HAS HIV. Would you consider this a good test?

Explanation / Answer

P(False Positive) = 0.02

P(False negative) = 0.06

Let the sample size is 100. Following table shows various number.

(a) is total patient with positive condition.

(b) is total patient with negative condition.

(c) is total number of positive results shown by test.

(d) is total number of negative results shown by test.

(e) is total number of true positives.

(f) is total number of false negatives.

(g) is total number of false positives.

(h) is total number of true negatives.

We need to calculate P(Has HIV | Shown positive) = (e)/(c)

= (2.82)/(4.76)

= 0.59

We should not consider it as a good test because about 41% of the time it showing a person as HIV positive even though that person is not HIV positive.

Test Result Total '+' '-' Condition 4.76 (c) 95.24 (d) 100 '+' 3 (a) 2.82 (e) 0.18 (f) '-' 97 (b) 1.94 (g) 95.06 (h) Total 100

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