Question

In Exercises 17–20, refer to the accompanying table showing results from a Chembio test for hepatitis C among HIV-infected patients (based on data from a variety of sources).

	Positive Test Result	Negative Test Result
Hepatitis C	335	10
No Hepatitis C	2	1153

Positive Predictive Value Find the positive predictive value for the test. That is, find the probability that a subject has hepatitis C, given that the test yields a positive result. Does the result make the test appear to be effective?

Short answer

The probability of selecting a subject who has Hepatitis C, given that the test is positive, is equal to 0.9941.

Since the probability is high, it can be concluded that the test is highly effective in diagnosing Hepatitis C.

Step-by-step solution

Given information

The data shows the number of people who have and do not have Hepatitis C.

Further, the number of subjects who tested positive and negative is also given.

Define conditional probability

The following is the expression to compute the conditional probability of B given A.

$P\left(B|A\right)=\frac{P\left(AandB\right)}{P\left(A\right)}$

It defines the occurrence of B with reference to a prior event A.

Computation for conditional probability

Let A be the event of selecting a subject who has Hepatitis C.

Let B be the event of selecting a subject who does not have Hepatitis C.

Let C be the event of selecting a subject who has a positive test result.

Let D be the event of selecting a subject who has a negative test result.

The following table consists of the total frequency under each category:

	Positive Test Result	Negative Test Result	Totals
Hepatitis C	335	10	345
No Hepatitis C	2	1153	1155
Totals	337	1163	1500

The total number of subjects is equal to 1500.

The number of subjects who tested positive is equal to 337.

The probability of selecting a subject who tested positive is given as follows:

$P\left(C\right)=\frac{337}{1500}$

The number of subjects who have Hepatitis C and tested positive is equal to 335.

The probability of selecting a subject who tested positive and has Hepatitis C is given by:

$P\left(AandC\right)=\frac{335}{1500}$

The probability of selecting a subject who has Hepatitis C, given that he/she tested positive, is calculated as follows:

$$\begin{gathered} P\left(A|C\right)=\frac{P\left(AandC\right)}{P\left(C\right)} \\ =\frac{\frac{335}{1500}}{\frac{337}{1500}} \\ =\frac{335}{337} \\ =0.9941 \end{gathered}$$

Therefore, the probability of selecting a subject who has Hepatitis C, given that he/she tested positive, is equal to 0.9941.

Interpret the effectiveness of the result 

The event of having Hepatitis C, given that the test result is positive, indicates the effectiveness of the test. As the probability value is approximately equal to 1, it can be said that the test is highly effective in detecting the disease.