SajjadAhmad wrote:
Source: McGraw Hill GMAT
Any test that has a probability of success less than 100 hundred percent leaves a researcher open to various types of errors. In one type of error, a test with some probability of success less than 100 percent has a chance of missing the true result equal to 100 percent minus the test’s percentage chance of success.
Which of the following inferences is most strongly supported by the information above?
A. If a test has a chance of success less than 100 percent it should not be used by a researcher.
B. There are no tests that have a success rate of 100 percent.
C. Researchers should use tests susceptible to error only when no perfectly reliable test is available.
D. If a test with a 95 percent chance of success indicates a negative result, there is a 5 percent chance the actual result is positive.
E. If a test with a 95 percent chance of success indicates a positive result, there is a 5 percent chance the test should have indicated a negative result.
Neither of D and E can be inferred.Let's calculate D.
Pr[actual result +ve| test shows -ve] =
Pr[shows -ve| actually +ve]*\(\frac{Pr[actually +ve]}{Pr[shows -ve]}\) = 5%*\(\frac{Pr[actually +ve]}{Pr[shows -ve]}\)
Now, the denominator, Pr[shows -ve] = Pr[shows -ve| actually -ve]*Pr[actually -ve] + Pr[shows -ve|actually +ve]*Pr[actually +ve]
=> Pr[shows -ve] = 95%*Pr[actually -ve] + 5%*Pr[actually +ve] = 95%*(1-Pr[actually +ve]) + 5% Pr[actually +ve] = 95% - 90%*Pr[actually +ve]
Putting, Pr[actually +ve] = p
Original equation becomes, Pr[actual result +ve| test shows -ve] = 5%*\(\frac{p}{(0.95- 0.90*p)}\)
Which essentially means,
[1] Pr[actual result +ve| test shows -ve] = 5% when +ve and -ve outcomes are equally likely (This is option D)
[2] Pr[actual result +ve| test shows -ve] > 5% when +ve outcome is more likely
[3] Pr[actual result +ve| test shows -ve] < 5% when -ve outcome is more likely
Hence we
cannot infer D with confidence unless we know which outcome is more likely.
BTW, on a similar note,
we cannot say E either.
Pr[test shows -ve]
= Pr[shows -ve|actually -ve]*Pr[actually -ve] + Pr[shows -ve| actually +ve]*Pr[actually +ve]
= 0.95*(1-p) + 0.05*p = 0.95 - 0.90*p
Which means, Pr[test shows -ve] = 5% when only +ve outcome can occur. Otherwise the probability will be greater than 5%.