Question

Diagnostic tests of medical conditions can have several types of results. The test result can be positive or negative, whether or not a patient has the condition. A positive test (+) indicates that the patient has the condition. A negative test (−) indicates that the patient does not have the condition. Remember, a positive test does not prove the patient has the condition. Additional medical work may be required. Consider a random sample of 200 patients, some of whom have a medical condition and some of whom do not. Results of a new diagnostic test for a condition are shown.

	Condition Present	Condition Absent	Row Total
Test Result +	100	30	130
Test Result −	15	55	70
Column Total	115	85	200

Assume the sample is representative of the entire population. For a person selected at random, compute the following probabilities. (Enter your answers as fractions.)

(a)    P(+ | condition present); this is known as the sensitivity of a test.


(b)    P(− | condition present); this is known as the false-negative rate.


(c)    P(− | condition absent); this is known as the specificity of a test.


(d)    P(+ | condition absent); this is known as the false-positive rate.


(e)    P(condition present and +); this is the predictive value of the test.


(f)    P(condition present and −).

Answer 1

(a)    P(+ | condition present); this is known as the sensitivity of a test.

Answer: =P(+ and condition present)/P(condition present)

=(100/200)/(115/200)=100/115

similarly

(b)    P(− | condition present); this is known as the false-negative rate.

=(15/200)/(115/200)=100/115

Answer: =15/115

(c) P(− | condition absent); this is known as the specificity of a test.

=(55/200)/(85/200)=55/85

Answer: =55/85

(d)    P(+ | condition absent); this is known as the false-positive rate.

=(30/200)/(85/200)=30/85

Answer: =30/85

(e)    P(condition present and +); this is the predictive value of the test.

Answer 100/200

(f)    P(condition present and −).

Answer: 15/200