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Homework answers / question archive / Diagnostic tests of medical conditions can have several types of results
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 + | 106 | 24 | 130 |
| Test Result − | 15 | 55 | 70 |
| Column Total | 121 | 79 | 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 −).
(a) P(+ | condition present)
106/121
(b) P(− | condition present)
15/121
(c) P(− | condition absent)
55/79
(d) P(+ | condition absent)
24/79
(e) P(condition present and +)
53/100
(f) P(condition present and −)
3/40
Step-by-step explanation
The rule for conditional probability for "probability of A given B" is P (A | B)
P (A | B) = P (A and B) / P(B)
(a) P(+ | condition present)
this will be
P (+ AND condition present) / P (condition present)
since we have a frequency chart we can do
N (+ AND condition present) / N (condition present)
from the chart we see
N (+ AND condition present) = 106 elements are positive and got the condition
N (condition present) = 121 , this is the subtotal of "condition present"
then
P(+ | condition present) = N (+ AND condition present) / N (condition present)
P(+ | condition present) = 106/ 121
b) P(− | condition present)
this will be
P (- AND condition present) / P (condition present)
since we have a frequency chart we can do
N (- AND condition present) / N (condition present)
from the chart we see
N (- AND condition present) = 15 elements are negative and got the condition
N (condition present) = 121 , this is the subtotal of "condition present"
then
P(- | condition present) = N (- AND condition present) / N (condition present)
P(- | condition present) = 15/121
c) P(− | condition absent)
this will be
P (- AND condition absent ) / P (condition absent )
since we have a frequency chart we can do
N (- AND condition absent) / N (condition absent)
from the chart we see
N (- AND condition absent ) = 55 elements are negative and did not get the condition
N (condition absent) = 79, this is the subtotal of "condition absent"
then
P(- | condition absent) = N (- AND condition absent) / N (condition present)
P(- | condition absent) = 55/79
d) P(+ | condition absent)
this will be
P (+ AND condition absent ) / P (condition absent )
since we have a frequency chart we can do
N (+ AND condition absent) / N (condition absent)
from the chart we see
N (+ AND condition absent ) = 24 elements are negative and did not get the condition
N (condition absent) = 79, this is the subtotal of "condition absent"
then
P(+ | condition absent) = N (+ AND condition absent) / N (condition present)
P(+ | condition absent) = 24/79
e) P(condition present and +)
use the formula
probability = #favorable outcomes / total outcomes
then
P(condition present and +) = #condition present and + / total outcomes
#condition present AND + according the chart those are 106
We see there are 200 outcomes
then
P(condition present and +) = 106/200
P(condition present and +) = 53/100
f) P(condition present and −)
use the formula
probability = #favorable outcomes / total outcomes
then
P(condition present and -) = #condition present and - / total outcomes
#condition present AND - according the chart those are 15
We see there are 200 outcomes
then
P(condition present and +) = 15/200
P(condition present and +) = 3/40
