It is estimated that approximately 8.46% Americans are afflicted with diabetes.
ID: 3306683 • Letter: I
Question
It is estimated that approximately 8.46% Americans are afflicted with diabetes. Suppose that a certain diagnostic evaluation for diabetes will correctly diagnose 95% of all adults over 40 with diabetes as having the disease and incorrectly diagnoses 4% of all adults over 40 without diabetes as having the disease.
a) Find the probability that a randomly selected adult over 40 does not have diabetes, and is diagnosed as having diabetes (such diagnoses are called "false positives").
b) Find the probability that a randomly selected adult of 40 is diagnosed as not having diabetes.
c) Find the probability that a randomly selected adult over 40 actually has diabetes, given that he/she is diagnosed as not having diabetes (such diagnoses are called "false negatives").
Explanation / Answer
The probability of all adults over 40 with diabetes as having the disease will correctly diagnose is P(A) = 0.95.
The probability of all adults over 40 with diabetes as having the disease will incorrectly diagnose is P(A') = 0.05.
The probability of all adults over 40 without diabetes as having the disease will correctly diagnose is P(B) = 0.96.
The probability of all adults over 40 without diabetes as having the disease will incorrectly diagnose is P(B') = 0.04.
a)
The probability that a randomly selected adult over 40 does not have diabetes, and is diagnosed as having diabetes (such diagnoses are called "false positives")
= P(A') / ( P(A') + P(B) ) = 0.05/(0.05+0.96) = 0..0495.
b)
The probability that a randomly selected adult of 40 is diagnosed as not having diabetes
= 1 - The probability of all adults over 40 without diabetes as having the disease will incorrectly diagnose
= 1 - 0.04 = 0.96
c)
The probability that a randomly selected adult over 40 actually has diabetes, given that he/she is diagnosed as not having diabetes (such diagnoses are called "false negatives").
= P(B') / ( P(A) + P(B') )
= 0.04 / ( 0.95 + 0.04)
= 0.040
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