Imagine there is a fatal contagious disease spreading in country A. Based on pre
ID: 3221719 • Letter: I
Question
Imagine there is a fatal contagious disease spreading in country A. Based on previous clinic data, there were 1000 patients who received the diagnostic test for the disease -- 200 of the patients actually had the disease and died, the other 800 did not have the disease. In other words, in reality, the probability of having the disease is 20%, and the probability of not having the disease is 80%. Fortunately, advanced medical technology makes it possible to detect the disease. If the diagnosis is positive, that implies that the person has the disease. If the diagnosis is negative, that implies that the person does not have the disease. This medical diagnostic test, however, is not perfect and does make errors. Based on the 1000 patients whom we have data on, among the 200 people who actually had the disease, only 195 of them received a positive result according to the diagnostic. Among the 800 people who actually did not have the disease, 30 of them also received a positive result. Answer the following: a. Are the diagnostic decisions and reality statistically independent events? Present applicable conditional probability calculations and state your answer. b. What is the probability of getting a positive diagnostic result when, in fact, you do not have the disease? Would the diagnosis commit a Type I or Type II error in this case? Explain. c. What is the probability of getting a negative diagnostic result when, in fact, you have the disease? Would the diagnostic commit a Type I or Type II error in this case? Explain.Explanation / Answer
(a) Independence between diagnostic events can be understood considering the following probabilities-
P (P+|ND) and and P(P+ | D ) where P+ denotes the event where result comes positive
and D represents event when person actually has the disease and ND represent the events when person doesn't have the disease.
Now calculating them we get,
P (P+|ND) = 30/800 =.0375 and P(P+ | D ) = 195 / 200 = .975
If the events would be independent then the above two probabilities must be equal. But they are not equal then the diagnostic events are dependent.
(b) Required probability = P ( P+| ND) = 30/800 = .0375
Considering the null hypothesis is- Person doesn't have the disease then null hypothesis is true and we reject it saying the positive diagnostic result , it means we have type one error.
(c) Required probability = P ( P- | D) = 5/200 = .025
Considering the null hypothesis is- Person doesn't have the disease then null hypothesis is false and we do not reject it saying the negative diagnostic result , it means we have type two error.
TY!