Question

A test to diagnose heart disease is correct 99% of the time if the patient has the disease, and 97% correct in its diagnosis if the patient does not have the disease. Only 2% of the population has this heart disease.

a).If a patient is randomly selected from the population to perform the test, what is the probability that the disease will be diagnosed? What do you think of that result?
b). If the disease is diagnostic, what is the probability that they actually have the disease? What do you think of the result?
c). What is the hypothesis linked to this test?
d). What is the significance level of the heart disease test? That is, what is the probability of making a Type I error?
and.

Answer 1

So

We can estimate the probability based on the given information:

p(person having disease) = 0.02

p(person having no disease) = 0.98

p(diagonised disease| having disease) = 0.99

p(not diagonised disease| having disease) = 0.01

p(diagonised disease| having not disease) = 0.03

p(not diagonised disease| having not disease) = 0.97

Part A:

Choosing the person randomly is and diagonsis is the independant test which consist the following eventes

= p(diagonised disease| having disease)*p(person having disease) + p(diagonised disease| having not disease)*p(person having no disease)

= 0.99*0.02 + 0.03*0.98

= 0.0198 + 0.0294

= 0.492

For randomly selected person, 0.492 probability that person will be diagonised with diseases.

Answer B:

If we have 1000 person, we will have 20 person will having disease and 80 will not have diseases.

Out of 20, test will be positive to 19.8 (20*0.99) and 0.02 (20*0.01) will be having negative test despite having disease.

and out of 80 person whom don't have any disease, test will show positive to 2.4 (80*0.03) and 77.6 will be having negative test.

So total positive test would be 22.2 (19.8 + 2.4)

so out of 22.2 positive diagonsed, 19.8 will have the disease so probability of positive diagonsed person having disease will be = 19.8 /22.2= 0.8919

So if person got positive test 0.8919 probability that he will have diseases.

Part D:

So type I error is the false positve error.

So if the result shows that person is having disease and test shows negative impact it is type I error.

So p(not diagonised disease| having disease) = 0.01

So given hyptothesis proves false but actually the hypothesis should be true.

so probability of Type I error is 0.01