Question

12. Alzheimer's. Testing for Alzheimer's disease can be a

long and expensive process, consisting of lengthy tests
and medical diagnosis. Recently, a group of researchers
(Solomon et al., 1998) devised a 7-minute test to serve as
a quick screen for the disease for use in the general pop-
ulation of senior citizens. A patient who tested positive
would then go through the more expensive battery of
tests and medical diagnosis. The authors reported a false
positive rate of 4% and a false negative rate of 8%.
a) Put this in the context of a hypothesis test. What are
the null and alternative hypotheses?
b) What would a Type I error mean?
c) What would a Type II error mean?
d) Which is worse here, a Type I or Type II error? Explain.
e) What is the power of this test?

Answer 1

Below...

Step-by-step explanation:

a) In the context of a hypothesis test, the null hypothesis would be that the patient does not have Alzheimer's disease, while the alternative hypothesis would be that the patient does have Alzheimer's disease.

b) A Type I error would occur if the test incorrectly identified a patient as having Alzheimer's disease (a false positive), when in fact, the patient does not have the disease.

c) A Type II error would occur if the test incorrectly identified a patient as not having Alzheimer's disease (a false negative), when in fact, the patient does have the disease.

d) In this scenario, a Type II error would be worse because a false negative result could delay diagnosis and treatment of the disease, leading to a worse prognosis for the patient. A false positive result, while concerning, would lead to further testing and diagnosis to confirm the presence of the disease, rather than delaying treatment.

e) The power of a test is the probability of correctly rejecting the null hypothesis when it is false. In this case, the power of the test is 1 minus the false negative rate (8%), or 92%. This means that there is a 92% chance of correctly identifying a patient who has Alzheimer's disease as positive on the screening test.