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Problem 1. (Mental Health) Estimates of the prevalence of Alzheimer’s disease have re-cently been provided by Pfeffer et al. The estimates are given in Table 3.5.
Table 3.5. Prevalence of Alzheimer’s disease (cases per 100 population)
Age group   Males Females
65-69            1.6       0.0
70-74            0.0       2.2
75-79            4.9       2.3
80-84            8.6       7.8
85+               35.0    27.9
Suppose an unrelated 77-year-old man and 88-year-old woman are selected from a com-munity. Please solve the following problems. This is not a JMP project. You need to showyour work.
To help to solve the problems, we may define the following events. Let A be the event that a 77-year-old man has Alzheimer’s disease, C be the event that a 88-year-old woman has Alzheimer’s disease.
1. What is the probability that at most one of the two selected individuals has Alzheimer’s disease?
2. What is the probability that exactly one of the two people has Alzheimer’s disease?
3. Suppose we know exactly one of the two people have Alzheimer’s disease.What is the conditional probability that the person is a woman?
4. Suppose the probability that both members of couple, each of whom is 88 years old, will have Alzheimer’s disease is 0.04. What is the conditional probability that the man will be affected given that the woman is affected? How does this value compare with the prevalence in Table 3.5?

Problem 2.The SEER Cancer Registry is an important resource for estimating cancer incidence rates and documenting changes in incidence rates over time.Cancer cases are accumulated over 17 cancer registries and are converted to cumulative incidence rates by dividing by U.S.Census population estimates within specific age groups. Suppose that the following data were obtained for breast cancer 1-year cumulative inci-dence rates for Caucasian women of age 50–74 from 1995 to 2000 (data is modified).
Age group                   1-year cumulative incidence per 100,000 women+
50–54                                           360
55–59                                           380
60–64                                           479
65–69                                           534
70–74                                           650
+For example, for 100,000 50–54 year-old women who are disease-free at baseline, 360 will develop breast cancer over a 1-year period.
1. Suppose that among women who are age 50–74, 22% are age 50–54, 21% are age 55–59, 20% are age 60–64, 19% are age 65–69 and 18% are age 70–74. What is the overall 1-year cumulative incidence among Caucasian women ages 50–74?
2. Suppose that a Caucasian woman ages 50–74 from 1995 to 2000 has breast cancer. What is the conditional probability that she is in the age group 50–54 years age group?

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1. Solution: ROW 2 + ROW 3 + ROW 4 = 0.9863
Solution, using formula :
P(A) = 4.9% and P(A)’=95.1%
P(A)’ is the event that a 77-year-old man has NOT Alzheimer’s disease
P(C) = 27.9% and P(C)’=72.1%...

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