Question

According to the CDC, the prevalence of diabetes among US adults varies with education: 12.6% of those with less than high school education have diabetes, 9.5% of those with high school education have diabetes, and 7.2% of those with more than high school education have diabetes.

A7. In Ohio, 10% of adults have less than high school education, 33% have high school education, and 57% have more than high school education. If the diabetes prevalence in each group matches the CDC data, what is the overall prevalence of diabetes among adults in Ohio?

A8. If we randomly sample 100 Ohio adults, what is the distribution of the number of diabetes patients in the sample? Give the values of all distribution parameters. [Hint: Because sampling ignores educational level, you can use the overall prevalence of diabetes from Question A7.]

A9. What are the mean and variance of the number of diabetes patients in the sample from Question A8?

Answer 1

Let D, LH, H, MH denotes the events of diabetes, having less than high school education, having high school education and having more than high school education respectively.

Given,

P(D | LH) = 0.126

P(D | H) = 0.095

P(D | MH) = 0.072

A7.

Given,

P(LH) = 0.1

P(H) = 0.33

P(MH) = 0.57

By law of total probability,

P(D) = P(LH) P(D | LH) + P(H) P(D | H) + P(MH) P(D | MH)

= 0.1 * 0.126 + 0.33 * 0.095 + 0.57 * 0.072

= 0.085 = 8.5%

A8.

The distribution of the number of diabetes patients in the sample will be Binomial distribution with parameters p = 0.085 and n = 100.

A9.

Mean number of diabetes patients = np= 100 * 0.085 = 8.5

Variance of the number of diabetes patients= np(1-p) = 100 * 0.085 * (1 - 0.085) = 7.78