The Arizona Health Care Cost Containment System (AHCCCS) provides medical assistance to low-income households in Arizona. Each county determines whether households are eligible for assistance. Sometimes, however, households are certified to be eligible when they are really not eligible. The Arizona Statutes, section
36-2905.01, mandate the collection of a “statistically valid quality control sample of the eligibility certifications made by each county.” The certification error rate for each county is to be determined “by dividing the number of members in the sample who were erroneously certified by the total number of members in the sample.” Quality control audits are done by sampling household records, however; once a household record is selected and audited, it costs the same amount to evaluate one person in the household as to evaluate all persons in the household.
a. Explain how to use cluster sampling to estimate the certification error rate for a county.
b. Suppose that a county certified 1572 households to be eligible for medical assistance in 1995. In past years, the certification error rate per household has been about 10%. How many households should be included in your sample so that the half-width of a 95% CI for estimating the per-person certification error rate is less than 0.03? What assumptions did you need to make to arrive at your sample size? Calculate the sample size for different values of M and homogeneity.