An individual is considered to be obese if his or her BMI is at least 30 (Kuczmarski et al., 1997). Consider the population of NHANES III data given in Table 15.2 (discussed in Example 15.4) and let p denote the proportion of individuals in this population who are obese.
(a) Using buttocks circumference as the auxiliary variable to perform the judgment rankings and set size k = 5, select an unbalanced RSS (see the "Unbalanced Ranked Set Sampling" discussion in this section) of size n = 100, where the BMI value X[5] is obtained for the largest judgment ordered subject in half (50) of these sets and the BMI value X[4] is obtained for the second largest judgment ordered subject in the other half (50) of these sets. Find the sample percentage, 
high, of the individuals in this RSS who are obese.
(b) Using buttocks circumference as the auxiliary variable to perform the judgment rankings and set size k = 5, select an unbalanced RSS (see the "Unbalanced Ranked Set Sampling" discussion in this section) of size n = 100, where the BMI value X[1] is obtained for the smallest judgment ordered subject in half (50) of these sets and the BMI value X[2] is obtained for the second smallest judgment ordered subject in the other half (50) of these sets. Find the sample percentage, low, of the individuals in this RSS who are obese.
(c) Using buttocks circumference as the auxiliary variable to perform the judgment rankings and set size k = 5, select an unbalanced RSS (see the "Unbalanced Ranked Set Sampling" discussion in this section) of size n = 100, where the BMI value X[3] is obtained for the median judgment ordered subject in all 100 of these sets. Find the sample percentage, middle, of the individuals in this RSS who are obese.
(d) Compare the sample percentages obtained in parts (a)-(c) of this problem in conjunction with the true proportion of obese individuals in the NHANES III population. (See Chen, Stasny, and Wolfe (2006b) for more discussion about the use of unbalanced RSSs in the estimation of population proportions.)
Example 15.4
The NHANES III survey, 1988-1994, was conducted by the National Center for Health Statistics, Centers for Disease Control and Prevention. This survey was designed to obtain nationally representative information on the health and nutritional status of the population of the United States. The data set contains information for 33,994 persons aged 2 months and older who participated in the survey. Specifically, it contains various body measurements and information on other health-related variables for the respondents. The survey used a complex, multistage cluster sample of households. (Since we are going to treat a subset of the NHANES sample as our population for this example, we ignore the complex nature of the sample design.)