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NAEP Technical DocumentationSampling of Private Schools for the 2008 Arts Assessment

When designing each school sample, there are five objectives that underlie the process of determining the probability of selection for each school and how many students are to be sampled from each selected school containing eighth-grade students:

• to meet the target student sample size;

• to select an equal-probability sample of students;

• to limit the number of students that are selected from a school;

• to ensure that the sample within a school does not include a very high percentage of the students in the school, unless all students are included; and

• to reduce the rate of sampling of small schools, in recognition of the greater cost and burden per student of conducting assessments in such schools.

The goal in determining the school's measure of size is to optimize across the last four objectives in terms of maintaining the accuracy of estimates and the cost effectiveness of the sample design.

Therefore, to meet the target student sample size objective and achieve a reasonable compromise among the other four objectives, the following algorithm was used to assign a measure of size to each school based on its eighth-grade enrollment as indicated on the sampling frame.

The measures of size vary by enrollment size. In general, very small schools were assigned a measure of size one quarter of the target sample size of 35 students. The initial measures of size (MOS) was set as follows:

where x_js is the estimated eighth-grade enrollment (j = 8) in school s, PSCHWT_S = the PSS area frame weight for school s, computed by the U.S. Census Bureau, and PSU_WT_S = the PSU weight for school s.

A school with 15 percent or more Black and Hispanic students in the eighth grade is considered in the high Black/Hispanic stratum for NAEP. The measures of size for schools in the high Black/Hispanic stratum are doubled to increase their chances of selection:

Also, note that the value of M_js (the preliminary measure of size) was doubled for all schools in the Honolulu, Hawaii PSU in order to ensure the inclusion of sample schools from Hawaii.

The next task in this development is to describe b_j, the constant of proportionality. It is a sampling parameter that, when multiplied with M_js, yields the school’s final measure of size. It is computed in such a way that, when used with the systematic sampling procedure, the target student sample size is achieved. For private schools, b_j is 0.000112322 for eighth grade.

The final measure of size, E_js, is defined as: 

The quantity u_j (the maximum number of times a school can be selected or “hits” allowed) in this formula is designed to put an upper bound on the burden for the sampled schools. For private schools, u_j is 1.

Schools were ordered within each jurisdiction using the serpentine sort described under the stratification of private schools. A systematic sample was then drawn using this serpentine sorted list and the measures of size. The numbers of private schools selected was approximately 60 for the eighth-grade arts assessment.

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