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NAEP Analysis and Scaling → Drawing Inferences from NAEP Results → Comparison of Two Groups → t Test for Partly Overlapping Groups

NAEP Technical Documentationt Test for Partly Overlapping Groups

When a state is compared to the nation, there is an overlap between the samples in the sense that the state is part of the nation. These are referred to as part-whole comparisons and usually occur at two levels: a state or region versus the nation (e.g., Alabama versus the Nation) or an urban district versus the associated state, region, or nation (e.g., Houston versus Texas). Such comparisons require that the standard error of the mean differences be adjusted to account for the overlap.

Let A_i be the statistics in question (e.g., the mean for group i) and let S(A_i) be the standard error of the statistic. Furthermore, group i is the whole (e.g., Nation) and group j is the part (e.g., Alabama). Groups i and j are said to be significantly different if

where p is the proportion of students from group j within group i.

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