Abstract (18.0: The reliability in psychological test)

Abstract: In this chapter, we will introduce the measurement theoretical approach to a problem of analyzing scores of tests for students. The obtained score is assumed to be the sum of a true value and a measurement error caused by the test, in which a student's score is subject to a systematic error (=noise) depending on his/her health or psychological condition at the test. In such cases, statistical measurements are convenient since these two errors (i.e., measurement error and systematic error) in measurement theory can be characterized in the different mathematical structures respectively. The goodness of a test is characterized by "reliability coefficient" such that \begin{align*} \mbox{ [reliability coefficient] }^2 = \frac{\mbox{[systematic error]}^2}{ \mbox{[systematic error]}^2+\mbox{[measurement error]}^2 } \end{align*} Now we have the following problem:

How do we calculate the "reliability coefficient"?
This will be answered in this chapter.

This chapter is extracted from the following.
 $(\sharp):$ K. Kikuchi, S. Ishikawa, "Psychological tests in Measurement Theory," Far east journal of theoretical statistics,} 32(1) 81-99, (2010)

Again recall that, as mentioned in $\S$1.1, the main purpose of this book is to assert the following figure 1.1:
Fig.1.1: the location of "quantum language" in the world-views
This(particularly, ⑦--⑨) implies that quantum language has the following three aspects: $$\left\{\begin{array}{ll} \mbox{ ⑦ :the standard interpretation of quantum mechanics} \\ \mbox{ \qquad (i.e., the true colors of the Copenhagen interpretation) } \\ \\ \mbox{ ⑧ : the final goal of the dualistic idealism (Descartes=Kant philosophy) } \\ \\ \mbox{ ⑨ : theoretical statistics of the future } \end{array}\right.$$