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Empirische Pädagogik

Zeitschrift zu Theorie und Praxis erziehungswissenschaftlicher Forschung

Zusammenfassung des Artikels: 2006, 20(3), 342-343

Willi Hager

On the importance of the t and the unit normal distribution in hypotheses testing. A comment on Wolf (2006)

The attempt to unify statistical sampling distributions such as z, t, c², and F into just one of these distributions, namely F, at first glance seems laudable. But at second glance, this attempt is not worthwhile. The type of hypotheses tested with the c² and with the F distributions are invariably bidirectional with respect to unsquared parameters such as the difference between two or more means m_k, relative frequencies/probabilities p and so on. But most of the statistical hypotheses encountered in educational psychology are directional (e.g., H₁: m₁ – m₂ > 0), because most of our psychological hypotheses to be examined are also directional, so that they must be examined by statistical tests that allow one-sided testing of directional hypotheses, that is by z and t tests, amongst others, where the t test may be replaced by the z test if N ³ 30; both tests are also applicable to ranked data and to testing (directional) hypotheses on probabilities p. Only when the psychological hypothesis is bidirectional, what rarely occurs (see Hager, 2004, 2005), it is adequate to examine it by tests that only test bidirectional statistical hypotheses (concerning unsquared parameters; e.g., H₁: m_k – m_k’ ¹ 0 for at least one pair k, k’ with k ¹ k’; squared version: H₁: S(m_k – m)² > 0), which is the case for the c² and F tests, where the latter may replace the former (cf. Wolf, 2006). But this can be done by a much easier way according to Bortz (2005, p. 83): F_(dfnum,¥) = c²/df_num (df_num: numerator degrees of freedom). This formula is less exact than Wolf’s formula, but it is exact enough for practical purposes.

Despite of what Wolf (2006) tells us, we need sampling distributions for tests that test directional hypotheses about mean differences, expected mean ranks, and probabilities, which may be the z test using the unit normal distribution as a sampling distribution, and in addition for some rather rare cases (and for global c² and F tests on unsquared parameters like m_k or p_jk) we need the sampling distributions of tests for bidirectional hypotheses, which may be the F distributions, which are closely related to the c² distributions, as Wolf (2006) shows as well as some textbooks on statistics. But both sampling distributions are so widely tabulated that it does not seem necessary to replace one by the other. – The c² and the F tests, by the way, test statistical hypotheses on squared parameters one-sidedly.

As usual, solving of statistical problems comes up to no good whenever the wider perspective of examining psychological hypotheses by statistical hypotheses is neglected.

Literature

Bortz, J. (2005). Statistik für Human- und Sozialwissenschaftler (6^th ed.). Berlin: Springer.

Hager, W. (2004). Testplanung zur statistischen Prüfung psychologischer Hypothesen. Göttingen: Hogrefe.

Hager, W. (2005). Vorgehensweisen in der deutschsprachigen psychologischen Forschung: Eine Analyse empirischer Arbeiten der Jahre 2001 und 2002. Psychologische Rundschau, 56, 191-200.

Wolf, B. (2006). Equivalents to z, t and c² within the F distribution. Empirische Pädagogik, 20, 203-206.

 Stand: 04.10.06