Experiment design for comparison of two normal population parameters

I. N. Volodin

Teoriâ veroâtnostej i ee primeneniâ, Tome 18 (1973) no. 1, pp. 206-210

Résumé

Two normal population with parameters ($m_1$, $\sigma_1$) and ($m_2$, $\sigma_2$) are given, three pairs of alternative hypotheses being considered: 1) $H_0\colon m_1-m_2=0$, $H_1\colon m_1-m_2\ge\Delta$; 2) $H_0\colon m_1-m_2=0$, $H_1\colon|m_1-m_2|\ge\Delta$; 3) $H_0\colon\sigma_1^2/\sigma_2^2\le k$, $H_1\colon\sigma_1^2/\sigma_2^2\ge k(1+\Delta)$. Given error probabilities of the first ($\alpha$) and the second kind ($\beta$), two-step procedures are constructed for the first two pairs of hypotheses which enable to determine how many extra observations are needed for the given procedures to have the strength ($\alpha$, $\beta$), the initial ($n_0$, $N_0$) observations being available. These tests have been obtained as a result of applying Stein's procedure to the Bartlette-Scheffe and Student's test. For the third pair of hypotheses, an asymptotic formula is proposed for the number of observations necessary for Fisher's test to have a given strength ($\alpha$, $\beta$).