Abstract:
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 ($\alpha$, $\beta$).
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