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Kieser and Friede (2007) revisited the performance of Tang and Tang's [ 14] asymptotic test statistics via simulation studies and derived approximate sample size formulae for achieving the desired power.
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The percentage of HRMax to obtain HRAT was 80.9% for achieved HRMax, 75.5% for HRMax estimated by formula "208 - 0.7 × age" and 76.2% for the formula "220 - age".
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Their proof was based on a neat formula for the payoffs achieved if both players use memory-one strategies.
where L ij is the (i, j th entry of the matrix L. Since the condition ∂C/∂p| p = 0 ≤ 0 is satisfied, the maximum value of (16) is less than or equal to 0. When β2 = 1, β j = 0 for j > 2, the formula (21) achieves the maximum value and so does (16).
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com