Section 7.3 Summary

• The sample size required to obtain a confidence interval with an expected margin of error no larger than $m$ for a population mean satisfies the constraint

  $m\ge t^{*}{s}^{*}/\sqrt{n}$

  where $t^{*}$ is the critical value for the desired level of confidence with $n-1$ degrees of freedom, and $s^{*}$ is the guessed value for the population standard deviation.

• The sample sizes necessary for a two-sample confidence interval can be obtained using a similar constraint, but we would now need to guess both standard deviations and decide on an estimate for the degrees of freedom. We suggest using the smaller of $n_1-1$ and $n_2-1$.

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• The power of the one-sample t test can be calculated like that of the $z$ test, using an approximate value for both $\sigma$ and $s$.
• The power of the two-sample t test is found by first finding the critical value for the significance test, the degrees of freedom, and the noncentrality parameter for the alternative of interest. These are used to calculate the power from a non-central $t$ distribution. A Normal approximation works quite well. Calculating margins of error for various study designs and conditions is an alternative procedure for evaluating designs.
• The sign test is a distribution-free test because it uses probability calculations that are correct for a wide range of population distributions.
• The sign test for “no treatment effect” in matched pairs counts the number of positive differences. The $P$-value is computed from the $B\left(n,1/2\right)$ distribution, where $n$ is the number of non-0 differences. The sign test is less powerful than the $t$ test in cases where use of the $t$ test is justified.