Well, I *am* a statistician…

To quote Henkle, Wiersma, & Jurs; Applied Statistics for the behavioral Sciences, p. 312

This means that the sample size is dependent on what you want to find. For example, if I want to see if a specific reading program increases test scores, I might need a different sample size than if I want to see if a specific drug has negative side effects. If what I want to find is hard to see, I need a bigger sample size than if the effect is easy to see.

The sample size is also dependent on the test/hypothesis tested. A one sample T-test has a different sample size equation for the same effect size than a 6 cell Chi squared test. Also, if the T-test is one tailed or two tailed the calculations are different (I think there is a difference between the two groups = two tailed test vs. I think that group B will be higher = one tailed test.)

In general, **Karl**’s equation is close. The equations all have the population error variance (sigma squared) and the squared difference between the tested and critical z scores in the numerator and the squared effect size in the denominator. Standardizing the effect size removes the population error variance. The Z value divided by 2 in **Karl**’s equation is only used for two tailed tests of H[sub]0[/sub].

The equation for the two-sample case in a one-tailed test of the hypothesis is as follows (provided that my coding and the {sym} code works…

```
2[sym]s[/sym][sup]2[/sup](Z[sub][sym]b[/sym][/sub] - Z[sub][sym]a[/sym][/sub])[sup]2[/sup]
n = --------------
(effect size)[sup]2[/sup]
```

where [sym]s[/sym][sup]2[/sup] is the population error variance,

Z[sub][sym]b[/sym][/sub] is the standard score in the sampling distribution with H[sub]a[/sub] corresponding to z[sub]a[/sub] for a given power (from look-up tables),

Z[sub][sym]a[/sym][/sub] is the critical value of the test statistic in the sampling distribution associated with H[sub]0[/sub] for a one tailed test at t given [sym]a[/sym], and ES is the effect size, as determined by the study.

(Definitions from same text.)

I chose the two-sample case for a one-tailed test because it is most common. If I wish to determine if Method A is better than Method B, I’m only looking for one outcome; I’m not too concerned about Method B being better than Method A. For example, I want to determine if taking Zinc ameliorates cold symptoms faster than not. I have one desired proposed outcome; I don’t suspect that Zinc will make the cold symptoms WORSE than doing nothing at all. This is a one tailed test.

To carry out this study, I’ll need two groups. One group will take the Zinc and one will take nothing. Hence, I have a two-sample, one-tailed test.

(In the real world, researchers would use multiple groups and a double blind methodology.)

Now, does this make any sense whatsoever?