We learned that the power function is symmetric about the null hypothesis value and increases to 1 as we move far away from that value.

**SAMPLE SIZE DETERMINATION FOR HYPOTHESIS TESTS**

In Section 8.10, we showed you how to determine the
required sample size based on a criterion for confidence intervals, namely, to
require the half-width or width of the confidence interval to be less than a
specified δ. For hypothesis testing, one can also set up a criterion for sample
size. Recall from Section 9.8 that we defined and illustrated in a particular
example the power function for a two-sided test. We showed that if the level of
a two-sided test (such as for a population mean or mean difference) is α, then the power of the test at the null hypothesis value (e.g., μ_{0} for a population mean) is equal to α and increases as we move away from the null hypothesis value.

We learned that the power function is symmetric
about the null hypothesis value and increases to 1 as we move far away from
that value. We also saw that when the sample size is increased, the power
function increases rapidly. This information suggests that we could specify a
level of power (e.g., 90%) and a separation such that for a true mean satisfying
| μ – μ_{0}| > δ, the
power of the test at that value of μ is at
least 90%.

For a given δ, this
will not be achieved for small sample sizes; however, as the sample size
increases there will be eventually a minimum value *n* at which the power will exceed 90% for the given δ. Various software packages including nQuery Advisor, PASS 2000, and
Power and Precision enable you to calculate the required *n *or to determine the power
that can be achieved at that δ for a specified* n*.

In the Tendril DX clinical trial, Chernick and
associates calculated the difference between the treatment and control group
means using an unpaired *t* test; the
sample size was *n _{t}* = 3

**TABLE 9.3. nQuery Advisor 4.0 Table for 3:1 and 1:1 Sample Size Ratios
for Tendril DX Trial Design**

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