Using the formulas for the normal approximation, sample sizes can be derived in a manner similar to that employed.

**SAMPLE SIZE DETERMINATION—CONFIDENCE INTERVALS AND HYPOTHESIS TESTS**

Using the formulas for the normal approximation,
sample sizes can be derived in a manner similar to that employed in Chapters 8
and 9. Again, these calculations would be based on the width of the confidence
interval or the power of a test at a specific alternative. The resulting
formulas are slightly different from those for continuous variables. In the
case of variance in the test of a single proportion, or calculating a
confidence interval about a proportion, we guess at *p* to find the necessary standard deviation. We make this estimate
because *W* is *p*(1 – *p*)/*n*, and we do not know *p* (the population parameter). We also
can be conservative in deter-mining the confidence interval, because for all 0 ≤ *p* ≤ 1, *p*(1 –
*p*) is largest at *p* = 1/2.

Therefore, the variance of *W* = *p*(1 – *p*)/*n*
≤ (1/2)(1/2)/*n* = 1/(4*n*). This upper
bound, 1/(4*n*), on the variance of *W* can be used in the formulas to obtain
a mini-mum sample size that will satisfy the condition for any value of *p*. We could not find such a bound for
the unknown variance of a normal distribution.

Again, software packages such as the ones reviewed
by Chernick and Liu (2002) provide solutions for all the cases (using both
exact and approximate methods).

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