Sample Size Determination-Confidence Intervals and Hypothesis Tests

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/(4n). This upper bound, 1/(4n), 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).