Final answer:
The sample size n for estimating proportions is derived using the error bound formula, incorporating desired margin of error, confidence level, and initial proportion estimates. By solving the formula n = (z² × p' × q') / E², researchers can calculate the necessary sample size for their study to achieve the specified precision.
Step-by-step explanation:
When researchers aim to estimate a population parameter with a certain degree of precision, they must calculate the sample size n based on the desired margin of error. To derive the formula for sample size using the relative error approach, we use the error bound formula, where E represents the margin of error, z represents the z-score corresponding to the desired confidence level, and p' and q' are the sample proportions, acting as estimates for the population proportions when these are unknown. The sample proportions are calculated using initial sample data or based on prior knowledge.
The general formula for determining the sample size n for estimating a proportion is given by:
Sample Size Formula
n = (z2 × p' × q') / E2
To solve for n, you square the z-score, multiply by the estimate of the population proportion p' and its complement q' = 1 - p', and then divide by the square of the margin of error E. If necessary, to accommodate a desired confidence level and maintain the specified error bound, the sample size may be increased.
According to this approach, if we increase the sample size n, we effectively decrease the margin of error, leading to a more precise estimate. In some cases, researchers add a continuity correction by increasing the sample size by 4 and adjusting the count of successes and failures accordingly.