Confidence Intervals for Proportions and Sample Size Determination

Lesson 36 of STA301 explores Confidence Intervals for Population Proportions and Sample Size Determination. In many scientific, social, and market research studies, the parameter of interest is the population proportion p possessing a specified qualitative characteristic. For a large sample (np >= 5 and nq >= 5), the sample proportion p-hat = x / n is approximately normally distributed. The 100*(1 - alpha)% Confidence Interval for a single population proportion p is given by: p-hat +- Z_(alpha/2) * sqrt[p-hat * q-hat / n], where q-hat = 1 - p-hat and the margin of error is E = Z_(alpha/2) * sqrt[p-hat * q-hat / n]. The lecture extends this formulation to comparative two-sample studies: estimating the difference between two independent population proportions (p1 - p2). When sample sizes n1 and n2 are large, the 100*(1 - alpha)% Confidence Interval for (p1 - p2) is: (p1-hat - p2-hat) +- Z_(alpha/2) * sqrt[(p1-hat * q1-hat / n1) + (p2-hat * q2-hat / n2)]. The second major segment of the lecture addresses Sample Size Determination, answering the practical research question: how large a sample must be drawn to estimate a parameter with specified precision? To estimate population mean mu with a maximum error bound E at confidence level 1 - alpha, the required sample size is derived from E = Z_(alpha/2) * (sigma / sqrt(n)) as: n = [Z_(alpha/2) * sigma / E]^2. To estimate population proportion p with margin of error E, the sample size formula is: n = [Z_(alpha/2)^2 * p * q] / E^2. If a preliminary estimate of p is available from past research, it is used; otherwise, the conservative worst-case value p = 0.5 (which maximizes the product p*q = 0.25) is deployed to guarantee that the sample size is sufficiently large. In Virtual University final term examinations, calculating required sample sizes for specified error bounds and constructing proportion confidence intervals are high-yield computational problems. Reviewing sta301 handouts and solved MCQs by Moaaz reinforces these techniques.