Sampling Distribution of Proportions and Difference of Means

Lesson 32 of STA301 explores the Sampling Distribution of the Sample Proportion p-hat and the Sampling Distribution of the Difference between Two Independent Sample Means (X1-bar - X2-bar). In survey research and qualitative data analysis, interest often centers on the proportion of elements possessing a specific attribute. If a random sample of size n is selected from a population with true parameter proportion p, the sample proportion is p-hat = X / n, where X is the number of successes. The expected value of the sample proportion equals the population proportion: E(p-hat) = mu_phat = p, confirming that p-hat is an unbiased estimator. The standard error of the sample proportion is given by sigma_phat = sqrt[p * q / n] for infinite populations or sampling with replacement, and sigma_phat = sqrt[p * q / n] * sqrt[(N - n) / (N - 1)] when sampling without replacement from finite populations. By the Central Limit Theorem, when np >= 5 and nq >= 5, the sampling distribution of p-hat is approximately normal, allowing Z-score computations via Z = (p-hat - p) / sigma_phat. The lecture then addresses comparative studies involving two independent populations. When independent random samples of sizes n1 and n2 are drawn from populations with means mu1, mu2 and variances sigma1^2, sigma2^2, the statistic of interest is the difference between sample means (X1-bar - X2-bar). The expected value of this difference equals the difference between population means: E(X1-bar - X2-bar) = mu1 - mu2. By the variance addition rule for independent variables, the standard error of the difference between means is: sigma_(x1 - x2) = sqrt[sigma1^2 / n1 + sigma2^2 / n2]. If both populations are normal, or if sample sizes are large (n1 >= 30, n2 >= 30), the sampling distribution of (X1-bar - X2-bar) follows a normal distribution: Z = [(X1-bar - X2-bar) - (mu1 - mu2)] / sigma_(x1 - x2). In Virtual University final term evaluations, two-sample standard error calculations and proportion Z-scores represent high-yield exam problems. Reviewing sta301 handouts and practicing solved MCQs by Moaaz reinforces these concepts.