Difference in Proportions and Student's t-Distribution

Lesson 39 of STA301 addresses two-sample proportion hypothesis testing and introduces the prominent small-sample distribution: Student's t-Distribution. In comparative proportion testing, the null hypothesis asserts that two population proportions are equal: H0: p1 = p2 (or p1 - p2 = 0). Under H0, both samples originate from populations with a common proportion p. Therefore, rather than keeping sample variances separate, sample successes are pooled: p-hat_pool = (x1 + x2) / (n1 + n2), and q-hat_pool = 1 - p-hat_pool. The pooled standard error is sqrt[p-hat_pool * q-hat_pool * (1/n1 + 1/n2)], yielding the Z-test statistic: Z = (p1-hat - p2-hat) / sqrt[p-hat_pool * q-hat_pool * (1/n1 + 1/n2)]. The second major segment of the lecture addresses the critical scenario where sample size is small (n < 30), population variance sigma^2 is unknown, and the population is normally distributed. Under these conditions, the sample standard deviation s exhibits substantial sampling variability, causing the ratio (X-bar - mu) / (s / sqrt(n)) to deviate from the standard normal curve. Discovered by William Sealy Gosset under the pseudonym 'Student' in 1908, this ratio follows Student's t-Distribution with degrees of freedom nu = n - 1: t = (X-bar - mu) / (s / sqrt(n)). The lecture details the mathematical properties of the t-distribution: it is continuous, bell-shaped, and symmetrical about t = 0; it has heavier tails and a lower central peak than the standard normal distribution (reflecting additional uncertainty from estimating sigma); its shape is uniquely determined by degrees of freedom nu; its variance is nu / (nu - 2) for nu > 2, which exceeds unity; and as degrees of freedom approach infinity (nu -> inf), the t-distribution converges asymptotically to the standard normal distribution Z. Students also learn to construct small-sample confidence intervals for mu: X-bar +- t_(alpha/2, n-1) * (s / sqrt(n)). In Virtual University final term examinations, pooled proportion tests and t-distribution properties appear consistently. Studying sta301 handouts and solved MCQs by Moaaz reinforces these concepts.