Hypothesis Testing Using Student's t-Distribution

Lesson 40 of STA301 explores small-sample hypothesis testing using Student's t-Distribution for single population means and two independent population means. When testing a claim about a single population mean mu where the population is normally distributed, sample size is small (n < 30), and population variance sigma^2 is unknown, the One-Sample t-Test is employed. The test statistic is: t = (X-bar - mu0) / (s / sqrt(n)), following a t-distribution with nu = n - 1 degrees of freedom. Students apply the five-step protocol, looking up critical t-values from Student's t-distribution tables based on alpha and degrees of freedom to establish one-tailed or two-tailed rejection regions. The lecture extends testing to Two Independent Samples when sample sizes are small (n1 < 30, n2 < 30) and population variances are unknown but assumed equal (sigma1^2 = sigma2^2 = sigma^2, known as the assumption of homoscedasticity). Because both samples estimate the common variance sigma^2, their sample variances s1^2 and s2^2 are combined into the Pooled Sample Variance s_p^2, weighted by their respective degrees of freedom: s_p^2 = [(n1 - 1)*s1^2 + (n2 - 1)*s2^2] / [n1 + n2 - 2]. The two-independent-sample t-test statistic testing H0: mu1 - mu2 = d0 is: t = [(X1-bar - X2-bar) - d0] / [s_p * sqrt(1/n1 + 1/n2)], which follows a t-distribution with degrees of freedom nu = n1 + n2 - 2. The lecture emphasizes the underlying assumptions required for valid inference: both sampled populations must be normally distributed, the samples must be independently selected, and the population variances must be equal. In Virtual University final term evaluations, calculating pooled variance s_p^2 and executing independent two-sample t-tests are frequent subjective computational questions. Diligent practice with sta301 handouts and solved past papers guarantees comprehensive exam readiness. In preparing for Virtual University assessments in STA301 Statistics and Probability, students should thoroughly review past paper subjective questions and verify key formulas relating to Hypothesis Testing Using Student's t-Distribution.