Demonstrating the sampling distribution of the difference between two independent sample means ($\bar{X}_2 - \bar{X}_1$). Compare how Standard Error is computed using Welch's vs Student's t-test under different variance and sample size conditions.
Student's t-test assumes equal variances. It calculates a pooled variance ($s_p^2$) which is a weighted average of the two sample variances based on their degrees of freedom ($n-1$).
Student's standard error formula: $$\text{SE} = \sqrt{s_p^2 \left(\frac{1}{n_1} + \frac{1}{n_2}\right)}$$ where $$s_p^2 = \frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}$$
Notice that because we draw unequal sample sizes ($n_1=15, n_2=10$) and variances ($\sigma_1=10, \sigma_2=20$), the pooled variance leans heavily toward the smaller variance of the larger sample (Pop 1). This causes Student's SE to consistently underestimate the true error!