Explain why the t-statistic helps

Imagine you fit a regression or run an A/B test, obtain an estimated effect size $\hat{\beta}$, and compute its standard error $SE(\hat{\beta})$. Explain why the t-statistic

$$t = \frac{\hat{\beta}}{SE(\hat{\beta})}$$

is frequently a helpful way to summarize evidence. What does it capture that the raw coefficient or mean difference alone does not? Discuss its connection to p-values and confidence intervals, the assumptions it requires, and the situations where relying on the t-statistic can mislead.

Overview: This question tests grasp of the t-statistic, standardized effect sizes, standard errors, hypothesis testing, and interpretation of statistical evidence in the Statistics & Math domain, and it emphasizes conceptual understanding with practical application implications.