Let X₁,X₂,… be iid with mean μ and variance 0<σ²<∞

Let \(X_1,X_2,\ldots\) be iid with mean \(\mu\) and variance \(0<\sigma^2<\infty\). For \(n\ge2\), define \[ \bar X_n=\frac1n\sum_{i=1}^nX_i,\qquad S_n^2=\frac1{n-1}\sum_{i=1}^n(X_i-\bar X_n)^2. \] Using the law of large numbers, the central limit theorem, and Slutsky's theorem, prove \[ \frac{\sqrt n(\bar X_n-\mu)}{S_n}\Rightarrow N(0,1). \] Define this statistic as zero when \(S_n=0\), and justify why that convention does not affect the limit. State the corresponding asymptotic confidence interval for \(\mu\). For a finite-sample warning, let \(n=5\) and \(X_i\sim\operatorname{Bernoulli}(0.1)\). Show that the interval \[ \left[\bar X_n-1.96S_n/\sqrt n,\ \bar X_n+1.96S_n/\sqrt n\right] \] has coverage at most \(0.40951\). Is the asymptotic theorem contradicted?

Rewrite the sample variance using the first two raw sample moments.