Why the CLT cannot be used for tail bounds

NumericalAdvanced⏱ 30 minProbabilitycltberry-esseenhoeffdingtails

Problem

Let $X_1,\dots,X_n$ be i.i.d. uniform on $\{0,1\}$ , $n = 100$ , and consider $\P(\bar X_n \ge 0.7)$ .

Compute the exact value from the binomial distribution.
Compute the CLT approximation.
Compute Hoeffding's upper bound.
Compute the Berry–Esseen error bound and compare it to the quantity being estimated.
State precisely which of (2) and (3) is a valid bound, and why.

Compute the requested quantities and say what they mean. A number without an interpretation is not an answer.

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Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

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