Explain Bootstrap and Prove Uniformity

Q: Explain Bootstrap and Prove Uniformity

This question evaluates mastery of statistical inference and probability theory, testing knowledge of resampling methods (bootstrap), interpretation of confidence intervals and p-values including significance level selection, and rigorous proof techniques for establishing uniform distribution modulo 1.

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You are interviewing for a Data Scientist role. Answer the following statistics-theory prompts:

What is the bootstrap, when would you use it, and can it be used to reduce variance?
What is a confidence interval, what is a p-value, and how would you explain both to a product manager? How would you choose the significance level α in practice?
Suppose you are given a sequence (x_n) in [0,1]. How would you rigorously prove that it is uniformly distributed modulo 1? State the formal definition and at least one proof technique.

Explain Bootstrap and Prove Uniformity

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