Central Limit Theorem (CLT) and Normal Distribution Questions
Understand the CLT: when you take multiple random samples and calculate their means, those sample means are normally distributed (bell-shaped) even if the underlying data isn't. Know that normal distribution is parameterized by mean and standard deviation. Appreciate why this matters: it allows you to estimate population characteristics from samples and construct confidence intervals.
MediumTechnical
37 practiced
For a sample of size n = 30 with 3 successes, compare the normal-approximation CI for the proportion with the Wilson (score) interval. Compute both intervals numerically and summarize which is preferable and why in small-sample proportion estimation.
HardTechnical
48 practiced
Provide a sketch of the classic proof of the CLT using characteristic functions (Fourier transforms). Outline the main steps, including standardizing sums, using independence to raise characteristic functions to the nth power, expanding via Taylor series, and invoking Lévy's continuity theorem. Keep the explanation at a level a senior data scientist can follow without writing full formal proofs.
MediumTechnical
51 practiced
In R, simulate the sampling distribution of the sample mean for a uniform distribution U(0,1). Draw 5,000 samples for each n in {1, 2, 5, 10, 30} and plot the means' histograms. Briefly interpret how the histogram changes with n. Provide the main R functions you would use and outline the code structure (no need to write full code).
HardSystem Design
32 practiced
Design an architecture to compute and publish running estimates of the population mean and an approximate 95% confidence interval for a metric in a high-throughput streaming system that processes 1M events per second. Explain trade-offs in terms of latency, memory, and statistical accuracy and list sketching or sampling techniques you would consider (e.g., reservoir sampling, t-digest, streaming variance algorithms).
MediumTechnical
36 practiced
You are forecasting expected daily revenue but the raw revenue per user is extremely heavy-tailed with occasional huge values. Discuss whether the CLT justifies using sample mean-based CIs and propose robust alternatives you would use as a data scientist to provide reliable uncertainty quantification.
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