Two Sigma·Data Scientist·Technical Phone Screen
- Why do statisticians treat the sample mean as approximately normally distributed for large samples? What exactly becomes normal, what theorem justifies it, why does it hold, and when does the approximation break down?
- Name a distribution where the sample mean never becomes approximately normal no matter how large the sample is, and explain why the theorem's conditions fail for it.
- How fast does the normal approximation improve as sample size grows, and what property of the underlying distribution controls that rate?
- Observations in a time series are dependent. Does a CLT-type result still hold, and what conditions are needed for it?
- If you suspect the normal approximation is poor for your actual sample size, what would you do to build a confidence interval for the mean instead?
“This question is deceptively broad and I fumbled the opening by saying 'the data becomes normal' which is wrong and they caught it immediately.”