Explain the Central Limit Theorem and its importance for hypothesis testing

WHAT THIS TESTS: The interviewer wants to know if you understand the mechanism that makes classical inference possible in real-world settings where population distributions are rarely normal. Specifically, they are checking whether you distinguish between the distribution of raw data and the sampling distribution of a statistic, and whether you know the conditions under which normality emerges.

A GOOD ANSWER COVERS: First, the core statement: for independent and identically distributed random variables with finite variance, the distribution of the sample mean approaches a normal distribution as sample size increases, even if the original variables are not normal. Second, the normalization detail: we are talking about a normalized version of the sample mean, typically subtracting the population mean and dividing by the standard error, converging to a standard normal distribution. Third, the practical importance: because of this convergence, we can construct confidence intervals and perform hypothesis tests using z-statistics or t-statistics without knowing the exact shape of the population distribution. Fourth, the conditions: mention finite variance, independence, and that the theorem describes large-sample behavior rather than small-sample exact distributions.

COMMON WRONG ANSWERS: A major red flag is saying the CLT means any data becomes normal if you collect enough of it; the theorem applies to the sample mean or sum, not to individual observations. Another error is ignoring sample size and claiming the CLT applies universally even with n equals five or heavily dependent data. Some candidates also forget the finite variance requirement or conflate the CLT with the Law of Large Numbers, which concerns convergence of the sample mean to the population mean rather than convergence of its distribution to a bell curve.

LIKELY FOLLOW-UPS: The interviewer may ask how large is large enough, which depends on skewness and kurtosis but often n greater than thirty is taught as a rule of thumb. They might ask what breaks the CLT, such as infinite variance distributions like the Cauchy distribution or strong serial dependence. They could also ask you to contrast the CLT with the Law of Large Numbers, or to explain why we use a t-distribution instead of the normal when the population variance is estimated from the sample.

ONE CONCRETE EXAMPLE: Suppose you are measuring website load times that are heavily right-skewed, with most pages loading in under two seconds but a few taking ten seconds. The distribution of individual load times is far from normal. However, if you repeatedly take random samples of one hundred load times and plot the average of each sample, those averages will form an approximately normal distribution centered around the true mean. This lets you calculate a ninety-five percent confidence interval for the true average load time using the sample standard deviation and the normal or t-distribution, even though you never assumed the underlying load times were normal.

Source: Wikipedia: Central limit theorem