Questions: Central Limit Theorem: Rigor and Applications

Question 1

A researcher samples n = 100 values from a heavily right-skewed income distribution. The CLT is invoked to justify a normal approximation. What is the CLT actually saying is approximately normal?

• A. Each individual income value is approximately normally distributed for large samples
• B. The sample mean X̄ computed from the 100 values is approximately normally distributed
• C. The population income distribution becomes approximately normal as sample size grows
• D. The CLT cannot apply because the population distribution is not symmetric

Answer: (B)

The CLT is a statement about the sampling distribution of the sample mean X̄, not about individual observations or the population distribution. Individual income values remain right-skewed regardless of n. The population distribution doesn't change as n grows. What changes is the distribution of X̄ across repeated samples of size n — that distribution converges to normal. Option D is a common misconception: the CLT explicitly applies to non-normal, even skewed populations, as long as the population has finite mean and variance.

Question 2

If you increase sample size from n = 25 to n = 100, how does the standard error of the sample mean change?

• A. It halves — the standard error is σ/√n, so √100 = 10 vs √25 = 5, a ratio of 2
• B. It quarters — larger n reduces variability more aggressively
• C. It doubles — more observations means more deviation from the true mean
• D. It remains unchanged — the standard error depends only on population variance σ², not n

Answer: (A)

The standard error is σ/√n. At n = 25, SE = σ/5. At n = 100, SE = σ/10. The ratio is 2: the standard error halves. This is the formal statement of the intuition that larger samples give more precise estimates. Note that to halve the SE again you'd need n = 400 — gains in precision require quadrupling sample size, diminishing returns that have real implications for study design.

Question 3

The Central Limit Theorem guarantees that for sufficiently large n, the sample mean X̄ is approximately normally distributed even when the population is discrete (e.g., a Poisson or Bernoulli distribution).

• T. True
• F. False

Answer: True

The CLT applies to any population distribution with finite mean and variance — discrete, continuous, skewed, bimodal, uniform. The sample mean of i.i.d. draws from a Poisson(λ) distribution, for example, converges to N(λ, λ/n) as n → ∞. Discreteness of the population is not an obstacle; the averaging operation smooths out the distribution.

Question 4

The Central Limit Theorem states that as sample size n grows, the population distribution approaches a normal distribution.

• T. True
• F. False

Answer: False

This is the most common misstatement of the CLT. The population distribution does not change — it stays right-skewed, bimodal, or whatever shape it has, regardless of n. What converges to normal is the sampling distribution of the sample mean X̄: the distribution you'd get by computing X̄ from many independent random samples of size n. The CLT is a theorem about statistics (functions of data), not about the underlying data-generating process.

Question 5

Why does the CLT not require the population to be normally distributed, and what two conditions on the population are actually required?

Model answer: The CLT does not require normality because it is a theorem about the behavior of sums and averages of many independent random variables. By the law of large numbers, the average converges to the true mean; the CLT adds that the *fluctuations* around that mean become normally distributed as n grows. The two required conditions are: (1) finite mean μ — the population must have a well-defined expected value; and (2) finite variance σ² — the population must not have infinite spread (heavy-tailed distributions like Cauchy, with undefined variance, do not satisfy the CLT).

The technical proof uses characteristic functions: the characteristic function of the standardized sample mean converges pointwise to e^{−t²/2}, the characteristic function of N(0,1). This convergence holds whenever the population has finite variance. The independence requirement (i.i.d. draws) can be relaxed by the Lindeberg-Feller CLT, which allows non-identically distributed observations as long as no single observation dominates the total variance.