A population has a heavily right-skewed distribution (e.g., household incomes). A researcher takes random samples of size n = 50. According to the Central Limit Theorem, which statement is correct?
AThe individual income observations will be approximately normally distributed within each sample
BThe population distribution will become more symmetric as more samples are drawn
CThe sample mean x̄ will be approximately normally distributed across repeated samples
DThe CLT does not apply here because the population is not normal
The CLT applies to the distribution of the *sample mean* x̄, not to individual observations. Individual incomes stay skewed — the CLT says nothing about them. What becomes approximately normal is x̄ computed across many repeated samples of size 50. Option D is wrong because the CLT explicitly applies *regardless* of the population's shape, provided n is large enough.
Question 2 Multiple Choice
A quality engineer reduces sampling cost by cutting sample size from n = 100 to n = 25. What happens to the standard error of the sample mean?
AIt doubles, because sample size was cut in half twice
BIt doubles, because standard error is σ/√n and √25 is half of √100
CIt stays the same, because σ (the population standard deviation) didn't change
DIt quadruples, because precision degrades proportionally to sample size reduction
Standard error = σ/√n. With n = 100, SE = σ/10. With n = 25, SE = σ/5 — exactly double. The square-root relationship means you must quadruple the sample size to halve the standard error, not double it. Option A confusingly describes the factor correctly but misstates why: n went from 100 to 25 (a factor of 4 reduction), and √4 = 2, so SE doubles.
Question 3 True / False
The Central Limit Theorem guarantees that for large n, the sampling distribution of the sample mean is approximately normal, regardless of the population's shape.
TTrue
FFalse
Answer: True
This is the core claim of the CLT. The population itself can be exponential, uniform, bimodal, or highly skewed — the distribution of x̄ across repeated samples of size n converges to a normal distribution (with mean μ and standard deviation σ/√n) as n grows. This is why normal-based inference methods work for non-normal populations.
Question 4 True / False
If the Central Limit Theorem applies to a dataset, the individual data points in that dataset are approximately normally distributed.
TTrue
FFalse
Answer: False
This is the most common CLT misconception. The CLT makes a claim about the *sample mean* x̄ — a statistic computed from a sample — not about the individual observations themselves. If you draw n = 50 values from an exponential distribution, each individual value is still exponential. Only the distribution of x̄ (computed across many such samples) becomes approximately normal.
Question 5 Short Answer
Why does the Central Limit Theorem apply to sample means but not to individual observations from a non-normal population?
Think about your answer, then reveal below.
Model answer: The sample mean is an average of n independent random variables. Averaging causes the idiosyncratic extremes and asymmetries of individual draws to cancel each other out — large values in one draw are offset by small values in others. This averaging-out process (mathematically, the convergence of the sum's characteristic function to that of a normal distribution) is what produces the bell shape. Individual observations have no such averaging — each one reflects the full shape of the population distribution directly.
The key is that averaging introduces a mathematical smoothing effect absent for individual values. This is why the CLT is about sums and means, not raw data. The standard deviation of that bell-shaped distribution (σ/√n) also shrinks with n, capturing how averaging reduces variability. Understanding this distinction separates students who can correctly apply CLT from those who misapply it to raw data.