Regarding the Central Limit Theorem:

Read the following statements.

(a) Regardless of the underlying population distribution,
𝑋
¯
is approximately Normal with mean
𝜇
and variance
𝜎
𝑛
.

(b) Regardless of the underlying population distribution,
𝑌
=
∑
𝑖
=
1
𝑛
𝑋
𝑖
is approximately distributed as
𝑁
(
𝑛
𝜇
,
𝑛
𝜎
2
)
.

(c) The central limit theorem can only be applied to continuous distributions.

(d) If
𝑛
is greater than 25 or 30 then normal approximations generally will be good.

Now choose the correct answer.

Question

Regarding the Central Limit Theorem:

Read the following statements.

(a) Regardless of the underlying population distribution, 
𝑋
¯
 is approximately Normal with mean 
𝜇
 and variance 
𝜎
𝑛
.

(b) Regardless of the underlying population distribution, 
𝑌
=
∑
𝑖
=
1
𝑛
𝑋
𝑖
 is approximately distributed as 
𝑁
(
𝑛
𝜇
,
𝑛
𝜎
2
)
.

(c) The central limit theorem can only be applied to continuous distributions.

(d) If 
𝑛
 is greater than 25 or 30 then normal approximations generally will be good.

Now choose the correct answer.

BlackTom AI · Accepted Answer

The sample mean X̄ has mean μ and variance σ^2/n (not σ/n), so statement (a) as written is incorrect.

Relevant concepts:
- For i.i.d. random variables Xi with mean μ and variance σ^2, E[X̄] = μ and Var(X̄) = Var( (1/n)∑Xi ) = (1/n^2) ∑Var(Xi) = σ^2/n.
- The sum Y = ∑_{i=1}^n Xi has E[Y] = nμ and Var(Y) = nσ^2.
- The Central Limit Theorem (CLT) states that, under mild conditions (i.i.d. with finite variance), the distribution of standardized sums (or means) approaches the Normal as n → ∞; this applies to both discrete and continuous underlying distributions.
- A common rule-of-thumb: for many practical purposes a Normal approximation is reasonable when n is about 25–30 or larger, though required n depends on skewness and tail behavior.

Step-by-step reasoning and intermediate steps:
1. Compute Var(X̄): Var(X̄) = Var( (1/n)∑_{i=1}^n Xi ) = (1/n^2) ∑_{i=1}^n Var(Xi) = (1/n^2)(nσ^2) = σ^2/n. Thus statement (a) is false because it claims variance σ/n (missing the square).
2. For Y = ∑Xi: E[Y] = ∑E[Xi] = nμ; Var(Y) = ∑Var(Xi) = nσ^2. CLT implies Y is approximately N(nμ, nσ^2) for large n. So (b) is correct.
3. Statement (c) is false: CLT applies equally to sums of discrete variables (e.g., Bernoulli) as long as assumptions hold; continuity of the original distribution is not required.
4. Statement (d) is a practical guideline: saying n > 25 or 30 generally gives good normal approximations is a widely used rule-of-thumb (with caveats for highly skewed or heavy-tailed distributions), so (d) is acceptable.

Why the other options are wrong:
- “All statements … are correct” is wrong because (a) and (c) are incorrect.
- “None … are correct” is wrong because (b) and (d) are correct.
- Any option including (a) or (c) as correct is wrong for reasons above.

Conclusion: Only statements (b) and (d) are correct. This matches the provided answer: "Statements (b) and (d) only are correct."

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