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Questions › IIT JAM › IIT JAM 2024 › Q36
2 marksStatistics › Sampling distributions and limit theorems › The central limit theorem and probability inequalitiesHardabout 3 min

IIT JAM 2024 Question 36: Sampling distributions and limit theorems

Let X1,X2,…,XnX_1, X_2, \dots, X_nX1​,X2​,…,Xn​ (n>1n > 1n>1) be a random sample from a distribution with mean μ\muμ and non-zero variance σ2\sigma^2σ2. Which of the following is/are CORRECT?

Select all that apply.

Answer and worked solution

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Plain-text version

Let X₁, X₂, …, Xₙ (n > 1) be a random sample from a distribution with mean μ and non-zero variance σ². Which of the following is/are CORRECT? (A) The sample mean has standard deviation σ/√n (B) (Σᵢ₌₁ⁿ(Xᵢ - μ))/σ√n tends to a standard normal distribution as n → ∞ (C) ((n-1)S²)/σ² follows a χ² distribution with (n-1) degrees of freedom, where S² is the sample variance (D) The sample mean is always a consistent estimator of μ

Also written as: Let X_1, X_2,..., X_n (n > 1) be a random sample from a distribution with mean mu and non-zero variance sigma^2. Which of the following is/are CORRECT?

Sampling distributions and limit theorems has come up 25 times in 7 exams since 2007.

What this question tests

This question is on The central limit theorem and probability inequalities, part of Sampling distributions and limit theorems in Statistics.

Why sample means are approximately normal, and bounds that hold for any distribution.

TestStop has 5 past-paper questions on this subtopic.

Take this paper as a timed mockMore on The central limit theorem and probability inequalities

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