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Questions › IIT JAM › IIT JAM 2022 › Q3
1 mark, −0.333Statistics › Sampling distributions and limit theorems › Sampling distributions of the mean and proportionModerateabout 1 min

IIT JAM 2022 Question 3: Sampling distributions and limit theorems

Which of the following CORRECTLY defines the relationship between the variances of sample means for simple random samples drawn with and without replacement from a normal population?

Tap an option to check your answer.

Answer and worked solution

Answer: (A)

For sampling with replacement the variance of the sample mean is σ2/n\sigma^2/nσ2/n. For sampling without replacement the same expression is multiplied by the finite population correction N−nN−1\dfrac{N-n}{N-1}N−1N−n​, which is less than 1 for 1<n<N1<n<N1<n<N.

Hence the without-replacement variance is smaller, i.e. σ2n>σ2n(N−nN−1)\dfrac{\sigma^2}{n} > \dfrac{\sigma^2}{n}\left(\dfrac{N-n}{N-1}\right)nσ2​>nσ2​(N−1N−n​). The answer is (A).

The trap: (D) σ2n=σ2n(N−nN−1)\dfrac{\sigma^2}{n} = \dfrac{\sigma^2}{n}\left(\dfrac{N-n}{N-1}\right)nσ2​=nσ2​(N−1N−n​)

Option (D) treats the two schemes as identical; the finite population correction is strictly below 1, so equality holds only in the limiting case n = 1.

Plain-text version

Which of the following CORRECTLY defines the relationship between the variances of sample means for simple random samples drawn with and without replacement from a normal population? (A) σ²/n > σ²/n((N-n)/(N-1)) (B) σ²/n ≤ σ²/n((N-n)/(N-1)) (C) σ²/n < σ²/n((N-n)/(N-1)) (D) σ²/n = σ²/n((N-n)/(N-1))

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

What this question tests

This question is on Sampling distributions of the mean and proportion, part of Sampling distributions and limit theorems in Statistics.

How sample statistics vary from sample to sample and how precisely they estimate.

TestStop has 16 past-paper questions on this subtopic; the nearest in year are below.

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