2 marks, −0.667StatisticsSampling distributions and limit theoremsSampling distributions of the mean and proportionHardabout 2 min

IIT JAM 2022 Question 17: Sampling distributions and limit theorems

Two random samples of sizes $n_1$ and $n_2$ are selected without replacement from two binomial populations with means $\mu_1 = n_1 p_1$ , $\mu_2 = n_2 p_2$ and variances $\sigma_1^2 = n_1 p_1 q_1$ , $\sigma_2^2 = n_2 p_2 q_2$ . The difference of sample proportions $\bar P_1$ and $\bar P_2$ is approximately normal with mean $(p_1 - p_2)$ . Then the standard deviation of $\bar P_1 - \bar P_2$ is

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Answer and worked solution

Answer: (A)

For each independent sample the variance of the sample proportion is $\dfrac{p q}{n}$ multiplied by its finite population correction $\dfrac{N-n}{N-1}$ , since sampling is without replacement.

The two samples are independent, so the variance of the difference is the sum of the two variances, and the standard deviation is the square root of that sum. Each term keeps its own $n$ in the denominator and its own correction, which is exactly option (A).

The trap: (B)

\sqrt{\left(\dfrac{p_1 q_1}{n_1}\right) + \left(\dfrac{p_2 q_2}{n_2}\right)}

Option (B) drops the finite population correction, which is only valid for sampling with replacement or an infinite population; option (D) wrongly pools the sample sizes as $n_1+n_2$ .

Plain-text version

Two random samples of sizes n₁ and n₂ are selected without replacement from two binomial populations with means μ₁ = n₁ p₁, μ₂ = n₂ p₂ and variances σ₁² = n₁ p₁ q₁, σ₂² = n₂ p₂ q₂. The difference of sample proportions P̄₁ and P̄₂ is approximately normal with mean (p₁ - p₂). Then the standard deviation of P̄₁ - P̄₂ is (A) √(((p₁ q₁)/n₁)((N₁-n₁)/(N₁-1)) + ((p₂ q₂)/n₂)((N₂-n₂)/(N₂-1))) (B) √(((p₁ q₁)/n₁) + ((p₂ q₂)/n₂)) (C) √((p₁ q₁ - p₂ q₂)/(n₁ + n₂)) (D) √(((p₁ q₁)/(n₁+n₂))((N₁-n₁)/(N₁-1)) + ((p₂ q₂)/(n₁+n₂))((N₂-n₂)/(N₂-1)))

Also written as: Two random samples of sizes n_1 and n_2 are selected without replacement from two binomial populations with means mu_1 = n_1 p_1, mu_2 = n_2 p_2 and variances sigma_1^2 = n_1 p_1 q_1, sigma_2^2 = n_2 p_2 q_2. The difference of sample proportions P_bar_1 and P_bar_2 is approximately normal with mean (p_1 - p_2). Then the standard deviation of P_bar_1 - P_bar_2 is

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.

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