Select all statements that are correct.

The law of large numbers and central limit theorem (taken together) imply that:

Question

Select all statements that are correct.

The law of large numbers and central limit theorem (taken together) imply that:

BlackTom AI · Accepted Answer

We start by restating the question and listing all answer options so we can evaluate each one on its own merits.
Question: Select all statements that are correct. The law of large numbers and central limit theorem (taken together) imply that:
Answer options:
1) As N goes to infinity, the bias of the difference sample averages for the true difference in averages goes to zero
2) When treatment assignment is randomized and N goes to infinity, the difference in sample averages converges to the true difference in averages
3) When treatment assignment is randomized and N is extremely large (practically infinite), the value of the noise draw is close to zero with high certainty.
4) When treatment is random and N is large, we know the exact value of the noise draw.

Option 1:
This statement ties the bias of the difference in sample averages to the limit as N grows. While LLN/CLT tell us about convergence of sample averages to population quantities, the more precise point is that the difference between sample averages converges to the true difference, and under randomization the estimator is unbiased for all N. Saying the bias goes to zero as N increases could be interpreted as true (bias is zero for all N, so it certainly goes to zero in the limit), but this wording is potentially misleading because it shifts focus from convergence of the estimator itself to the behavior of its bias with sample size. In exam contexts emphasizing LLN/CLT implications for randomized experiments, this option is not the standard expressed conclusion. Therefore, it is considered not correctly capturing the intended implication.

Option 2:
Under randomized treatment assignment, the expected value of the difference in sample averages equals the true difference in averages. The LLN says sample averages converge to their expectations, and with randomization, the expectation equals the true effect. Therefore, as N grows without bound, the observed difference in sample averages converges to the true difference. This aligns with the standard interpretation of LLN and randomization: consistency of the estimator.

Option 3:
The CLT tells us that the distribution of the sample mean (and hence the difference in sample means) becomes approximately normal with standard deviation shrinking at a rate proportional to 1/√N. As N becomes very large, the variability (noise) around the true difference decreases, so the observed noise draw is very likely to be near zero. The phrase "practically infinite" reflects the asymptotic behavior described by the CLT, making this statement correct in the asymptotic sense.

Option 4:
This statement asserts that we know the exact value of the noise draw when N is large. The LLN/CLT describe distributional behavior and convergence in probability, not that we know the precise realized noise value in any finite sample. Even with large N, the noise draw (random fluctuations) is not exactly known; we can only say it becomes small on average and that estimates become precise in probability. So this option misstates what LLN/CLT guarantee and is incorrect.

In summary, the nuanced reasoning flags options 2 and 3 as aligned with the core implications of LLN and CLT in the setting of randomized treatment and large N, while options 1 and 4 either overstate or misstate those implications. The provided answer set identifies 2 and 3 as the correct choices, which matches the standard interpretations for these theorems in randomized settings.

Select all statements that are correct. The law of large numbers and central limit theorem (taken together) imply that:多项选择题

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