Moyu has access to a data set about the COVID-19 vaccine study. There are 120 patients in the study.

In the study, there are two types of vaccines A and B. The doctor rolls two dice and uses the following rule to assign the vaccine to each individual :

If the sum of the two dice is equal or greater than 8, the patient is assigned vaccine A. In the data, this is denoted as
Otherwise, the patient is assigned vaccine B. In the data, this individual is denoted by

Select all answers that are correct.

Question

Moyu has access to a data set about the COVID-19 vaccine study. There are 120 patients in the study.

In the study, there are two types of vaccines A and B. The doctor rolls two dice and uses the following rule to assign the vaccine to each individual :

If the sum of the two dice is equal or greater than 8, the patient is assigned vaccine A. In the data, this is denoted as 
Otherwise, the patient is assigned vaccine B. In the data, this individual is denoted by

Select all answers that are correct.

BlackTom AI · Accepted Answer

The problem describes a setup where two dice are rolled to assign vaccines, with a threshold at sum >= 8 to receive vaccine A; otherwise vaccine B. Let X(i) denote the indicator variable for receiving vaccine A on the i-th patient, so X(i) = 1 if the sum is ≥ 8 and X(i) = 0 otherwise. First, determine p, the probability that X(i) = 1. The sums that lead to A are 8, 9, 10, 11, and 12. The number of outcomes for each sum on two fair dice is: 8→5, 9→4, 10→3, 11→2, 12→1, totaling 5+4+3+2+1 = 15 outcomes out of 36. Therefore p = 15/36 = 5/12.

Option 1: The standard deviation of the sample mean of X(i) is approximately 0.045 (rounded to 3 decimals).
To evaluate this, recall that for a Bernoulli population with parameter p, the variance of a single X(i) is Var(X(i)) = p(1 − p). Here p = 5/12, so Var(X(i)) = (5/12)(7/12) = 35/144. The sample mean 
	n = 120 patients, so Var(

X̄) = Var(X(i)) / n = (35/144) / 120 = 35/17280 ≈ 0.002025. The standard deviation of the sample mean is SD(X̄) = sqrt(0.002025) ≈ 0.045. Rounding to three decimals gives 0.045. This line of reasoning shows the statement is correct.

Option 2: The variance of X(i) is 35/144.
As established above, Var(X(i)) = p(1 − p) with p = 5/12, so Var(X(i)) = (5/12)(7/12) = 35/144. This matches the stated value exactly, confirming this option is also correct.

In summary, both computed quantities align with the underlying Bernoulli model for X(i) with p = 5/12: the per-trial variance is 35/144, and the standard deviation of the sample mean for n = 120 is approximately 0.045 when expressed to three decimals.

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