Consider 3 box models as follows, where the draws are made with replacement, and the sum of the draws is calculated.

BoxModel1: 100 draws from the box with 1,2,3,4,5;
BoxModel2: 100 draws from the box with 2,4,6,8,10;
BoxModel3: 10,000 draws from the box with 1,2,3,4,5

Denote the standard error (SE) for the sum of draws of the 3 models by SE1, SE2 and SE3 respectively.

Which of the following statements is TRUE?

Question

Consider 3 box models as follows, where the draws are made with replacement, and the sum of the draws is calculated.

BoxModel1: 100 draws from the box with 1,2,3,4,5;
BoxModel2: 100 draws from the box with 2,4,6,8,10;
BoxModel3: 10,000 draws from the box with 1,2,3,4,5

Denote the standard error (SE) for the sum of draws of the 3 models by SE1, SE2 and SE3 respectively.

Which of the following statements is TRUE?

BlackTom AI · Accepted Answer

For draws with replacement, the standard error (SE) of the sum equals sqrt(n) times the standard deviation of the box. Use variance-scaling for multiplied values.

1. Relevant knowledge points and concepts
- For independent draws with replacement from a box, Sum S of n draws has SD(S) = sqrt(n) · σ_box (σ_box is the SD of one draw).
- Multiplying every entry in a box by a constant c multiplies the box SD by |c| (Var scales by c^2).
- Compute box mean and variance by averaging over the box values.

2. Step-by-step reasoning
- BoxModel1 values: 1,2,3,4,5. Mean μ1 = (1+2+3+4+5)/5 = 15/5 = 3.
- Compute variance for Box1: Var1 = [(1−3)^2+(2−3)^2+(3−3)^2+(4−3)^2+(5−3)^2]/5 = (4+1+0+1+4)/5 = 10/5 = 2.
- So σ1 = sqrt(Var1) = sqrt(2).

- BoxModel2 values: 2,4,6,8,10 = 2·(1,2,3,4,5). Therefore σ2 = 2·σ1 = 2·sqrt(2).

- BoxModel3 uses the same box as BoxModel1, so σ3 = σ1 = sqrt(2).

- Use SD of sum formula:
  - SE1 (sum of 100 draws from Box1) = sqrt(100)·σ1 = 10·σ1 = 10·sqrt(2).
  - SE2 (sum of 100 draws from Box2) = sqrt(100)·σ2 = 10·(2·sqrt(2)) = 20·sqrt(2).
  - SE3 (sum of 10,000 draws from Box1) = sqrt(10,000)·σ1 = 100·σ1 = 100·sqrt(2).

3. Intermediate arithmetic shown
- σ1 = sqrt(2)
- SE1 = 10·sqrt(2)
- SE2 = 10·(2·sqrt(2)) = 20·sqrt(2) = 2·(10·sqrt(2)) = 2·SE1
- SE3 = 100·sqrt(2) = 10·(10·sqrt(2)) = 10·SE1

4. Why the correct answer is correct
- SE scales with the box SD and with sqrt(n). Doubling every box entry doubles the box SD, so for the same n (100) SE doubles: SE2 = 2·SE1. Increasing n from 100 to 10,000 multiplies sqrt(n) by 10, so SE3 = 10·SE1.

5. Why other possible answers would be incorrect
- Claims that SE scales with the mean, with n linearly, or that SE stays equal between Box1 and Box2 are wrong because they ignore variance scaling and the sqrt(n) dependence of the sum’s SD.

6. Conclusion
The computed relations SE2 = 2 SE1 and SE3 = 10 SE1 are correct, matching the provided answer.

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