Statistics Question - None

Question

Let $x_1, x_2, \ldots, x_n$ be $n$ observations, and let $\bar{x}$ be their arithmetic mean and $\sigma^2$ be their variance.
Statement-1: Variance of $2x_1, 2x_2, \ldots, 2x_n$ is $4\sigma^2$.
Statement-2: Arithmetic mean of $2x_1, 2x_2, \ldots, 2x_n$ is $4\bar{x}$.

Accepted Answer

Step 1: Analyze Statement-1 (Variance)

For transformed data {2x₁, 2x₂, ..., 2xₙ}, the variance is:

Var(2xᵢ) = E[(2xᵢ - 2x̄)²] = E[4(xᵢ - x̄)²] = 4E[(xᵢ - x̄)²] = 4σ²

✓ Statement-1 is TRUE

Step 2: Analyze Statement-2 (Arithmetic Mean)

For transformed data {2x₁, 2x₂, ..., 2xₙ}, the mean is:

Mean = (2x₁ + 2x₂ + ... + 2xₙ)/n = 2(x₁ + x₂ + ... + xₙ)/n = 2x̄

✗ Statement-2 is FALSE (claims 4x̄ but should be 2x̄)

Step 3: Conclusion

Only Statement-1 is correct. The key rule: Var(kX) = k²Var(X) but Mean(kX) = k·Mean(X)

∴ Answer: A

Question:

Step-by-Step Solution

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