Given

The observations are $x_1, x_2, \dots, x_n$ with:

• arithmetic mean $\overline{x}$
• variance $\sigma^2$

We must check the two statements for the new observations: $2x_1, 2x_2, \dots, 2x_n$

Check Statement 1: Variance of $2x_1,2x_2,\dots,2x_n$ is $4\sigma^2$

Recall: $\mathrm{Var}(aX)=a^2\mathrm{Var}(X)$

So if each observation is multiplied by $2$, the variance becomes: $\mathrm{Var}(2x)=2^2\mathrm{Var}(x)=4\sigma^2$

Therefore, Statement 1 is true.

Check Statement 2: Arithmetic mean of $2x_1,2x_2,\dots,2x_n$ is $4\overline{x}$

Mean of the new observations: $\frac{2x_1+2x_2+\cdots+2x_n}{n}$

Factor out $2$: $=2\cdot \frac{x_1+x_2+\cdots+x_n}{n}$

Since $\overline{x}=\frac{x_1+x_2+\cdots+x_n}{n},$ we get $\text{new mean}=2\overline{x}$

not $4\overline{x}$.

Therefore, Statement 2 is false.

Evaluate options

• A: Statement 1 false, Statement 2 true — incorrect
• B: Both true, and 2 explains 1 — incorrect
• C: Both true, but 2 not explanation — incorrect
• D: Statement 1 true, Statement 2 false — correct

Final answer

The correct option is: $\boxed{D}$