Question
Let $${x_1},{x_2},.....,{x_n}$$ be $$n$$ observations, and let $$\overline x $$ be their arithmetic mean and $${\sigma ^2}$$ be the variance.
Statement - 1 : Variance of $$2{x_1},2{x_2},.....,2{x_n}$$ is $$4{\sigma ^2}.$$
Statement - 2 : Arithmetic mean $$2{x_1},2{x_2},.....,2{x_n}$$ is $$4\overline x .$$
A.
Statement - 1 is false, Statement - 2 is true.
B.
Statement - 1 is true, statement - 2 is true; statement - 2 is a correct explanation for Statement - 1 .
C.
Statement - 1 is true, statement - 2 is true; statement - 2 is not a correct explanation for Statement - 1.
D.
Statement - 1 is true, statement - 2 is false.
Answer :
Statement - 1 is true, statement - 2 is false.
Solution :
KEY CONCEPT : If each observation is multiplied by $$k,$$ mean gets multiplied by $$k$$ and variance gets multiplied by $${k^2}.$$ Hence the new mean should be $$2\overline x $$ and new variance should be $${k^2}{\sigma ^2}.$$