Question

The mean of n observations is X. If each observation is multiplied by k, the mean of new observations is

Options

• `k bar(X) `

• `bar(X)/k`

• `bar(X)  +k`

• `bar(X)- k`

Answer 1

Let us take n observations `X_i,...,X_n.`

If `bar(X)` be the mean of the n observations, then we have

`bar(X) = 1/n sum _(i=1)^n X_i`

`⇒ sum_(i=1)^n X_i = n bar(X)`

Multiply a constant k to each of the observations. Then the observations becomes `kX_i,...,kX_n.`

If `bar(Y)` be the mean of the new observations, then we have

`bar(Y) = 1/n sum _(i=1)^n kX_i`

`= k/n sum _(i=1)^n X_i`

`= k . 1/n sum _(i=1)^n X_i`

`= kbar(X)`