Question

Assertion (A): Mean of n observations is x and mean of another set of n observations is y, the combined mean of all the observations is `(x + y)/2`.

Reason (R): Total of all the observations = nx + ny ∴ Mean of all the observations = `(nx + ny)/(2n)`.

Options

• A is true, R is false.

• A is false, R is true.

• Both A and R are true and R is the correct reason for A.

• Both A and R are true and R is the incorrect reason for A.

Answer 1

Both A and R are true and R is the correct reason for A.

Explanation:

The sum of the first set of n observations with mean x is nx, and the sum of the second set of n observations with mean y is ny. The total sum of all observations combined is nx + ny, and dividing this total sum by the total number of observations (n + n = 2n) gives the combined mean:

`frac{nx + ny}{2n} = \frac{n(x + y)}{2n} = \frac{x + y}{2}`

Thus, both the assertion and the reason are completely true, and the reason accurately explains how the combined mean is derived.