Assertion (A): The mean of 5 observations is 30. On excluding one of these observations, the mean of the remaining observation is increased to 31. The excluded observation is

= 5 x 30 - 4 x 31 = 150 - 124 = 26

Reason (R): The excluded observation is = 5 x 31 - 4 x 30 = 155 - 120 = 35

1. A is true, R is false.
2. A is false, R is true.
3. Both A and R are true.
4. Both A and R are false.

A is true, R is false.

Explanation

Given,

Mean = 30

Number of all observations = 5

Mean = $\dfrac{\text{Sum of all observations}}{\text{Number of all observations}}$

⇒ 30 = $\dfrac{\text{Sum of all observations}}{5}$

⇒ Sum of all 5 observations = 30 x 5 = 150

After excluding one observation, the new mean of the remaining 4 observations is 31:

Number of all observations = 4

⇒ 31 = $\dfrac{\text{Sum of all 4 observations}}{4}$

⇒ Sum of all 4 observations = 31 x 4 = 124

Sum of all 5 observations = Sum of all 4 observations + Excluded observation

⇒ 150 = 124 + Excluded observation

⇒ Excluded observation = 150 - 124

⇒ Excluded observation = 26

∴ Assertion (A) is true.

From the above calculation,

The excluded observation is 26.

According to Reason, the excluded observation is = 5 x 31 - 4 x 30 = 155 - 120 = 35 (≠ 26)

∴ Reason (R) is false.

Hence, Assertion (A) is true, Reason (R) is false.