Question

The mean of some observations is 54. If each observation is increased by 8 and its sum is divided by 2, then what is the mean of the resulting observations?

• A. 27
• B. 31
• C. 35
• D. 62

Hint:

Mean is a linear operator. If you do \(y_i = mx_i + c\), then the new mean is \(\bar{y} = m\bar{x} + c\). Here, \(y_i = \frac{1}{2}(x_i + 8)\), so \(\bar{y} = \frac{1}{2}(54 + 8) = 31\).

Answer 1

The Correct Option is B

To solve this problem, we need to determine the mean of the resulting observations after performing a specified operation on each of the initial observations.

1. Initially, the mean of some observations is given as 54. If we denote the number of observations as \(n\), the sum of these observations is \(54 \times n\).
2. According to the problem, each observation is increased by 8. Thus, the new value of each observation becomes the old value plus 8.
3. Therefore, the total increase in the sum of all observations is \(8 \times n\). Thus, the new sum of the observations is:
   • New sum = Old sum + Total increase = \(54n + 8n = 62n\).
4. The problem further states that this new sum is divided by 2 to get the resulting observations. Thus, the sum of the resulting observations is:
   • Resulting sum = \(\frac{62n}{2} = 31n\).
5. The mean of the resulting observations is given by dividing the resulting sum by the number of observations, \(n\):
   • Mean = \(\frac{31n}{n} = 31\).

Hence, the mean of the resulting observations is 31.