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

Step 1: Understanding the Question:
We are given an initial mean. The data undergoes two operations: an addition of a constant to each term, and a division of the sum by a constant. We need to find the new mean. If an operation is applied linearly to all observations, the same operation applies to the mean.


Step 2: Key Formula or Approach:
If a random variable \(X\) has mean \(\bar{x}\), and a new variable \(Y\) is formed by the linear transformation \(Y = aX + b\), then the mean of \(Y\) is \(\bar{y} = a\bar{x} + b\).


Step 3: Detailed Explanation:
Let the initial observations be \(x_1, x_2, \dots, x_n\). The initial mean is \(\bar{x} = 54\).
Operation 1: Each observation is increased by 8. The new observations are \(x_i' = x_i + 8\). The new mean becomes \(\bar{x}' = \bar{x} + 8 = 54 + 8 = 62\).
Operation 2: The text says "and its sum is divided by 2". Dividing the total sum by 2 is mathematically equivalent to dividing each individual observation by 2, which in turn divides the mean by 2. So, the final observations are \(y_i = \frac{x_i + 8}{2}\). The final mean is \(\bar{y} = \frac{\bar{x}'}{2} = \frac{62}{2} = 31\).


Step 4: Final Answer:
The mean of the resulting observations is 31.