Question 17

Explanation

Identify Information

Given:

• Two study groups, let's call them $A$ and $B$.

• Number of members is the same: $n_A = n_B = 4$.

• Initial average weights: $\bar{x}_A = 30$ and $\bar{x}_B = 33$.

• Initial total weights:

  $$\sum x_A = n_A \cdot \bar{x}_A = 4 \cdot 30 = 120$$

  $$\sum x_B = n_B \cdot \bar{x}_B = 4 \cdot 33 = 132$$

Analyze the Exchange

Let the exchanged children be $x_A$ from group $A$ and $x_B$ from group $B$. After the exchange, the number of members remains the same ($n = 4$) and the averages become equal, which means the new total weights are also equal.

New total weight of group $A$ ($\sum x_A'$):

New total weight of group $B$ ($\sum x_B'$):

Calculate Weight Difference

Since the new averages are equal, the new total sums are also equal:

Group similar variables:

Thus, the difference in weight between the two exchanged children is $6 \text{ kg}$.