Question 2

Explanation

First, let's find the sum of the initial data.

Let $x$ be the datum that is replaced. Since it is replaced by the number $34$, the new sum of data becomes:

The number of data points ($n$) remains $7$. The formula for the new average ($\bar{x}_{new}$) is:

It is given that the new average is equal to twice the new median ($\bar{x}_{new} = 2 \times \text{Median}_{new}$). We need to check the possible values for the new median.

The initial data is ordered: $1, 2, 3, 4, 5, 6, 7$. The initial median is $4$. The replacement number $34$ will definitely be in the last position because it is the largest.

Case 1: If the replaced number is 1, 2, 3, or 4

For example, if we replace $1$, the data becomes $2, 3, 4, 5, 6, 7, 34$. The median becomes $5$. The same applies if we replace $2, 3$, or $4$. The data will shift so that the median becomes $5$.

Let's test this in the equation:

The result $x = -8$ is invalid because the data must be from the initial set ($1$ to $7$).

Case 2: If the replaced number is 5, 6, or 7

For example, if we replace $7$, the data becomes $1, 2, 3, 4, 5, 6, 34$. The median remains $4$. The same applies if we replace $5$ or $6$. The median remains $4$.

Let's test this in the equation:

The result $x = 6$ is valid because $6$ is in the initial data.

So, the replaced number is $6$. Thus, $P = 6$ and $Q = 6$.

In conclusion, $P = Q$.