Question 17

Explanation

The first step is to determine the value of $a$ so that the data in Group II is complete.

Determining the value of a

• Mode of Group I: The value that appears most frequently in the data $3, 5, 5, 8, 9$ is $5$.
• Average of Group II: It is given as two more than the mode of Group I, which is $5 + 2 = 7$.

We calculate the value of $a$ using the average formula:

So, the complete data is:

• Group I: $3, 5, 5, 8, 9$ (Sum = $30$)
• Group II: $4, 6, 8, 8, 9$ (Sum = $35$)

Replacing the lowest data

We will replace the lowest data in each group.

• The lowest data in Group I is $3$. Let's replace it with $x$.
• The lowest data in Group II is $4$. Let's replace it with $y$.

The goal is for the averages of both groups to be equal. Since the number of data points is the same ($n=5$), the sum of the data in both groups must also be equal.

Calculating the difference between replacement numbers

New Sum of Group I = Initial Sum - Old Data + New Data

New Sum of Group II = Initial Sum - Old Data + New Data

Condition for equal averages (thus equal sums):

The difference between the two replacement numbers is $4$.