Problem 6

Number 6

Given the following data:

2, 3, 5, 7, 8, 9, 11

Two identical natural numbers are added to the data, resulting in a new data set with an average of $9$ .

$P$	$Q$
The difference between the median of the new data and the mean of the new data	$-12$

$P > Q$

$P < Q$

$P = Q$

$P + Q = 1$

The information provided is not sufficient to decide one of the options.

We will find the value of the added numbers, determine the new median, and then compare the values of $P$ and $Q$ .

Let the two added natural numbers be $x$ . The initial data consists of $7$ numbers: $2, 3, 5, 7, 8, 9, 11$ . Sum of initial data:

2 + 3 + 5 + 7 + 8 + 9 + 11 = 45

After adding $2$ numbers of value $x$ , the count of data becomes $7 + 2 = 9$ . The new average is given as $9$ .

\bar{x}_{new} = \frac{\text{Sum of Initial Data} + 2x}{\text{Count of New Data}}

9 = \frac{45 + 2x}{9}

81 = 45 + 2x

2x = 81 - 45

2x = 36

x = 18

So, the added number is $18$ .

The new data after adding $18$ and $18$ is:

2, 3, 5, 7, 8, 9, 11, 18, 18

The data is already sorted. Since the number of data points $n = 9$ (odd), the median is the $\frac{9+1}{2} = 5$ -th data point.

\text{Median} = \text{5th Data} = 8

The value of $P$ is defined as the difference between the new median and the new mean. The new mean is known to be $9$ .

P = \text{Median} - \text{Mean}

P = 8 - 9

P = -1

Given $Q = -12$ .

P = -1

Q = -12

Since $-1 > -12$ , then:

P > Q

Thus, the correct relationship is $P > Q$ .