Let x be the median of the data 13, 8, 15, 14, 17, 9, 14, 16, 13, 17,14, 15, 16,15,14.
If 8 is replaced by 18, then the median of the data is y. What is the sum of the values of x and y ?

Options

A. 30
B. 27
C. 28
D. 29

Answer

To solve the problem, we need to find the median of the given data set and then determine how the median changes when one value is replaced. Finally, we will compute the sum of the two medians.

### Step 1: Organize the Data
First, we need to arrange the given data in ascending order. The original data set is:
\[ 13, 8, 15, 14, 17, 9, 14, 16, 13, 17, 14, 15, 16, 15, 14 \]

Arranging this in ascending order:
\[ 8, 9, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 16, 17, 17 \]

### Step 2: Find the Median (x)
The median is the middle value of a data set. Since there are 15 values (which is odd), the median is the value at position \( \frac{n+1}{2} \), where \( n \) is the number of values.

Calculating the position:
\[ \text{Position} = \frac{15 + 1}{2} = \frac{16}{2} = 8 \]

Now, we find the 8th value in the ordered list:
- The ordered list is: \( 8, 9, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 16, 17, 17 \)
- The 8th value is \( 14 \).

Thus, \( x = 14 \).

### Step 3: Replace 8 with 18 and Reorganize the Data
Now, we replace 8 with 18 in the data set:
\[ 18, 9, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 16, 17, 17 \]

Arranging this new data in ascending order:
\[ 9, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 16, 17, 17, 18 \]

### Step 4: Find the New Median (y)
Again, we find the median of this new data set. The number of values is still 15, so we again use the position \( \frac{n+1}{2} \):
\[ \text{Position} = \frac{15 + 1}{2} = \frac{16}{2} = 8 \]

Now, we find the 8th value in the new ordered list:
- The ordered list is: \( 9, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 16, 17, 17, 18 \)
- The 8th value is \( 15 \).

Thus, \( y = 15 \).

### Step 5: Calculate the Sum of x and y
Now, we compute the sum of \( x \) and \( y \):
\[ x + y = 14 + 15 = 29 \]

### Final Answer
The sum of the values of \( x \) and \( y \) is \( 29 \).

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