To solve the problem step by step, we need to find the value of \( k \) such that the median of the given data set is 11. The data set is:
\[ 3, 21, 10, 7, 6, 9, (k+6), 15, 20, 16 \]
### Step 1: Arrange the Data in Ascending Order
First, we need to arrange the data in ascending order. However, since \( k + 6 \) is not a fixed number, we need to consider its possible values.
### Step 2: Determine the Range for \( k + 6 \)
To find the range for \( k + 6 \), we can analyze the possible values of \( k \). Let's assume:
- The minimum value for \( k \) is 4, which gives \( k + 6 = 10 \).
- The maximum value for \( k \) is 7, which gives \( k + 6 = 13 \).
Thus, \( k + 6 \) can vary between 10 and 13.
### Step 3: Arrange the Data with Possible Values of \( k + 6 \)
Now we can arrange the data in ascending order considering the possible values of \( k + 6 \):
- If \( k + 6 = 10 \): The data becomes \( 3, 6, 7, 9, 10, 10, 15, 16, 20, 21 \)
- If \( k + 6 = 11 \): The data becomes \( 3, 6, 7, 9, 10, 11, 15, 16, 20, 21 \)
- If \( k + 6 = 12 \): The data becomes \( 3, 6, 7, 9, 10, 12, 15, 16, 20, 21 \)
- If \( k + 6 = 13 \): The data becomes \( 3, 6, 7, 9, 10, 13, 15, 16, 20, 21 \)
### Step 4: Count the Number of Terms
The total number of terms \( n \) is 10, which is an even number.
### Step 5: Find the Median
For an even number of terms, the median is calculated as:
\[
\text{Median} = \frac{\text{n/2th term} + \text{(n/2 + 1)th term}}{2}
\]
Here, \( n = 10 \), so:
\[
\text{Median} = \frac{\text{5th term} + \text{6th term}}{2}
\]
### Step 6: Identify the 5th and 6th Terms
- If \( k + 6 = 10 \): 5th term = 10, 6th term = 10
- If \( k + 6 = 11 \): 5th term = 10, 6th term = 11
- If \( k + 6 = 12 \): 5th term = 10, 6th term = 12
- If \( k + 6 = 13 \): 5th term = 10, 6th term = 13
### Step 7: Set Up the Equation
We know the median is given as 11. Therefore, we set up the equation:
\[
\frac{10 + (k + 6)}{2} = 11
\]
### Step 8: Solve for \( k \)
Multiply both sides by 2:
\[
10 + (k + 6) = 22
\]
Simplifying gives:
\[
k + 16 = 22
\]
Subtract 16 from both sides:
\[
k = 6
\]
### Final Answer
Thus, the value of \( k \) is \( 6 \).
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