If the mean of the following data is 9, then find the value of k. 11, (k-2), 7, (k-1), 11, 16, 12, 15, (k-1), 13

Options

A. 4
B. 5
C. 3
D. 2

Answer

To find the value of \( k \) given that the mean of the data set is 9, we can follow these steps:

### Step 1: Understand the Mean Formula
The mean (average) of a set of numbers is calculated using the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
Given that the mean is 9, we can set up the equation:
\[ 9 = \frac{\text{Sum of observations}}{\text{Number of observations}} \]

### Step 2: Identify the Observations
The observations given are:
- 11
- \( k - 2 \)
- 7
- \( k - 1 \)
- 11
- 16
- 12
- 15
- \( k - 1 \)
- 13

### Step 3: Calculate the Number of Observations
Count the total number of observations:
There are 10 observations in total.

### Step 4: Set Up the Equation for the Sum of Observations
The sum of the observations can be expressed as:
\[ \text{Sum} = 11 + (k - 2) + 7 + (k - 1) + 11 + 16 + 12 + 15 + (k - 1) + 13 \]
Simplifying this:
\[ \text{Sum} = 11 + 7 + 11 + 16 + 12 + 15 + 13 + (k - 2) + (k - 1) + (k - 1) \]
Combine the constant terms:
\[ \text{Sum} = 11 + 7 + 11 + 16 + 12 + 15 + 13 - 2 - 1 - 1 + 3k \]
Calculating the constant terms:
\[ = 11 + 7 + 11 + 16 + 12 + 15 + 13 - 4 = 71 \]
Thus, the sum becomes:
\[ \text{Sum} = 71 + 3k \]

### Step 5: Set Up the Equation for the Mean
Using the mean formula:
\[ 9 = \frac{71 + 3k}{10} \]

### Step 6: Solve for \( k \)
Multiply both sides by 10:
\[ 90 = 71 + 3k \]
Subtract 71 from both sides:
\[ 90 - 71 = 3k \]
\[ 19 = 3k \]
Now divide by 3:
\[ k = \frac{19}{3} \approx 6.33 \]

### Step 7: Conclusion
Since \( k \) must be a whole number, we can check if there was an error in the question or if we need to round. However, based on the calculations, \( k \) does not yield a whole number in this case.