The mean the five observations
`x,x + 2 , x + 4, x + 6, x + 8` is 11 Then, the mean of the first three observations is

Options

A. 12
B. 13
C. 15
D. 9

Answer

To solve the problem step by step, we will follow the instructions outlined in the video transcript.

### Step 1: Write down the observations
The five observations are:
- \( x \)
- \( x + 2 \)
- \( x + 4 \)
- \( x + 6 \)
- \( x + 8 \)

### Step 2: Calculate the sum of the observations
The sum of the observations can be calculated as follows:
\[ \text{Sum} = x + (x + 2) + (x + 4) + (x + 6) + (x + 8) \]
Combining like terms:
\[ \text{Sum} = x + x + 2 + x + 4 + x + 6 + x + 8 = 5x + 20 \]

### Step 3: Use the mean formula
The mean is given by the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
Here, the mean is given as 11, and the number of observations is 5. Therefore, we can set up the equation:
\[ 11 = \frac{5x + 20}{5} \]

### Step 4: Solve for \( x \)
To eliminate the fraction, multiply both sides by 5:
\[ 55 = 5x + 20 \]
Now, isolate \( 5x \) by subtracting 20 from both sides:
\[ 55 - 20 = 5x \implies 35 = 5x \]
Now, divide both sides by 5 to find \( x \):
\[ x = \frac{35}{5} = 7 \]

### Step 5: Find the first three observations
Now that we have \( x = 7 \), we can find the first three observations:
- First observation: \( x = 7 \)
- Second observation: \( x + 2 = 7 + 2 = 9 \)
- Third observation: \( x + 4 = 7 + 4 = 11 \)

### Step 6: Calculate the mean of the first three observations
Now we can find the mean of the first three observations:
\[ \text{Mean of first three observations} = \frac{7 + 9 + 11}{3} \]
Calculating the sum:
\[ 7 + 9 + 11 = 27 \]
Now divide by the number of observations:
\[ \text{Mean} = \frac{27}{3} = 9 \]

### Final Answer
The mean of the first three observations is **9**.

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