If the mean of five observations x, (x+2) , (x+4) ,(x+6) and (x+8) is 11 , then the mean of first three observations is

Options

A. 9
B. 11
C. 13
D. None of these

Answer

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

### Step 1: Understand the Mean Formula
The mean (average) of a set of observations is calculated using the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]

### Step 2: Set Up the Observations
The five observations given are:
- \( x \)
- \( x + 2 \)
- \( x + 4 \)
- \( x + 6 \)
- \( x + 8 \)

### Step 3: Calculate the Sum of the Observations
Now, we calculate the sum of these five observations:
\[ \text{Sum} = x + (x + 2) + (x + 4) + (x + 6) + (x + 8) \]
Combining like terms:
\[ \text{Sum} = x + x + 2 + x + 4 + x + 6 + x + 8 = 5x + 20 \]

### Step 4: Set Up the Equation for the Mean
According to the problem, the mean of these observations is 11:
\[ \frac{5x + 20}{5} = 11 \]

### Step 5: Solve for \( x \)
Multiply both sides by 5 to eliminate the fraction:
\[ 5x + 20 = 55 \]
Now, subtract 20 from both sides:
\[ 5x = 35 \]
Finally, divide by 5:
\[ x = 7 \]

### Step 6: Find the First Three Observations
Now that we have \( x \), 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 7: Calculate the Mean of the First Three Observations
Now we calculate the mean of the first three observations:
\[ \text{Mean of first three} = \frac{7 + 9 + 11}{3} \]
Calculating the sum:
\[ 7 + 9 + 11 = 27 \]
Now divide by 3:
\[ \text{Mean} = \frac{27}{3} = 9 \]

### Final Answer
The mean of the first three observations is \( 9 \).

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