If the mean of five observations x, x + 2, x + 4, x + 6, x + 8 is 11, then the mean of the 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 mathematical definitions and calculations as outlined in the video transcript.

### Step 1: Understand the Mean Formula
The mean (average) of a set of numbers is calculated by taking the sum of all the numbers and dividing it by the count of the numbers.

### Step 2: Set Up the Equation
Given the observations:
- \( x \)
- \( x + 2 \)
- \( x + 4 \)
- \( x + 6 \)
- \( x + 8 \)

The mean of these five observations is given as 11. Therefore, we can set up the equation:
\[ \frac{x + (x + 2) + (x + 4) + (x + 6) + (x + 8)}{5} = 11 \]

### Step 3: Simplify the Equation
Now, simplify the left side:
\[ \frac{x + x + 2 + x + 4 + x + 6 + x + 8}{5} = \frac{5x + (2 + 4 + 6 + 8)}{5} \]
Calculating the sum inside the parentheses:
\[ 2 + 4 + 6 + 8 = 20 \]
Thus, the equation becomes:
\[ \frac{5x + 20}{5} = 11 \]

### Step 4: Eliminate the Denominator
Multiply both sides by 5 to eliminate the denominator:
\[ 5x + 20 = 55 \]

### Step 5: Solve for \( x \)
Now, isolate \( x \):
\[ 5x = 55 - 20 \]
\[ 5x = 35 \]
\[ x = 7 \]

### Step 6: Find the Mean of the First Three Observations
The first three observations are:
- \( x \)
- \( x + 2 \)
- \( x + 4 \)

Substituting \( x = 7 \):
- First observation: \( 7 \)
- Second observation: \( 7 + 2 = 9 \)
- Third observation: \( 7 + 4 = 11 \)

Now, calculate the mean of these three observations:
\[ \text{Mean} = \frac{7 + 9 + 11}{3} \]
Calculating the sum:
\[ 7 + 9 + 11 = 27 \]
Thus, the mean is:
\[ \frac{27}{3} = 9 \]

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

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