Question

The mean and variance of $5$ observations are $5$ and $8$ respectively If $3$ observations are $1,3,5$, then the sum of cubes of the remaining two observations is

• A. 1216
• B. 1456
• C. 1072
• D. 1792

Answer 1

The Correct Option is C

To solve the problem, we need to use the given information about the mean and variance, and find the sum of the cubes of the remaining two observations.

Step 1: Understand the mean and variance formulas

• \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}}.
• \text{Variance} = \frac{\sum (x_i - \bar{x})^2}{n}, where \bar{x} is the mean.

Step 2: Find the sum of all five observations

Let the five observations be x_1, x_2, x_3, x_4, x_5. It's given:

• x_1 = 1, x_2 = 3, x_3 = 5
• The mean of the five observations is 5.

From the mean, we have:

5 = \frac{1 + 3 + 5 + x_4 + x_5}{5}

5 \times 5 = 9 + x_4 + x_5

25 = 9 + x_4 + x_5 \Rightarrow x_4 + x_5 = 16

Step 3: Use the variance to find another equation for the observations

Variance is given as 8, so:

8 = \frac{(1-5)^2 + (3-5)^2 + (5-5)^2 + (x_4-5)^2 + (x_5-5)^2}{5}

Simplifying, we get:

{\begin{align*} 8 \times 5 &= 16 + 4 + 0 + (x_4-5)^2 + (x_5-5)^2 \\ &= 40 = 20 + (x_4-5)^2 + (x_5-5)^2 \\ \\ &\Rightarrow (x_4-5)^2 + (x_5-5)^2 = 20 \end{align*}}

Step 4: Solve the system of equations

Now, we have two equations:

• x_4 + x_5 = 16
• (x_4-5)^2 + (x_5-5)^2 = 20

Using x_4 = 16 - x_5 in the second equation, and simplifying, we find:

The two roots that satisfy both equations are x_4 = 9, x_5 = 7.

Step 5: Calculate the sum of cubes of the remaining two observations

Third observations being x_4 = 9 and x_5 = 7.

Sum of cubes is computed as follows:

x_4^3 + x_5^3 = 9^3 + 7^3 = 729 + 343 = 1072

Thus, the sum of cubes of the remaining two observations is 1072.