Question

The mean of five observations is 4 and their variance is 5.2. If three of these observations are 1, 2 and 6, then the other two are

• A. 2 and 9
• B. 3 and 8
• C. 4 and 7
• D. 5 and 6

Hint:

Save time by checking the options against your first derived relation ($a + b = 11$):
Option (A): $2 + 9 = 11$
Option (B): $3 + 8 = 11$
Option (C): $4 + 7 = 11$
Option (D): $5 + 6 = 11$
Since all options sum to 11, do a quick mental sum-of-squares check ($a^2 + b^2 = 65$):
For option (C): $4^2 + 7^2 = 16 + 49 = 65$. It matches instantly!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given statistical summary data (mean and variance) for a set of five numbers. Three data points are explicitly provided, and we must determine the values of the remaining two unknown observations.

Step 2: Key Formula or Approach:
Let the two unknown numbers be denoted as $a$ and $b$.
1. The arithmetic mean formula for $n=5$ elements is:
$$\bar{x} = \frac{\sum x_i}{5}$$ 2. The statistical population variance formula is defined as:
$$\sigma^2 = \frac{\sum x_i^2}{5} - (\bar{x})^2$$ We will set up a system of two algebraic equations to solve for variables $a$ and $b$.

Step 3: Detailed Explanation:
Using the given mean value ($\bar{x} = 4$):
$$4 = \frac{1 + 2 + 6 + a + b}{5}$$ $$20 = 9 + a + b \implies a + b = 11 \quad \text{--- (Equation 1)}$$ Now, let's substitute the parameters into the variance expression ($\sigma^2 = 5.2$):
$$5.2 = \frac{1^2 + 2^2 + 6^2 + a^2 + b^2}{5} - (4)^2$$ $$5.2 = \frac{1 + 4 + 36 + a^2 + b^2}{5} - 16$$ Add 16 to both sides of the equation:
$$21.2 = \frac{41 + a^2 + b^2}{5}$$ Multiply both sides by 5:
$$106 = 41 + a^2 + b^2 \implies a^2 + b^2 = 65 \quad \text{--- (Equation 2)}$$ From Equation 1, we know that $b = 11 - a$. Substitute this relation into Equation 2:
$$a^2 + (11 - a)^2 = 65$$ $$a^2 + 121 - 22a + a^2 = 65$$ $$2a^2 - 22a + 56 = 0$$ Divide the entire quadratic equation by 2:
$$a^2 - 11a + 28 = 0$$ Factoring this quadratic equation yields:
$$(a - 4)(a - 7) = 0$$ Thus, $a$ can be either 4 or 7. If $a = 4$, then $b = 7$, and vice versa.

Step 4: Final Answer:
The other two missing numerical observations are 4 and 7, matching option (C).