Question

Consider a data consisting of 10 observations \[ x_1, x_2, \dots, x_{10}, \] whose mean is \(5\) and variance is \(7\). If the mean and the variance of the first 8 observations \[ x_1, x_2, \dots, x_8 \] are \(4\) and \(3.5\), respectively, and \[ x_9<x_{10}, \] then the value of \[ 3x_9 + 2x_{10} \] is:

Hint:

To find values of missing observations, always write equations for the sum and the sum of squares. This converts the problem into a quadratic equation where the missing observations are the roots.

Answer 1

Correct Answer: 44

Step 1: Understanding the Question:
We are given statistical parameters for a set of 10 observations and a subset of 8 observations. We need to find the values of the remaining two observations, $x_9$ and $x_{10}$, and calculate a linear combination of them.

Step 2: Key Formula or Approach:

• Sum of observations: $\sum x_i = n \bar{x}$.

• Sum of squares of observations: $\sum x_i^2 = n(\sigma^2 + \bar{x}^2)$.

Step 3: Detailed Explanation:

• For $n=10$, $\bar{x}=5, \sigma^2=7 \implies \sum_{i=1}^{10} x_i = 10 \times 5 = 50$.
$\sum_{i=1}^{10} x_i^2 = 10(7 + 5^2) = 10(32) = 320$.

• For the first 8 observations, $\bar{x}_1=4, \sigma_1^2=3.5 \implies \sum_{i=1}^8 x_i = 8 \times 4 = 32$.
$\sum_{i=1}^8 x_i^2 = 8(3.5 + 4^2) = 8(19.5) = 156$.

• Now, find equations for $x_9$ and $x_{10}$:
$x_9 + x_{10} = \sum_{1}^{10} x_i - \sum_{1}^8 x_i = 50 - 32 = 18$.
$x_9^2 + x_{10}^2 = \sum_{1}^{10} x_i^2 - \sum_{1}^8 x_i^2 = 320 - 156 = 164$.

• Use the identity $(x_9 + x_{10})^2 - 2x_9x_{10} = x_9^2 + x_{10}^2$:
$18^2 - 2x_9x_{10} = 164 \implies 324 - 164 = 2x_9x_{10} \implies x_9x_{10} = 80$.

• $x_9$ and $x_{10}$ are roots of $t^2 - 18t + 80 = 0 \implies (t-10)(t-8) = 0$.

• Since $x_9 < x_{10}$, we have $x_9 = 8$ and $x_{10} = 10$.

• Value $= 3(8) + 2(10) = 24 + 20 = 44$.

Step 4: Final Answer:
The value of $3x_9 + 2x_{10}$ is 44.