Question

If the mean and the variance of 6, 4, a, 8, b, 12, 10, 13 are 9 and 9.25 respectively, then \(a + b + ab\) is equal to:

• A. 105
• B. 103
• C. 100
• D. 106

Hint:

In solving problems involving variance and mean, use the formula for variance to derive the necessary equations. Then, use the system of equations to solve for unknowns such as \( a + b + ab \).

Answer 1

The Correct Option is B

Given 8 observations with two unknown values, \(a\) and \(b\), and their mean and variance, calculate the expression \( a + b + ab \).

Concept Used:

The solution utilizes the definitions of the mean and variance of a dataset.

1. Mean (\( \bar{x} \)): The mean is the sum of observations divided by the count of observations (\( n \)). \[ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \]
2. Variance (\( \sigma^2 \)): Variance is the mean of squared observations minus the square of the mean. \[ \sigma^2 = \frac{\sum_{i=1}^{n} x_i^2}{n} - (\bar{x})^2 \]

These formulas will be used to establish two equations with \(a\) and \(b\) as unknowns.

Step-by-Step Solution:

Step 1: Determine the sum \( a + b \) using the mean formula.

Observations: {6, 4, a, 8, b, 12, 10, 13}. Number of observations, \( n = 8 \). Given mean, \( \bar{x} = 9 \).

Sum of observations: \( \sum x_i = 6 + 4 + a + 8 + b + 12 + 10 + 13 = 53 + a + b \).

Applying the mean formula:

\[9 = \frac{53 + a + b}{8}\]

Solving for \( a + b \):

\[72 = 53 + a + b \implies a + b = 19\]

Step 2: Determine the sum of squares \( a^2 + b^2 \) using the variance formula.

Given variance, \( \sigma^2 = 9.25 \).

Sum of squares of observations: \( \sum x_i^2 = 6^2 + 4^2 + a^2 + 8^2 + b^2 + 12^2 + 10^2 + 13^2 = 529 + a^2 + b^2 \).

Applying the variance formula:

\[9.25 = \frac{529 + a^2 + b^2}{8} - (9)^2\]\[9.25 = \frac{529 + a^2 + b^2}{8} - 81\]

Solving for \( a^2 + b^2 \):

\[90.25 = \frac{529 + a^2 + b^2}{8} \implies 722 = 529 + a^2 + b^2 \implies a^2 + b^2 = 193\]

Step 3: Calculate the product \( ab \) using \( a + b \) and \( a^2 + b^2 \).

Using the identity \( (a+b)^2 = a^2 + b^2 + 2ab \):

\[(19)^2 = 193 + 2ab\]\[361 = 193 + 2ab \implies 2ab = 168 \implies ab = 84\]

Final Computation & Result

Calculate the expression \( a + b + ab \).

Using the derived values \( a + b = 19 \) and \( ab = 84 \):

\[a + b + ab = 19 + 84 = 103\]

The value of \( a + b + ab \) is 103.