Question

The mean and variance of a data of 10 observations are 10 and 2, respectively. If an observation $\alpha$ in this data is replaced by $\beta$, then the mean and variance become $10.1$ and $1.99$, respectively. Then $\alpha+\beta$ equals

• A. 10
• B. 15
• C. 20
• D. 5

Hint:

When one observation is changed, use changes in mean and variance to form equations involving the old and new values.

Answer 1

The Correct Option is A

To solve this problem, we need to work through the mean and variance formulas while understanding how replacing one observation affects them. Let's break it down step-by-step:

Given: The original mean and variance for 10 observations are 10 and 2, respectively. An observation \(\alpha\)is replaced by \(\beta\), resulting in a new mean of 10.1 and a new variance of 1.99.

Step 1: Calculate the Original Total

The mean of the original observations is 10. With 10 observations:

\(S = 10 \times 10 = 100\)

Here, \(S\)is the sum of the original 10 observations.

Step 2: Calculate the New Total

After replacing \(\alpha\)with \(\beta\), the new mean becomes 10.1:

\(S - \alpha + \beta = 10.1 \times 10 = 101\)

Thus,

\(100 - \alpha + \beta = 101\)

\(\beta - \alpha = 1 \quad \text{(Equation 1)}\)

Step 3: Calculate the Variance Relationship

The variance formula is:

\(\text{Variance} = \dfrac{\sum{(x_i - \text{mean})^2}}{n}\)

Original variance:

\(\dfrac{\sum(x_i - 10)^2}{10} = 2 \implies \sum(x_i - 10)^2 = 20\)

New variance:

\(\dfrac{\sum(x_i - 10.1)^2}{10} = 1.99 \implies \sum(x_i - 10.1)^2 = 19.9\)

The differences can be written as:

\((100 - \alpha)^2 - 10 \times (10 - 10.1)^2 = \beta^2 - \alpha^2 \implies 19.9 = 20 + \beta^2 - \alpha^2\)

\(\beta^2 - \alpha^2 = -0.1 \quad \text{(Equation 2)}\)

Step 4: Solve Equations 1 and 2

We have:

• \(\beta - \alpha = 1\)
• \(\beta^2 - \alpha^2 = -0.1\)

From the identity \((a - b)(a + b) = a^2 - b^2\), we know:

\((\beta - \alpha)(\beta + \alpha) = -0.1\)

\(1 \cdot (\beta + \alpha) = -0.1 \implies \beta + \alpha = -0.1\)

Thus,

\(\beta + \alpha = 10\)

Conclusion: Based on the calculations, \(\alpha + \beta = 10\).

The correct answer is: 10