Mean Variance Data Change: Alpha Plus Beta Calculation

This solution details the calculation of $\alpha + \beta$ based on changes in the mean and variance of a dataset after an observation is replaced.

Initial Data Conditions

We are given a dataset with 10 observations ($n=10$).

• Initial Mean ($\bar{x}$) = 10
• Initial Variance ($\sigma^2$) = 2

Calculating Initial Sum and Sum of Squares

The sum of the original 10 observations ($\sum x_i$) is calculated using the formula $\sum x_i = n \times \bar{x}$.

Calculation: $\sum x_i = 10 \times 10 = 100$.

The variance formula is $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$. We use this to find the sum of squares ($\sum x_i^2$).

Calculation: $2 = \frac{\sum x_i^2}{10} - (10)^2$ $2 = \frac{\sum x_i^2}{10} - 100$ $102 = \frac{\sum x_i^2}{10}$ $\sum x_i^2 = 1020$

Changes in Data

One observation, $\alpha$, is replaced by $\beta$. The number of observations remains $n=10$.

• New Mean ($\bar{x}'$) = 10.1
• New Variance ($\sigma'^2$) = 1.99

Calculating New Sum and Deriving $\beta - \alpha$

The new sum of observations ($\sum x_i'$) is $\sum x_i' = (\sum x_i - \alpha) + \beta$.

Using the new mean, $\sum x_i' = n \times \bar{x}' = 10 \times 10.1 = 101$.

Equating the two expressions for the new sum:

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

This gives us our first equation:

$\beta - \alpha = 1$ (Equation 1)

Calculating New Sum of Squares and Deriving $\beta^2 - \alpha^2$

The new sum of squares ($\sum x_i'^2$) is $\sum x_i'^2 = (\sum x_i^2 - \alpha^2) + \beta^2$.

Using the new variance formula: $\sigma'^2 = \frac{\sum x_i'^2}{n} - (\bar{x}')^2$.

Substituting the known values:

$1.99 = \frac{1020 - \alpha^2 + \beta^2}{10} - (10.1)^2$ $1.99 = \frac{1020 - \alpha^2 + \beta^2}{10} - 102.01$ $1.99 + 102.01 = \frac{1020 - \alpha^2 + \beta^2}{10}$ $104 = \frac{1020 - \alpha^2 + \beta^2}{10}$ $1040 = 1020 - \alpha^2 + \beta^2$

Rearranging gives our second equation:

$\beta^2 - \alpha^2 = 1040 - 1020$

$\beta^2 - \alpha^2 = 20$ (Equation 2)

Solving for $\alpha + \beta$

We have two equations:

1. $\beta - \alpha = 1$
2. $\beta^2 - \alpha^2 = 20$

Factorizing Equation 2 using the difference of squares formula ($a^2 - b^2 = (a-b)(a+b)$):

$(\beta - \alpha)(\beta + \alpha) = 20$.

Substitute the value of $(\beta - \alpha)$ from Equation 1 into the factorized Equation 2:

$(1)(\beta + \alpha) = 20$.

Therefore:

$\alpha + \beta = 20$.

Final Result

The value of $\alpha + \beta$ is 20.