If mean and variance of observations 60, 60, 44, 58, 68, 56, `alpha,beta` are 58 and 66.2 respectively then `alpha^2+beta^2` is equal to

Options

A. 6150.2
B. 7181.6
C. 9532.8
D. 3252.6

Answer

To solve the problem, we need to find the values of \( \alpha^2 + \beta^2 \) given that the mean and variance of the observations \( 60, 60, 44, 58, 68, 56, \alpha, \beta \) are 58 and 66.2 respectively.

### Step-by-step Solution:

1. **Calculate the number of observations (n)**:
\[ n = 8 \quad (\text{since there are 8 observations: } 60, 60, 44, 58, 68, 56, \alpha, \beta) \]

2. **Set up the equation for the mean**:
The mean \( \bar{x} \) is given by:
\[ \bar{x} = \frac{\text{Sum of all observations}}{n} \]
Given that the mean is 58, we can write:
\[ \frac{60 + 60 + 44 + 58 + 68 + 56 + \alpha + \beta}{8} = 58 \]

3. **Calculate the sum of the known observations**:
\[ 60 + 60 + 44 + 58 + 68 + 56 = 336 \]
Therefore, the equation becomes:
\[ \frac{336 + \alpha + \beta}{8} = 58 \]

4. **Multiply both sides by 8**:
\[ 336 + \alpha + \beta = 464 \]

5. **Solve for \( \alpha + \beta \)**:
\[ \alpha + \beta = 464 - 336 = 128 \]

6. **Set up the equation for the variance**:
The variance \( \sigma^2 \) is given by:
\[ \sigma^2 = \frac{\sum x_i^2}{n} - \bar{x}^2 \]
Given that the variance is 66.2, we can write:
\[ 66.2 = \frac{\sum x_i^2}{8} - 58^2 \]

7. **Calculate \( 58^2 \)**:
\[ 58^2 = 3364 \]
Thus, the equation becomes:
\[ 66.2 = \frac{\sum x_i^2}{8} - 3364 \]

8. **Multiply both sides by 8**:
\[ 529.6 = \sum x_i^2 - 3364 \]

9. **Solve for \( \sum x_i^2 \)**:
\[ \sum x_i^2 = 529.6 + 3364 = 3893.6 \]

10. **Calculate the sum of squares of the known observations**:
\[ 60^2 + 60^2 + 44^2 + 58^2 + 68^2 + 56^2 = 3600 + 3600 + 1936 + 3364 + 4624 + 3136 = 20260 \]

11. **Set up the equation for the sum of squares**:
\[ 20260 + \alpha^2 + \beta^2 = 3893.6 \]

12. **Solve for \( \alpha^2 + \beta^2 \)**:
\[ \alpha^2 + \beta^2 = 3893.6 - 20260 = 7181.6 \]

Thus, the value of \( \alpha^2 + \beta^2 \) is \( 7181.6 \).

### Final Answer:
\[ \alpha^2 + \beta^2 = 7181.6 \]