Question

Consider 10 observations $x_1, x_2, \ldots, x_{10}$ such that \[\sum_{i=1}^{10} (x_i - \alpha) = 2 \quad \text{and} \quad \sum_{i=1}^{10} (x_i - \beta)^2 = 40,\]where $\alpha, \beta$ are positive integers. Let the mean and the variance of the observations be $\frac{6}{5}$ and $\frac{84}{25}$, respectively. The value of $\frac{\beta}{\alpha}$ is equal to:

• A. 2
• B. $\frac{3}{2}$
• C. $\frac{5}{2}$
• D. 1

Answer 1

The Correct Option is A

To solve the given problem, we need to decipher the information about the 10 observations \(x_1, x_2, \ldots, x_{10}\) in terms of their mean and variance, and how they relate to \(\alpha\) and \(\beta\).

1. We are given: 

\[\sum_{i=1}^{10} (x_i - \alpha) = 2\]

1.  and 

\[\sum_{i=1}^{10} (x_i - \beta)^2 = 40\]

1. .
2. The mean of the observations is 

\[\bar{x} = \frac{6}{5}\]

1. . We know: 

\[\bar{x} = \frac{1}{10} \sum_{i=1}^{10} x_i = \frac{6}{5}\]

1.  which gives 

\[\sum_{i=1}^{10} x_i = 12\]

1. .
2. According to the formula for mean: 

\[\sum_{i=1}^{10} (x_i - \alpha) = \sum_{i=1}^{10} x_i - 10\alpha = 2\]

1. . Substituting for \(\sum_{i=1}^{10} x_i = 12\): 

\[12 - 10\alpha = 2 \Rightarrow \alpha = 1\]

1. .
2. The variance of the observations is: 

\[\text{Variance} = \frac{84}{25}\]

1. . For the variance: 

\[\frac{1}{10} \sum_{i=1}^{10} (x_i - \bar{x})^2 = \frac{84}{25}\]

1. , leading to: 

\[\sum_{i=1}^{10} (x_i - \bar{x})^2 = \frac{84}{2.5} = 33.6\]

1. .
2. We are also given: 

\[\sum_{i=1}^{10} (x_i - \beta)^2 = 40\]

1. . Hence: 

\[10 \times \text{Variance for } (\beta) = 40\]

1. , which simplifies to: 

\[40 = 33.6 + 10 (\beta - \bar{x})^2\]

1. . Solving: 

\[10 (\beta - \frac{6}{5})^2 = 6.4 \Rightarrow (\beta - \frac{6}{5})^2 = 0.64\]

1.  

\[\Rightarrow \beta - \frac{6}{5} = \pm \frac{4}{5}\]

1. .
2. This yields: 

\[\beta = \frac{6}{5} \pm \frac{4}{5}\]

1. . We conclude: 

\[\beta = 2 \text{ (since both \(\beta\) and \(\alpha\) are integers)}\]

1. .
2. Finally, the ratio is: 

\[\frac{\beta}{\alpha} = \frac{2}{1} = 2\]

1. .

Hence, the correct answer is: 2.

Answer 2

We are given:

$\sum_{i=1}^{10} X_i - 10A = 2 \implies \sum_{i=1}^{10} X_i = 10A + 2$.

$\sum_{i=1}^{10} X_i - 10B = 40 \implies \sum_{i=1}^{10} X_i = 10B + 40$.

Equating both expressions for $\sum_{i=1}^{10} X_i$, we get:

$10A + 2 = 10B + 40 \implies 10A - 10B = 38 \implies A - B = 3.8$.

Since A and B are integers, $A = 4$ and $B = 2$.

Thus, $B = 2$.