1. Key Concepts and Formulas

To tackle this problem effectively, we'll rely on these fundamental statistical definitions and properties for a set of $n$ observations $x_1, x_2, \ldots, x_n$:

• Mean ($\bar{x}$): The average of the observations. $\bar{x} = \frac{\sum_{i=1}^n x_i}{n}$
• Variance ($\sigma^2$): A measure of data spread. The computational formula is often most useful: $\sigma^2 = \frac{\sum_{i=1}^n x_i^2}{n} - (\bar{x})^2$
• Properties of Mean and Variance under Linear Transformation: If new observations $y_i$ are related to $x_i$ by $y_i = ax_i + b$ (where $a, b$ are constants), then:
  • Mean of $y_i$: $\bar{y} = a\bar{x} + b$
  • Variance of $y_i$: $\sigma_y^2 = a^2 \sigma_x^2$ (Note: adding a constant $b$ does not affect variance).

2. Step-by-Step Solution

We are given information about 10 observations ($n=10$) and need to find the value of $\frac{\beta\mu}{\sigma^2}$.

Step 1: Calculate the Mean ($\bar{x}$) of the Original Observations

• What we are doing: We use the first given summation to find the sum of $x_i$ and then the mean $\bar{x}$.
• Why we are doing this: The mean $\bar{x}$ is a fundamental parameter required for subsequent calculations, including the variance and the mean of the transformed data. We are given: $\sum_{i=1}^{10} (x_i - 2) = 30$ Expand the summation: $\sum_{i=1}^{10} x_i - \sum_{i=1}^{10} 2 = 30$ Since $\sum_{i=1}^{10} 2 = 10 \times 2 = 20$: $\sum_{i=1}^{10} x_i - 20 = 30$ $\sum_{i=1}^{10} x_i = 50$ Now, calculate the mean $\bar{x}$: $\bar{x} = \frac{\sum_{i=1}^{10} x_i}{n} = \frac{50}{10} = 5$ So, the mean of the original observations is $\bar{x} = 5$.

Step 2: Calculate the Sum of Squares ($\sum x_i^2$) of the Original Observations

• What we are doing: We use the given variance of $x_i$ and the calculated mean $\bar{x}$ to find $\sum x_i^2$.
• Why we are doing this: The value of $\sum x_i^2$ is necessary to solve for $\beta$ in a later step. We are given that the variance of $x_i$, denoted as $\sigma_x^2$, is $\frac{4}{5}$. Using the computational formula for variance: $\sigma_x^2 = \frac{\sum_{i=1}^{10} x_i^2}{n} - (\bar{x})^2$ Substitute the known values $\sigma_x^2 = \frac{4}{5}$, $n=10$, and $\bar{x}=5$: $\frac{4}{5} = \frac{\sum_{i=1}^{10} x_i^2}{10} - (5)^2$ $\frac{4}{5} = \frac{\sum_{i=1}^{10} x_i^2}{10} - 25$ Rearrange to solve for $\frac{\sum x_i^2}{10}$: $\frac{\sum_{i=1}^{10} x_i^2}{10} = 25 + \frac{4}{5} = \frac{125}{5} + \frac{4}{5} = \frac{129}{5}$ Multiply by 10 to find $\sum x_i^2$: $\sum_{i=1}^{10} x_i^2 = 10 \times \frac{129}{5} = 2 \times 129 = 258$ So, $\sum_{i=1}^{10} x_i^2 = 258$.

Step 3: Determine the Value of $\beta$

• What we are doing: We use the second given summation, along with $\sum x_i$ and $\sum x_i^2$, to form and solve a quadratic equation for $\beta$.
• Why we are doing this: $\beta$ is a crucial constant needed for the definition of the transformed observations. We are given: $\sum_{i=1}^{10} (x_i - \beta)^2 = 98$ Expand the term $(x_i - \beta)^2 = x_i^2 - 2\beta x_i + \beta^2$: $\sum_{i=1}^{10} (x_i^2 - 2\beta x_i + \beta^2) = 98$ Distribute the summation: $\sum_{i=1}^{10} x_i^2 - 2\beta \sum_{i=1}^{10} x_i + \sum_{i=1}^{10} \beta^2 = 98$ Substitute the values $\sum x_i^2 = 258$, $\sum x_i = 50$, and $\sum_{i=1}^{10} \beta^2 = 10\beta^2$: $258 - 2\beta(50) + 10\beta^2 = 98$ $258 - 100\beta + 10\beta^2 = 98$ Rearrange into a standard quadratic equation: $10\beta^2 - 100\beta + 258 - 98 = 0$ $10\beta^2 - 100\beta + 160 = 0$ Divide by 10 to simplify: $\beta^2 - 10\beta + 16 = 0$ Factor the quadratic equation: $(\beta - 2)(\beta - 8) = 0$ This yields two possible values for $\beta$: $\beta = 2$ or $\beta = 8$. The problem states that $\beta > 2$. Therefore, we choose: $\beta = 8$

Step 4: Calculate the Mean ($\mu$) of the Transformed Observations

• What we are doing: We first simplify the expression for the new observations and then apply the linear transformation property for the mean.
• Why we are doing this: $\mu$ is one of the components required for the final expression. The new observations are $Y_i = 2(x_i - 1) + 4\beta$. Substitute $\beta=8$: $Y_i = 2(x_i - 1) + 4(8)$ $Y_i = 2x_i - 2 + 32$ $Y_i = 2x_i + 30$ This is in the form $Y_i = ax_i + b$, where $a=2$ and $b=30$. Using the property for the mean of transformed data $\mu = a\bar{x} + b$: $\mu = 2(\bar{x}) + 30$ Substitute $\bar{x}=5$: $\mu = 2(5) + 30 = 10 + 30 = 40$ So, the mean of the transformed observations is $\mu = 40$.

Step 5: Calculate the Variance ($\sigma^2$) of the Transformed Observations

• What we are doing: We apply the linear transformation property for variance.
• Why we are doing this: $\sigma^2$ is the final component needed for the problem's expression. Using the property for the variance of transformed data $\sigma^2 = a^2 \sigma_x^2$: $\sigma^2 = (2)^2 \times \sigma_x^2$ Substitute $a=2$ and the original variance $\sigma_x^2 = \frac{4}{5}$: $\sigma^2 = 4 \times \frac{4}{5} = \frac{16}{5}$ So, the variance of the transformed observations is $\sigma^2 = \frac{16}{5}$.

Step 6: Calculate the Final Expression $\frac{\beta\mu}{\sigma^2}$

• What we are doing: We substitute the values of $\beta$, $\mu$, and $\sigma^2$ we found into the target expression.
• Why we are doing this: This is the final quantity requested by the problem. Substitute $\beta = 8$, $\mu = 40$, and $\sigma^2 = \frac{16}{5}$: $\frac{\beta\mu}{\sigma^2} = \frac{8 \times 40}{\frac{16}{5}}$ $\frac{\beta\mu}{\sigma^2} = \frac{320}{\frac{16}{5}}$ Multiply by the reciprocal of the denominator: $\frac{\beta\mu}{\sigma^2} = 320 \times \frac{5}{16}$ $\frac{\beta\mu}{\sigma^2} = (320 \div 16) \times 5$ $\frac{\beta\mu}{\sigma^2} = 20 \times 5$ $\frac{\beta\mu}{\sigma^2} = 100$

3. Common Mistakes & Tips

• Linear Transformation for Variance: Remember that adding a constant ($b$) to each observation does NOT affect the variance. Only the scaling factor ($a$) matters, and it's squared ($a^2$). A common mistake is to add $b$ or $b^2$ to the variance.
• Summation of Constants: When summing a constant $C$ for $n$ terms, $\sum_{i=1}^n C = nC$. Forgetting this can lead to errors in expanding summations.
• Using Conditions: Always pay attention to conditions provided in the problem, such as $\beta > 2$. These are crucial for selecting the correct solution from multiple possibilities.

4. Summary

This problem required a systematic application of statistical definitions and properties. We first extracted the mean ($\bar{x}$) and sum of squares ($\sum x_i^2$) for the original observations using the given summations and variance. Then, we used another summation to form and solve a quadratic equation for $\beta$, carefully applying the given condition $\beta > 2$. Finally, we simplified the linear transformation for the new observations and applied the properties of mean and variance under linear transformations to find $\mu$ and $\sigma^2$, leading to the final result of 100.

The final answer is $\boxed{100}$, which corresponds to option (A).