Key Concepts and Formulas

To solve this problem, we'll rely on the fundamental definitions and computational formulas for the mean and variance of a set of observations.

• Mean ($\overline{x}$): The mean of $n$ observations $x_1, x_2, \dots, x_n$ is the sum of the observations divided by the number of observations. $\overline{x} = \frac{\sum_{i=1}^{n} x_i}{n}$ This formula allows us to easily find the sum of observations ($\sum x_i$) if we know the mean and the number of observations ($n$).

• Variance ($\sigma^2$): The variance measures the spread of data points around the mean. For computational efficiency, we often use the formula: $\sigma^2 = \frac{\sum_{i=1}^{n} x_i^2}{n} - (\overline{x})^2$ This formula is extremely useful as it connects the variance, the mean, and the sum of squares of the observations ($\sum x_i^2$). If any two are known, the third can be determined.

Step-by-Step Solution

Our goal is to find the value of $(4a + x_5)$. We are given information about a set of 5 observations and a subset of its first 4 observations. We will systematically use the mean and variance formulas to extract sums and sums of squares, eventually solving for $x_5$ and $a$.

Step 1: Analyze the Data for 5 Observations

We are given the following for the full set of 5 observations ($x_1, x_2, x_3, x_4, x_5$):

• Mean ($\overline{x}_5$) = $\frac{24}{5}$
• Variance ($\sigma_5^2$) = $\frac{194}{25}$

1.1. Calculate the Sum of the 5 Observations ($\sum_{i=1}^{5} x_i$)

• Reasoning: Using the mean formula, we can find the total sum of all observations, which will be essential for finding $x_5$ later.
• Applying the formula: $\overline{x}_5 = \frac{\sum_{i=1}^{5} x_i}{5}$ $\frac{24}{5} = \frac{\sum_{i=1}^{5} x_i}{5}$
• Calculation: Multiplying both sides by 5, we get: $\sum_{i=1}^{5} x_i = x_1 + x_2 + x_3 + x_4 + x_5 = 24 \quad \text{(Equation 1)}$

1.2. Calculate the Sum of Squares of the 5 Observations ($\sum_{i=1}^{5} x_i^2$)

• Reasoning: The variance formula involves the sum of squares. Calculating this sum will be crucial for finding the variance 'a' of the first 4 observations later.
• Applying the formula: $\sigma_5^2 = \frac{\sum_{i=1}^{5} x_i^2}{5} - (\overline{x}_5)^2$ $\frac{194}{25} = \frac{\sum_{i=1}^{5} x_i^2}{5} - \left(\frac{24}{5}\right)^2$
• Calculation: First, calculate the square of the mean: $\left(\frac{24}{5}\right)^2 = \frac{576}{25}$. Substitute this back into the equation: $\frac{194}{25} = \frac{\sum_{i=1}^{5} x_i^2}{5} - \frac{576}{25}$ Isolate the term with the sum of squares: $\frac{\sum_{i=1}^{5} x_i^2}{5} = \frac{194}{25} + \frac{576}{25}$ $\frac{\sum_{i=1}^{5} x_i^2}{5} = \frac{194 + 576}{25} = \frac{770}{25}$ Now, multiply both sides by 5: $\sum_{i=1}^{5} x_i^2 = \frac{770}{25} \times 5 = \frac{770}{5} = 154 \quad \text{(Equation 2)}$

Step 2: Analyze the Data for the First 4 Observations

We are given the following for the first 4 observations ($x_1, x_2, x_3, x_4$):

• Mean ($\overline{x}_4$) = $\frac{7}{2}$
• Variance ($\sigma_4^2$) = $a$

2.1. Calculate the Sum of the First 4 Observations ($\sum_{i=1}^{4} x_i$)

• Reasoning: Similar to Step 1.1, finding this sum is essential for isolating $x_5$ when combined with the total sum of 5 observations.
• Applying the formula: $\overline{x}_4 = \frac{\sum_{i=1}^{4} x_i}{4}$ $\frac{7}{2} = \frac{\sum_{i=1}^{4} x_i}{4}$
• Calculation: Multiplying both sides by 4: $\sum_{i=1}^{4} x_i = \frac{7}{2} \times 4 = 14 \quad \text{(Equation 3)}$

2.2. Express the Sum of Squares of the First 4 Observations ($\sum_{i=1}^{4} x_i^2$) in terms of 'a'

• Reasoning: We need to find 'a', the variance of these 4 observations. Using the variance formula, we can establish a relationship between 'a' and the sum of squares of these 4 observations. This will be used to solve for 'a' after we find $x_5$.
• Applying the formula: $\sigma_4^2 = \frac{\sum_{i=1}^{4} x_i^2}{4} - (\overline{x}_4)^2$ $a = \frac{\sum_{i=1}^{4} x_i^2}{4} - \left(\frac{7}{2}\right)^2$
• Calculation: First, calculate the square of the mean: $\left(\frac{7}{2}\right)^2 = \frac{49}{4}$. Substitute this back into the equation: $a = \frac{\sum_{i=1}^{4} x_i^2}{4} - \frac{49}{4}$ Isolate the term with the sum of squares: $\frac{\sum_{i=1}^{4} x_i^2}{4} = a + \frac{49}{4}$ Multiply both sides by 4: $\sum_{i=1}^{4} x_i^2 = 4a + 49 \quad \text{(Equation 4)}$

Step 3: Determine the Values of $x_5$ and 'a'

Now we have established relationships that allow us to find the unknown values.

3.1. Calculate $x_5$

• Reasoning: The sum of all 5 observations is the sum of the first 4 observations plus the fifth observation ($x_5$). By subtracting the sum of the first 4 from the total sum, we can find $x_5$.
• Applying the concept: $\sum_{i=1}^{5} x_i = \sum_{i=1}^{4} x_i + x_5$
• Calculation: Substitute values from Equation 1 and Equation 3: $24 = 14 + x_5$ $x_5 = 24 - 14$ $x_5 = 10$

3.2. Calculate 'a'

• Reasoning: Similarly, the sum of squares of all 5 observations is the sum of squares of the first 4 observations plus the square of the fifth observation ($x_5^2$). Using this relationship, we can substitute our known values to solve for 'a'.
• Applying the concept: $\sum_{i=1}^{5} x_i^2 = \sum_{i=1}^{4} x_i^2 + x_5^2$
• Calculation: Substitute values from Equation 2, Equation 4, and the calculated $x_5$: $154 = (4a + 49) + (10)^2$ $154 = 4a + 49 + 100$ $154 = 4a + 149$ Now, isolate $4a$: $4a = 154 - 149$ $4a = 5$ (Note: We don't need to find 'a' itself, as the final expression is $4a + x_5$.)

Step 4: Calculate the Final Expression ($4a + x_5$)

• Reasoning: We have successfully found the value of $4a$ and $x_5$. Now, we simply substitute these values into the required expression.
• Calculation: $4a + x_5 = 5 + 10$ $4a + x_5 = 15$

Common Mistakes & Tips

• Distinguish between sums: Always be clear whether you are calculating the sum of observations ($\sum x_i$) or the sum of squares of observations ($\sum x_i^2$). A common error is to mix these up.
• Correct Variance Formula: Ensure you use the correct computational formula for variance: $\sigma^2 = \frac{\sum x_i^2}{n} - (\overline{x})^2$. An alternative definition $\sigma^2 = \frac{\sum (x_i - \overline{x})^2}{n}$ is less efficient for calculations involving large datasets or when only sums are known.
• Systematic Approach: Break down the problem into smaller, manageable steps. First, analyze the full dataset, then the subset, and finally combine the information. This prevents errors and clarifies the path to the solution.

Summary

We systematically utilized the definitions of mean and variance for both the set of 5 observations and its subset of 4 observations. By first calculating the sum and sum of squares for the 5 observations, and then for the first 4 observations, we were able to isolate the value of the fifth observation, $x_5 = 10$. Subsequently, we used the sum of squares relationship to determine $4a = 5$. Finally, we combined these values to find the required expression $(4a + x_5)$, which resulted in 15.

The final answer is $\boxed{15}$, which corresponds to option (B).