1. Key Concepts and Formulas

This problem involves fundamental statistical concepts: the mean and variance of a set of observations, combined with essential algebraic identities.

1. Mean ($\bar{x}$): The average of a set of $n$ observations $x_1, x_2, \ldots, x_n$ is defined as the sum of all observations divided by the number of observations: $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$ From this, the sum of observations can be expressed as $\sum_{i=1}^{n} x_i = n \bar{x}$.

2. Variance ($\sigma^2$): Variance quantifies the spread of data points in a dataset. For a population of $n$ observations, the computational formula for variance is: $\sigma^2 = \frac{\sum_{i=1}^{n} x_i^2}{n} - (\bar{x})^2$ This formula is particularly useful as it allows calculation without finding deviations from the mean for each observation. It can be rearranged to find the sum of squares: $\sum_{i=1}^{n} x_i^2 = n(\sigma^2 + (\bar{x})^2)$.

3. Algebraic Identities: These are crucial for manipulating expressions involving sums, products, and differences of numbers.

   • $(a+b)^2 = a^2 + b^2 + 2ab$
   • $(a-b)^2 = a^2 + b^2 - 2ab$
   • A derived identity, very useful when you have the sum and product of two numbers and need their difference: $(a-b)^2 = (a+b)^2 - 4ab$.

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2. Step-by-Step Solution

Understanding the Problem: We are given five observations. Three are known (3, 4, 4), and two ($x_4, x_5$) are unknown. We have the mean ($\bar{x}=4$) and variance ($\sigma^2=5.20$) of all five observations. Our objective is to determine the absolute difference between the two unknown observations, i.e., $|x_4 - x_5|$.

Step 1: Use the Mean to find the sum of all observations and the sum of the unknown observations.

• Why this step? The mean formula directly relates the sum of all observations to the total count and the mean value. By calculating the total sum, we can then isolate the sum of the two unknown observations ($x_4 + x_5$) since the other three are known.

• Formula Application: Using the formula $\sum x_i = n \bar{x}$: Given $n=5$ and $\bar{x}=4$. $\sum x_i = 5 \times 4 = 20$

• Calculation for unknown observations: The sum of all observations is $x_1 + x_2 + x_3 + x_4 + x_5$. Substituting the known values: $3 + 4 + 4 + x_4 + x_5 = 20$ $11 + x_4 + x_5 = 20$ Subtracting 11 from both sides gives us the sum of the two unknown observations: $x_4 + x_5 = 9 \quad \ldots \text{(Equation 1)}$

Step 2: Use the Variance to find the sum of squares of all observations and the sum of squares of the unknown observations.

• Why this step? The variance formula involves the sum of squares of all observations. By using the given variance, mean, and number of observations, we can calculate the total sum of squares. From this, we can find the sum of squares of the two unknown observations ($x_4^2 + x_5^2$) by subtracting the squares of the known observations.

• Formula Application: Using the computational formula for variance: $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$. Given $\sigma^2=5.20$, $n=5$, and $\bar{x}=4$. $5.20 = \frac{\sum x_i^2}{5} - (4)^2$

• Calculation for sum of squares: First, calculate $(4)^2 = 16$: $5.20 = \frac{\sum x_i^2}{5} - 16$ Add 16 to both sides: $\frac{\sum x_i^2}{5} = 5.20 + 16 = 21.20$ Multiply both sides by 5 to find the sum of squares of all observations: $\sum x_i^2 = 21.20 \times 5 = 106$

• Calculation for sum of squares of unknown observations: The sum of squares of all observations is $x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2$. Substitute the known squares: $3^2=9$, $4^2=16$, $4^2=16$. $9 + 16 + 16 + x_4^2 + x_5^2 = 106$ $41 + x_4^2 + x_5^2 = 106$ Subtract 41 from both sides: $x_4^2 + x_5^2 = 65 \quad \ldots \text{(Equation 2)}$

Step 3: Find the product of the unknown observations ($x_4 x_5$).

• Why this step? We now have the sum of the unknown observations ($x_4+x_5$) from Equation 1 and the sum of their squares ($x_4^2+x_5^2$) from Equation 2. These two pieces of information, along with the algebraic identity $(a+b)^2 = a^2+b^2+2ab$, allow us to determine the product $x_4x_5$. This product is essential for finding their difference.

• Formula Application: Using the identity $(x_4+x_5)^2 = x_4^2 + x_5^2 + 2x_4x_5$. Substitute values from Equation 1 ($x_4+x_5=9$) and Equation 2 ($x_4^2+x_5^2=65$): $(9)^2 = 65 + 2x_4x_5$

• Calculation: $81 = 65 + 2x_4x_5$ Subtract 65 from both sides: $2x_4x_5 = 81 - 65 = 16$ Divide by 2: $x_4x_5 = 8$

Step 4: Find the absolute difference of the unknown observations ($|x_4 - x_5|$).

• Why this step? We have the sum ($x_4+x_5=9$) and the product ($x_4x_5=8$) of the two unknown observations. The algebraic identity $(a-b)^2 = (a+b)^2 - 4ab$ directly allows us to calculate the square of their difference. Taking the square root and then the absolute value will give us the final answer.

• Formula Application: Using the identity $(x_4-x_5)^2 = (x_4+x_5)^2 - 4x_4x_5$. Substitute the values we found: $(x_4-x_5)^2 = (9)^2 - 4(8)$

• Calculation: $(x_4-x_5)^2 = 81 - 32$ $(x_4-x_5)^2 = 49$ Taking the square root of both sides: $x_4 - x_5 = \pm\sqrt{49}$ $x_4 - x_5 = \pm 7$ The absolute value of the difference is: $|x_4 - x_5| = 7$

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3. Common Mistakes & Tips

• Typo in Variance Definition: Ensure you use the correct formula for variance. For JEE, it's typically $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$. Confusion with sample variance (using $n-1$ in the denominator) or sum of squared deviations can lead to errors.
• Algebraic Errors: Be careful with basic arithmetic and algebraic manipulations, especially when squaring numbers, distributing terms, and solving for unknowns. A small error in any step can propagate through the entire solution.
• Units and Decimals: Although not directly applicable here, always be mindful of units in physics problems and decimal places in calculations. For 5.20, ensure precise multiplication.
• Order of Operations: Follow the correct order of operations (PEMDAS/BODMAS) diligently.
• Checking for Consistency: After finding $x_4$ and $x_5$ (which would be 1 and 8 in this case), you can quickly check if they satisfy all original conditions. For example, $1+8=9$ (correct sum), $1^2+8^2 = 1+64=65$ (correct sum of squares).

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4. Summary

The problem requires a systematic application of statistical definitions for mean and variance, combined with algebraic identities. First, the mean was used to find the sum of the two unknown observations. Next, the variance formula was employed to determine the sum of squares of these observations. With both the sum and sum of squares, algebraic identities were then used to find the product of the unknown observations. Finally, another algebraic identity relating the sum, product, and difference allowed us to calculate the absolute difference between the two unknown observations. The calculations consistently led to an absolute difference of 7.

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5. Final Answer

The absolute value of the difference of the other two observations is 7.

The final answer is $\boxed{7}$, which corresponds to option (B).