Key Concepts and Formulas

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

2. Variance ($\sigma^2$): A measure of how spread out the numbers in a data set are from their mean. The computational formula for variance is particularly useful: $\sigma^2 = \frac{\sum_{i=1}^{n} x_i^2}{n} - (\bar{x})^2$ Here, $\sum_{i=1}^{n} x_i^2$ is the sum of the squares of all observations. This formula allows us to find the sum of squares directly.

3. Algebraic Identity: The square of a sum of two numbers: $(a+b)^2 = a^2 + b^2 + 2ab$ This identity is crucial for finding the product ($ab$) when the sum ($a+b$) and sum of squares ($a^2+b^2$) are known, which then helps in determining the individual numbers.

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Step-by-Step Solution

We are given:

• Number of observations ($n$) = $5$
• Mean ($\bar{x}$) = $5$
• Variance ($\sigma^2$) = $9.20$
• Three observations are $1, 3, 8$. Let the other two unknown observations be $a$ and $b$. The complete set of observations is $1, 3, 8, a, b$.

Step 1: Using the Mean to Determine the Sum of Unknown Observations

The definition of the mean states that the sum of all observations divided by the number of observations equals the mean. $\bar{x} = \frac{1 + 3 + 8 + a + b}{5}$ Substitute the given mean value: $5 = \frac{12 + a + b}{5}$ Multiply both sides by $5$ to find the total sum of observations: $5 \times 5 = 12 + a + b$ $25 = 12 + a + b$ Now, isolate the sum of the unknown observations: $a + b = 25 - 12$ $\boxed{a + b = 13} \quad \ldots (1)$ Explanation: This step uses the fundamental definition of the mean to establish a linear relationship between the two unknown observations. Knowing the mean and the total number of observations allows us to find the sum of all observations, and by subtracting the sum of the known observations, we deduce the sum of the unknown ones.

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Step 2: Using the Variance to Determine the Sum of Squares of Unknown Observations

We use the computational formula for variance: $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$ Substitute the given variance ($\sigma^2 = 9.20$) and mean ($\bar{x} = 5$): $9.20 = \frac{(1^2 + 3^2 + 8^2 + a^2 + b^2)}{5} - (5)^2$ Calculate the squares of the known observations and the mean:

• $1^2 = 1$
• $3^2 = 9$
• $8^2 = 64$
• $(\bar{x})^2 = 5^2 = 25$ Substitute these values back into the variance equation: $9.20 = \frac{1 + 9 + 64 + a^2 + b^2}{5} - 25$ $9.20 = \frac{74 + a^2 + b^2}{5} - 25$ Add $25$ to both sides to isolate the fraction term: $9.20 + 25 = \frac{74 + a^2 + b^2}{5}$ $31.8 = \frac{74 + a^2 + b^2}{5}$ Multiply both sides by $5$: $31.8 \times 5 = 74 + a^2 + b^2$ $159 = 74 + a^2 + b^2$ Finally, isolate the sum of the squares of the unknown observations: $a^2 + b^2 = 159 - 74$ $\boxed{a^2 + b^2 = 85} \quad \ldots (2)$ Explanation: The variance provides information about the sum of the squares of all observations. By plugging in the given variance, mean, and the sum of squares of the known observations, we establish a second relationship, this time involving the squares of $a$ and $b$.

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Step 3: Solving for the Unknown Observations using Algebraic Identities

Now we have a system of two equations with two unknowns:

1. $a + b = 13$
2. $a^2 + b^2 = 85$

We use the algebraic identity $(a+b)^2 = a^2 + b^2 + 2ab$ to find the product $ab$: Substitute equations (1) and (2) into the identity: $(13)^2 = 85 + 2ab$ $169 = 85 + 2ab$ Solve for $2ab$: $2ab = 169 - 85$ $2ab = 84$ $\boxed{ab = 42}$ Now we have the sum ($a+b=13$) and the product ($ab=42$) of the two unknown observations. These values are the roots of a quadratic equation of the form $t^2 - (a+b)t + ab = 0$. $t^2 - 13t + 42 = 0$ To find the values of $t$, we factorize the quadratic equation. We look for two numbers that multiply to $42$ and add up to $-13$. These numbers are $-6$ and $-7$. $(t - 6)(t - 7) = 0$ This yields two possible values for $t$: $t = 6 \quad \text{or} \quad t = 7$ Therefore, the two unknown observations are $6$ and $7$. Explanation: This step combines the results from the mean and variance calculations using a fundamental algebraic identity. By finding both the sum and product of the unknown numbers, we can construct and solve a quadratic equation whose roots are those numbers, thereby revealing their individual values.

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Step 4: Determining the Ratio of the Other Two Observations

The two other observations are $6$ and $7$. The question asks for "a ratio of other two observations". This means either $6:7$ or $7:6$ would be a valid ratio. Checking the given options: (A) 6 : 7 (B) 10 : 3 (C) 4 : 9 (D) 5 : 8

Our result $6:7$ matches option (A).

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Common Mistakes & Tips

• Arithmetic Precision: Even small calculation errors (especially with decimals or squares) can lead to an incorrect final answer. Always double-check your arithmetic.
• Correct Variance Formula: Ensure you use the appropriate variance formula. The computational form $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$ is generally more efficient for problems involving unknown values.
• Algebraic Manipulation: Be proficient with algebraic identities (like $(a+b)^2 = a^2+b^2+2ab$) and forming/solving quadratic equations. These are critical tools in such problems.

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Summary

This problem demonstrates a common strategy in statistics: using the definitions of mean and variance to create a system of algebraic equations. By first finding the sum of the unknown observations ($a+b=13$) from the mean, and then their sum of squares ($a^2+b^2=85$) from the variance, we could use an algebraic identity to determine their product ($ab=42$). With the sum and product, we formed a quadratic equation, whose roots revealed the individual values of the unknown observations as $6$ and $7$. Finally, we expressed these as a ratio.

The final answer is $\boxed{\text{6:7}}$, which corresponds to option (A).