1. Key Concepts and Formulas

For a set of $n$ observations $x_1, x_2, \ldots, x_n$:

• Mean ($\bar{x}$): The arithmetic average of all observations. It is calculated as the sum of all observations divided by the number of observations. $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$

• Variance ($\sigma^2$): A measure of the spread or dispersion of the data points around the mean. It quantifies how much the individual observations deviate from the mean. The most fundamental definition is the average of the squared differences from the mean: $\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}$ An alternative computational formula for variance, which is often useful, is: $\sigma^2 = \frac{\sum_{i=1}^{n} x_i^2}{n} - (\bar{x})^2$ For this problem, since the mean is an integer, the first formula (sum of squared deviations from the mean) will be more direct and often simpler to calculate.

2. Step-by-Step Solution

We are given 6 observations: $a, b, 68, 44, 48, 60$. The number of observations ($n$) = $6$. The mean ($\bar{x}$) = $55$. The variance ($\sigma^2$) = $194$. We are also given the condition $a > b$. Our goal is to find the value of $a+3b$.

Step 1: Formulating the First Equation using the Mean

• What we are doing: We use the given mean to establish a linear relationship between the unknown variables $a$ and $b$.
• Why this step: The mean formula provides the most straightforward initial connection between the sum of observations and their average, allowing us to form our first equation.

Using the mean formula: $\bar{x} = \frac{\sum x_i}{n}$ Substitute the given values: $55 = \frac{a+b+68+44+48+60}{6}$ First, sum the known constant observations: $68 + 44 + 48 + 60 = 220$. Substitute this sum back into the equation: $55 = \frac{a+b+220}{6}$ Multiply both sides by 6: $55 \times 6 = a+b+220$ $330 = a+b+220$ Isolate $a+b$ by subtracting 220 from both sides: $a+b = 330 - 220$ $a+b = 110 \quad \ldots (1)$ This is our first equation relating $a$ and $b$.

Step 2: Formulating the Second Equation using the Variance

• What we are doing: We use the given variance to establish a second, quadratic relationship between $a$ and $b$.
• Why this step: The variance formula provides an independent equation involving $a$ and $b$, which, when solved simultaneously with the mean equation, will allow us to find the specific values of $a$ and $b$. We will use the definition involving deviations from the mean, as $\bar{x}=55$ is an integer, simplifying calculations.

Using the variance formula $\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}$: Substitute the given variance $\sigma^2 = 194$ and mean $\bar{x} = 55$: $194 = \frac{(a-55)^2 + (b-55)^2 + (68-55)^2 + (44-55)^2 + (48-55)^2 + (60-55)^2}{6}$ Now, calculate the squared deviations for the known observations:

• $(68-55)^2 = (13)^2 = 169$
• $(44-55)^2 = (-11)^2 = 121$
• $(48-55)^2 = (-7)^2 = 49$
• $(60-55)^2 = (5)^2 = 25$ Substitute these values into the variance equation: $194 = \frac{(a-55)^2 + (b-55)^2 + 169 + 121 + 49 + 25}{6}$ Sum the constant terms in the numerator: $169 + 121 + 49 + 25 = 364$. $194 = \frac{(a-55)^2 + (b-55)^2 + 364}{6}$ Multiply both sides by 6: $194 \times 6 = (a-55)^2 + (b-55)^2 + 364$ $1164 = (a-55)^2 + (b-55)^2 + 364$ Isolate the terms involving $a$ and $b$: $(a-55)^2 + (b-55)^2 = 1164 - 364$ $(a-55)^2 + (b-55)^2 = 800 \quad \ldots (2)$ This is our second equation.

Step 3: Solving the System of Equations to Find $a$ and $b$

• What we are doing: We solve the system formed by equation (1) and equation (2) to determine the unique values of $a$ and $b$.
• Why this step: We now have two independent equations with two unknowns. Solving them is necessary to find the specific values of $a$ and $b$ required for the final calculation. A substitution strategy will simplify the algebra significantly.

Our system of equations is:

1. $a+b = 110$
2. $(a-55)^2 + (b-55)^2 = 800$

To simplify equation (2), let's introduce new variables representing the deviations from the mean: Let $A = a-55$ and $B = b-55$. This implies $a = A+55$ and $b = B+55$.

Substitute these expressions for $a$ and $b$ into equation (1): $(A+55) + (B+55) = 110$ $A+B+110 = 110$ Subtracting 110 from both sides gives: $A+B = 0 \quad \ldots (3)$ From this, we get $B = -A$. This shows that the deviations of $a$ and $b$ from the mean are equal in magnitude but opposite in sign.

Now, substitute $A$ and $B$ into equation (2): $A^2 + B^2 = 800 \quad \ldots (4)$ Substitute $B = -A$ from equation (3) into equation (4): $A^2 + (-A)^2 = 800$ $A^2 + A^2 = 800$ $2A^2 = 800$ $A^2 = 400$ Taking the square root of both sides, we get two possible values for $A$: $A = \pm \sqrt{400} \Rightarrow A = \pm 20$

Now we consider the two cases for $A$: Case 1: $A = 20$

• If $A=20$, then $a-55=20 \Rightarrow a = 75$.
• Since $B=-A$, then $B=-20$. So $b-55=-20 \Rightarrow b = 35$. This gives the pair $(a,b) = (75, 35)$.

Case 2: $A = -20$

• If $A=-20$, then $a-55=-20 \Rightarrow a = 35$.
• Since $B=-A$, then $B=20$. So $b-55=20 \Rightarrow b = 75$. This gives the pair $(a,b) = (35, 75)$.

Step 4: Applying the Condition $a>b$ and Calculating $a+3b$

• What we are doing: We use the given condition $a>b$ to select the correct pair of values for $a$ and $b$, and then substitute these values into the expression $a+3b$.
• Why this step: The condition $a>b$ is crucial for uniquely determining $a$ and $b$ from the two possible pairs derived in the previous step. Once $a$ and $b$ are uniquely identified, we can calculate the final required expression.

Let's check each pair against the condition $a>b$:

• From Case 1: $a=75, b=35$. Here, $75 > 35$. This pair satisfies the condition.
• From Case 2: $a=35, b=75$. Here, $35 \ngtr 75$. This pair does not satisfy the condition.

Therefore, the only valid values for $a$ and $b$ are $a=75$ and $b=35$.

Now, substitute these values into the expression $a+3b$: $a+3b = 75 + 3(35)$ $a+3b = 75 + 105$ $a+3b = 180$

3. Common Mistakes & Tips

• Forgetting the condition: Always remember to use all conditions given in the problem statement (e.g., $a>b$) to filter out incorrect solutions.
• Calculation errors: Be meticulous with arithmetic, especially when squaring numbers or dealing with negative signs.
• Choosing the right variance formula: While both variance formulas are correct, using $\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}$ is often simpler when the mean $\bar{x}$ is an integer, as it avoids calculating large squares of $x_i$ values and then subtracting a potentially large $(\bar{x})^2$.
• Algebraic Simplification: The substitution $A = x_i - \bar{x}$ is a powerful technique in statistics problems involving mean and variance for simplifying algebraic manipulations, particularly when dealing with two unknown observations.

4. Summary

This problem required a systematic application of the definitions of mean and variance. First, we used the mean to establish a linear equation ($a+b=110$). Next, we used the variance formula, specifically the sum of squared deviations from the mean, to establish a quadratic equation involving $a$ and $b$. A strategic substitution ($A=a-55, B=b-55$) transformed the system into a simpler form, leading to two possible pairs for $(a,b)$. Finally, the given condition $a>b$ was crucial in selecting the unique correct pair $(a=75, b=35)$, which allowed us to calculate the required expression $a+3b = 180$.

The final answer is $\boxed{\text{180}}$, which corresponds to option (A).