If the mean and variance of 6 observations a, b, 68, 44, 48, 60 are 55 and 194 respectively and `a > b` then `a + 3b` is

Options

A. 190
B. 180
C. 200
D. 210

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) given the mean and variance of the observations \( a, b, 68, 44, 48, 60 \). We will then compute \( a + 3b \).

### Step-by-Step Solution:

1. **Calculate the Mean**:
The mean of the observations is given by:
\[ \text{Mean} = \frac{a + b + 68 + 44 + 48 + 60}{6} = 55 \]
Simplifying the sum of the known values:
\[ 68 + 44 + 48 + 60 = 220 \]
Therefore, we can write:
\[ \frac{a + b + 220}{6} = 55 \]
Multiplying both sides by 6:
\[ a + b + 220 = 330 \]
Rearranging gives us:
\[ a + b = 110 \quad \text{(Equation 1)} \]

2. **Calculate the Variance**:
The variance is given by:
\[ \text{Variance} = \frac{(a - \text{Mean})^2 + (b - \text{Mean})^2 + (68 - \text{Mean})^2 + (44 - \text{Mean})^2 + (48 - \text{Mean})^2 + (60 - \text{Mean})^2}{6} = 194 \]
Substituting the mean (55):
\[ \frac{(a - 55)^2 + (b - 55)^2 + (68 - 55)^2 + (44 - 55)^2 + (48 - 55)^2 + (60 - 55)^2}{6} = 194 \]
Calculating the squared differences:
\[ (68 - 55)^2 = 169, \quad (44 - 55)^2 = 121, \quad (48 - 55)^2 = 49, \quad (60 - 55)^2 = 25 \]
Thus, we have:
\[ \frac{(a - 55)^2 + (b - 55)^2 + 169 + 121 + 49 + 25}{6} = 194 \]
Summing the known squares:
\[ 169 + 121 + 49 + 25 = 364 \]
Therefore:
\[ \frac{(a - 55)^2 + (b - 55)^2 + 364}{6} = 194 \]
Multiplying both sides by 6:
\[ (a - 55)^2 + (b - 55)^2 + 364 = 1164 \]
Rearranging gives:
\[ (a - 55)^2 + (b - 55)^2 = 800 \quad \text{(Equation 2)} \]

3. **Solve the Equations**:
We have the system of equations:
- \( a + b = 110 \) (Equation 1)
- \( (a - 55)^2 + (b - 55)^2 = 800 \) (Equation 2)

From Equation 1, we can express \( b \) in terms of \( a \):
\[ b = 110 - a \]
Substituting into Equation 2:
\[ (a - 55)^2 + ((110 - a) - 55)^2 = 800 \]
Simplifying the second term:
\[ (110 - a - 55)^2 = (55 - a)^2 \]
Thus:
\[ (a - 55)^2 + (55 - a)^2 = 800 \]
This simplifies to:
\[ 2(a - 55)^2 = 800 \]
Dividing by 2:
\[ (a - 55)^2 = 400 \]
Taking the square root:
\[ a - 55 = 20 \quad \text{or} \quad a - 55 = -20 \]
Thus:
\[ a = 75 \quad \text{or} \quad a = 35 \]
Since \( a > b \), we take \( a = 75 \). Then:
\[ b = 110 - 75 = 35 \]

4. **Calculate \( a + 3b \)**:
Now we can find \( a + 3b \):
\[ a + 3b = 75 + 3 \times 35 = 75 + 105 = 180 \]

### Final Answer:
\[ \boxed{180} \]