Question

Let the mean and the variance of 6 observation a,b, 68, 44, 48, 60 be 55 and 194, respectively if a > b, then a + 3b is

• A. 200
• B. 190
• C. 180
• D. 210

Answer 1

The Correct Option is C

The objective is to ascertain the values of \(a\) and \(b\) by leveraging the provided conditions, and subsequently compute \(a + 3b\). The solution proceeds with a methodical examination of the data.

Step 1: Mean Calculation

The mean of the dataset \(a, b, 68, 44, 48, 60\) is established as 55. The mean is computed using the formula:

\(\text{Mean} = \frac{a + b + 68 + 44 + 48 + 60}{6}\)

With the mean given as 55:

\(55 = \frac{a + b + 220}{6}\)

Multiplying both sides by 6 yields:

\(330 = a + b + 220\)

This simplifies to the equation:

\(a + b = 110\) (Equation 1)

Step 2: Variance Calculation

The variance for the given observations is 194. The variance is calculated as:

\(\text{Variance} = \frac{(a-55)^2 + (b-55)^2 + (68-55)^2 + (44-55)^2 + (48-55)^2 + (60-55)^2}{6}\)

The squared differences for the known values are:

• \((68-55)^2 = 169\)
• \((44-55)^2 = 121\)
• \((48-55)^2 = 49\)
• \((60-55)^2 = 25\)

Substituting these into the variance formula:

\(194 = \frac{(a-55)^2 + (b-55)^2 + 169 + 121 + 49 + 25}{6}\)

Simplifying the sum of the known squared differences:

\(194 = \frac{(a-55)^2 + (b-55)^2 + 364}{6}\)

Multiplying both sides by 6:

\(1164 = (a-55)^2 + (b-55)^2 + 364\)

This results in the equation:

\((a-55)^2 + (b-55)^2 = 800\) (Equation 2)

Step 3: System of Equations Resolution

The problem is reduced to solving the following system of two equations:

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

From Equation 1, we express \(a\) as \(a = 110 - b\) and substitute it into Equation 2:

\((110-b-55)^2 + (b-55)^2 = 800\)

\((55-b)^2 + (b-55)^2 = 800\)

Since \((55-b)^2 = (b-55)^2\), the equation becomes:

\((55-b)^2 + (55-b)^2 = 800\)

\(2(55-b)^2 = 800\)

Dividing by 2 and taking the square root gives:

\(55-b = \pm \sqrt{400}\)

\(55-b = \pm 20\)

This yields two possible values for \(b\):

\(b = 55 - 20 = 35\) or \(b = 55 + 20 = 75\).

Given the condition \(a > b\), we determine the corresponding values for \(a\):

• If \(b = 35\), then \(a = 110 - 35 = 75\).
• If \(b = 75\), then \(a = 110 - 75 = 35\).

The condition \(a > b\) dictates that we select \(a = 75\) and \(b = 35\).

Step 4: Final Computation

The final calculation for \(a + 3b\) is performed:

\(a + 3b = 75 + 3(35) = 75 + 105 = 180\)

The result is 180.