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

To solve this problem, we need to determine the values of \(a\) and \(b\) based on the given conditions, and then find the value of \(a + 3b\). Let's analyze the information step-by-step:

Step 1: Understanding the Mean

The mean of the observations \(a, b, 68, 44, 48, 60\) is given as 55. The mean is calculated by the formula:

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

Substituting the given mean:

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

Multiplying through by 6 gives:

\(330 = a + b + 220\)

Thus,

\(a + b = 110\) (Equation 1)

Step 2: Understanding the Variance

The variance of the observations is given as 194. The formula for variance is:

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

Substituting the known values and simplifying:

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

Therefore, substituting these values in:

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

Simplifying further:

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

Multiplying through by 6 gives:

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

Thus,

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

Step 3: Solving the Equations

We have two equations:

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

Substitute \(a = 110 - b\) into Equation 2:

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

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

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

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

55-b = \pm 20

Hence, \(b=35\) or \(b=75\).

Because \(a > b\), we assign:

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

Thus, given \(a > b\), we take \(a = 75\) and \(b = 35\).

Step 4: Calculation

Finally, compute \(a + 3b\):

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

The correct answer is 180.

Answer 2

Set up the equation for the mean. The mean of the six observations is given as 55. So,

\[ \frac{a + b + 68 + 44 + 48 + 60}{6} = 55. \]

Multiply both sides by 6 to eliminate the denominator:

\[ a + b + 68 + 44 + 48 + 60 = 330. \]

Simplify to get:

\[ a + b = 110 \quad \text{(Equation 1)}. \]

Set up the equation for the variance. The variance of the six observations is given as 194.

Recall that the variance formula for a set of observations \( x_1, x_2, \ldots, x_n \) with mean \( \overline{x} \) is:

\[ \text{Variance} = \frac{1}{n} \sum_{i=1}^{n} (x_i - \overline{x})^2. \]

Here, the mean \( \overline{x} \) is 55. Applying this to our observations:

\[ \frac{(a - 55)^2 + (b - 55)^2 + (68 - 55)^2 + (44 - 55)^2 + (48 - 55)^2 + (60 - 55)^2}{6} = 194. \]

Calculate known terms in the variance expression. Evaluate each squared term involving the known observations:

\[ (68 - 55)^2 = 13^2 = 169, \quad (44 - 55)^2 = (-11)^2 = 121, \quad (48 - 55)^2 = (-7)^2 = 49, \quad (60 - 55)^2 = 5^2 = 25. \]

Substitute these values into the variance equation:

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

Simplify:

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

Multiply both sides by 6:

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

Subtract 364 from both sides:

\[ (a - 55)^2 + (b - 55)^2 = 800 \quad \text{(Equation 2)}. \]

Solve the system of equations. We have the following two equations: 1. \( a + b = 110 \). 2. \( (a - 55)^2 + (b - 55)^2 = 800 \).

From Equation 1, express \( a \) in terms of \( b \):

\[ a = 110 - b. \]

Substitute \( a = 110 - b \) into Equation 2:

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

Simplify each term:

\[ (55 - b)^2 + (b - 55)^2 = 800. \]

Since \( (55 - b)^2 = (b - 55)^2 \), we can write:

\[ 2(b - 55)^2 = 800. \]

\[ (b - 55)^2 = 400. \]

Taking the square root of both sides:

\[ b - 55 = \pm 20. \]

This gives: 1. \( b = 75 \) (if \( b - 55 = 20 \)), 2. \( b = 35 \) (if \( b - 55 = -20 \)).

Since \( a > b \), we choose \( b = 35 \). Substitute \( b = 35 \) into Equation 1:

\[ a + 35 = 110. \] \[ a = 75. \]

Calculate \( a + 3b \)

\[ a + 3b = 75 + 3 \cdot 35 = 75 + 105 = 180. \]

Thus, the answer is: 180.