Question

If dataset $A=\{1,2,3,\ldots,19\}$ and dataset $B=\{ax+b;\,x\in A\}$. If mean of $B$ is $30$ and variance of $B$ is $750$, then sum of possible values of $b$ is

• A. 30
• B. 90
• C. 20
• D. 60

Hint:

Variance is unaffected by addition of constants but is multiplied by the square of the scaling factor.

Answer 1

The Correct Option is D

To solve this problem, we need to analyze the datasets \( A \) and \( B \) and use the given statistical properties.

1. First, understand the datasets:
   • Dataset \( A = \{1, 2, 3, \ldots, 19\} \).
   • Dataset \( B = \{ax+b; \, x\in A\} \) means \( B \) is a linear transformation of \( A \) with slope \( a \) and intercept \( b \).
2. Calculate the mean of dataset \( A \):
   • The mean of an arithmetic sequence is the average of the first and last term. So,
   • \(\text{Mean of } A = \frac{1 + 19}{2} = 10\).
3.  Given the mean of \( B \) is 30, set up the equation:
   • The mean of \( B \) is given by substituting the mean of \( A \) into the transformation:
   • \(a \times 10 + b = 30\), simplifying to \(10a + b = 30\). (Equation 1)
4. Calculate the variance of dataset \( A \):
   • The variance of an arithmetic sequence from 1 to \( n \) is \(\frac{(n^2-1)}{12}\).
   • For \( A \), where \( n = 19 \), \(\text{Variance of } A = \frac{(19^2 - 1)}{12} = \frac{360}{12} = 30\).
5. Given the variance of \( B \) is 750, relate it with \( a^2 \times \text{Variance of } A \):
   • The variance of \( B \) is transformed as \( a^2 \times \text{Variance of } A = 750 \).
   • \(a^2 \times 30 = 750\), simplifying to \(a^2 = 25\).
   • Solve for \( a \): \(a = 5\) or \(a = -5\).
6. Find possible values of \( b \) using \( a = 5 \) and \( a = -5 \):
   • For \( a = 5 \), substitute into Equation 1: \(10 \times 5 + b = 30\) giving \(b = -20\).
   • For \( a = -5 \), substitute into Equation 1: \(10 \times (-5) + b = 30\) giving \(b = 80\).
7. Sum the possible values of \( b \):
   • Possible values of \( b \) are \(-20\) and \(80\).
   • The sum of these values is \(-20 + 80 = 60\).

Conclusion: The sum of possible values of \( b \) is 60.