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

Hint:

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

Answer 1

Correct Answer: 60

Step 1: Mean and variance of dataset $A$.
Dataset $A=\{1,2,3,\ldots,19\}$
Mean of first $n$ natural numbers is \[ \bar{x}=\frac{1+19}{2}=10 \] Variance of $\{1,2,\ldots,n\}$ is \[ \sigma^2=\frac{n^2-1}{12} \] \[ \sigma^2=\frac{19^2-1}{12}=30 \] Step 2: Mean of dataset $B$.
Given $B=ax+b$
\[ \text{Mean of }B=a(10)+b=30 \Rightarrow 10a+b=30 \quad (1) \] Step 3: Variance of dataset $B$.
Variance scales as \[ \sigma_B^2=a^2\sigma_A^2 \] \[ 750=a^2\times30 \] \[ a^2=25 \Rightarrow a=5 \text{ or } a=-5 \] Step 4: Find corresponding values of $b$.
From equation (1):
If $a=5$, \[ b=30-50=-20 \] If $a=-5$, \[ b=30+50=80 \] Step 5: Final calculation.
\[ \text{Sum of possible values of }b=-20+80=60 \] Final conclusion.
The required sum is 60.