Let X = {1, 2, 3, … , 19}. If new data Y = {y_i} where y_i = ax_i + b, x_i ∈ X. Such that mean and variance of y is 30 and 120 respectively, then the sum of value(s) of b is

The correct option is 60.

Let us analyze the given data step-by-step.

We are given the set of data points:
$X=\{1,2,3,\cdots ,19\}$
The number of elements in the set is $N=19$.

First, let us calculate the mean of set $X$ (denoted by $\overline{x}$):
$\overline{x}=\frac{1+2+3+\cdots +19}{19}$
Using the formula for the sum of the first $n$ natural numbers, $\sum i=\frac{n(n+1)}{2}$:
$\overline{x}=\frac{19\times 20}{2\times 19}=10$

Next, let us calculate the variance of set $X$ (denoted by ${\sigma }_x^2$). For the first $n$ natural numbers, the variance is given by the formula:
${\sigma }_x^2=\frac{n^2-1}{12}$
Substituting $n=19$:
${\sigma }_x^2=\frac{{19}^2-1}{12}=\frac{361-1}{12}=\frac{360}{12}=30$

We are given a new dataset $Y=\left\{y_i\right\}$ where the elements are related to $X$ by the linear transformation:
$y_i=a{x}_i+b$

By the properties of linear transformations on statistical measures, the mean of the new dataset is:
$\overline{y}=a\overline{x}+b$
We are given that the mean of $y$ is 30. Substituting the value of $\overline{x}=10$:
$30=10a+b$ --- (Equation 1)

Similarly, the variance of the new dataset under a linear transformation is affected only by the scale factor $a$:
${\sigma }_y^2=a^2{\sigma }_x^2$
We are given that the variance of $y$ is 120. Substituting ${\sigma }_x^2=30$:
$120=a^2\left(30\right)$
Solving for $a^2$:
$a^2=4\Rightarrow a=2\text{or}a=-2$

Now, let us find the corresponding values of $b$ for each case of $a$ using Equation 1:

Case 1: When $a=2$
$10\left(2\right)+b=30\Rightarrow 20+b=30\Rightarrow b=10$

Case 2: When $a=-2$
$10(-2)+b=30\Rightarrow -20+b=30\Rightarrow b=50$

Therefore, the possible values of $b$ are 10 and 50.

The sum of the value(s) of $b$ is:
$10+50=60$