Let and for some If the mean and variance of the elements of Y are 30 and 750, respectively, then the sum of all possible values of is [2026]

Question

Let $X = {x \in ℕ : 1 \leq x \leq 19}$ and for some $a, b \in ℝ, Y = {a x + b : x \in X} .$ If the mean and variance of the elements of Y are 30 and 750, respectively, then the sum of all possible values of $b$ is [2026]

Smart Achievers · Accepted Answer

(4)

$\sum y_{i} = a \sum x_{i} + \sum b$

$= a (1 + 2 + \dots + 19) + 19 b$

$\frac{\sum y_{i}}{19} = \frac{a \cdot 19 \cdot 20}{2 \cdot 19} + b$

$30 = 10 a + b . . . (1)$

$Variance of X = \frac{\sum x_{i}^{2}}{19} - {(\frac{\sum x_{i}}{19})}^{2}$

$= \frac{19 \cdot 20 \cdot 39}{19 \cdot 6} - {(10)}^{2} = 30$

$Variance of Y = a^{2} (variance of X)$

$750 = a^{2} \cdot 30$

$a^{2} = 25 \Rightarrow a = \pm 5$

$if a = + 5 \Rightarrow b = 30 - 50 = - 20 from (1)$

$if a = - 5 \Rightarrow b = 30 + 50 = 80 from (1)$

$sum of values of b = 80 - 20 = 60$

$option (4)$

Solution

Q.
Let $X = {x \in ℕ : 1 \leq x \leq 19}$ and for some $a, b \in ℝ, Y = {a x + b : x \in X} .$ If the mean and variance of the elements of Y are 30 and 750, respectively, then the sum of all possible values of $b$ is [2026]

1 100

2 20

3 80

4 60

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