Given data $60, 60, 44, 58, 68, α, β, 56$ has mean $58$, variance = $66.2$, then find $α^2 + β^2$.

Question

Let $X=\{x\in\mathbb{N}:1\le x\le19\}$ and for some $a,b\in\mathbb{R}$, $Y=\{ax+b:x\in X\}$. 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

Answer 1

$\text{The formula for Variance is:}$ $\frac{\sum x^2}{n}-(\bar x)^2$

$\text{Using the provided data, we calculate:}$

$\frac{7200+1936+3364+4624+3136+\alpha^2+\beta^2}{8}-3364$ $=66.2$

$2532.5+\frac{\alpha^2+\beta^2}{8}-3364=66.2$

$=7181.6$ $\approx7182$

$\text{The final answer is 7182.}$

Given data \(60, 60, 44, 58, 68, α, β, 56\) has mean \(58\), variance = \(66.2\), then find \(α^2 + β^2\).

Solution and Explanation

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