Let the mean and the variance of 20 observations x_1,x_2,,x_20 be 15 and 9, respectively.

Question

Let the mean and the variance of 20 observations $x_1,x_2,\ldots,x_{20}$ be 15 and 9, respectively. For $\alpha\in\mathbb{R}$, if the mean of $(x_1+\alpha)^2,(x_2+\alpha)^2,\ldots,(x_{20}+\alpha)^2$ is 178, then the square of the maximum value of $\alpha$ is equal to \_\_\_\_\_\_.

Answer: 4

Accepted Answer

$\sum x_i = 300$; $\sigma^2=9 \implies \sum x_i^2 = 20(9+225)=4680$

$$\frac{\sum(x_i+\alpha)^2}{20} = \frac{\sum x_i^2+2\alpha\sum x_i+20\alpha^2}{20} = 178$$

$$4680+600\alpha+20\alpha^2 = 3560 \implies \alpha^2+30\alpha+56=0$$

$$(\alpha+28)(\alpha+2)=0 \implies \alpha = -28 \text{ or } -2$$

Maximum value $\alpha_{max}=-2 \implies \alpha_{max}^2=4$

Solutions · 1