Question

Find the probability that the sum of the $40$ values is less than $7,000$.

Short answer

The probability that the sum of the $40$ values is less than $7000$is$\mathrm{P}(X\le 7000)=0.0569$.

Step-by-step solution

Given Information

A mean (${\mu }_x$) is$180$and a standard deviation $\left(\sigma \right)$is $20$.

Explanation

To find the probability that the sums of the $40$values is less than$7000$:

localid="1649346312817" $\sum X~N\left(n{\mu }_x,\sqrt{n}{\sigma }_x\right)$

$\sum X~N\left(\right(40\left)\right(180),(\sqrt{40}\left)\right(20\left)\right)$

localid="1649346327071" $$\begin{gathered} \mathrm{P}(X\le 7000)=\mathrm{P}\left(Z\le \frac{X-n{\mu }_x}{\sqrt{n}{\sigma }_x}\right) \\ =\mathrm{P}\left(Z\le \frac{7000-7200}{126.4911}\right) \\ =\mathrm{P}(Z\le -1.5811) \end{gathered}$$

$\mathrm{P}(X\le 7000)=0.0569$