Question

Lightning strikes The number of lightning strikes on a square kilometer of open ground in a year has mean 6 and standard deviation 2.4. The National Lightning Detection Network (NLDN) uses automatic sensors to watch for lightning in 1-square-kilometer plots of land. Find the probability that the total number of lightning strikes in a random sample of 50 square-kilometer plots of land is less than $250$.

Short answer

The resultant value is$P(\overline{X}<5)=0.0016$

Step-by-step solution

Given information 

Given:

$$\begin{gathered} mu=6 \\ \sigma =2.4 \\ n=50 \\ \sum xi=250 \end{gathered}$$

Formula used:

$z=\frac{x-\mu }{\sigma }$

Calculation

Simplify

$\begin{array}{rcl}\overline{x}& =& \frac{\sum xi}{n}=\frac{250}{50}=5\\ z& =& \frac{\overline{x}-{\mu }_{\overline{x}}}{\sigma \overline{x}}=\frac{\overline{x}-\mu }{\sigma /\sqrt{n}}=\frac{5-6}{2.4/\sqrt{50}}\approx -2.95\\ z& =& \frac{\overline{x}-{\mu }_{\overline{x}}}{\sigma \overline{x}}=\frac{\overline{x}-\mu }{\sigma /\sqrt{n}}=\frac{5-6}{2.4/\sqrt{50}}\approx -2.95\\ P(\overline{X}& <& 5)=P(Z<-2.95)\\ & =& 0.0016\\ & =& 0.16\%\end{array}$