Chapter 7.3, Problem 72E

To determine

To find: the probability that the total number of lightning strikes in a random sample of 50 square-kilometre plots of land is less than 250.

Answer to Problem 72E

  $P\left(\overline{X}<5\right)=0.0016$

Explanation of Solution

Given:

  $\begin{array}{l}\mu =\text{6}\\ \sigma \text{=2}\text{.4}\\ n=\text{50}\\ {\displaystyle \sum xi=250}\end{array}$

Formula used:

  $z=\frac{x-\mu }{\sigma }$

Calculation:

  $\overline{x}=\frac{{\displaystyle \sum xi}}{n}=\frac{250}{50}=5$

  $z=\frac{\overline{x}-{\mu }_{\overline{x}}}{\sigma \overline{x}}=\frac{\overline{x}-\mu }{\sigma /\sqrt{n}}=\frac{5-6}{2.4/\sqrt{50}}\approx -2.95$

  $z=\frac{\overline{x}-{\mu }_{\overline{x}}}{\sigma \overline{x}}=\frac{\overline{x}-\mu }{\sigma /\sqrt{n}}=\frac{5-6}{2.4/\sqrt{50}}\approx -2.95$

  $\begin{array}{c}P\left(\overline{X}<5\right)=P\left(Z<-2.95\right)\\ =0.0016\\ =0.16\%\end{array}$