Question

Airline passengers get heavier In response to the increasing weight of airline passengers, the Federal Aviation Administration (FAA) told airlines to assume that passengers average 190 pounds in the summer, including clothes and carry-on baggage. But passengers vary, and the FAA did not specify a standard deviation. A reasonable standard deviation is 35 pounds. A commuter plane carries 30 passengers. Find the probability that the total weight of 30 randomly selected passengers exceeds $6000$ pounds.

Short answer

The required value is$P(\overline{X}\ge 200)=5.94\%$

Step-by-step solution

Given information 

Given,

$$\begin{gathered} \mu =190 \\ \sigma =35 \\ n=30 \\ \sum xi=6000 \end{gathered}$$

Calculation

Formula used:

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

Lets calculate,

localid="1657630239006" $\begin{array}{rcl}\overline{x}& =& \frac{\sum xi}{n}=\frac{6000}{30}=200\\ z& =& \frac{\overline{x}-{\mu }_{\overline{x}}}{\sigma \overline{x}}=\frac{\overline{x}-\mu }{\sigma l\sqrt{n}}=\frac{200-190}{35/\sqrt{30}}\approx 1.56\\ P(\overline{X}& \ge & 200)=P(Z>1.56)\\ & =& 1-P(Z>1.56)\\ & =& 1-0.9406\\ & =& 0.0594\\ & =& 5.94\%\end{array}$