Loading AircraftBefore every flight, the pilot must verify that the total weight of the load is less than the maximum allowable load for the aircraft. The Bombardier Dash 8 aircraft can carry 37 passengers, and a flight has fuel and baggage that allows for a total passenger load of 6200 lb. The pilot sees that the plane is full and all passengers are men. The aircraft will be overloaded if the mean weight of the passengers is greater than 6200 lb/37 = 167.6 lb. What is the probability that the aircraft is overloaded? Should the pilot take any action to correct for an overloaded aircraft? Assume that weights of men are normally distributed with a mean of 189 lb and a standard deviation of 39 lb (based on Data Set 1 “Body Data” in Appendix B).

The weights of the men are normally distributed with a mean $\left(\mu \right)$equal to 189 lb and a standard deviation $\left(\sigma \right)$equal to 39 lb.

The aircraft gets overloaded if the mean weight of the passengers exceeds 167.6 lb.

Let $\overline{x}$denote the sample weight of the men.

The sample mean weight of the men follows a normal distribution with a mean equal to ${\mu }_{\overline{x}}=\mu$and a standard deviation equal to ${\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}$.

The sample size is equal to n=37.

The probability that the aircraft will be overloaded is computed using the standard normal table, as shown below.

$$\begin{gathered} P\left(\overline{x}>167.6\right)=1-P\left(\overline{x}<167.6\right) \\ =1-P\left(\frac{\overline{x}-\mu }{\frac{\sigma }{\sqrt{n}}}<\frac{167.6-\mu }{\frac{\sigma }{\sqrt{n}}}\right) \\ =1-P\left(z<\frac{167.6-189}{\frac{39}{\sqrt{37}}}\right) \end{gathered}$$

$$\begin{gathered} =1-P\left(z<-3.34\right) \\ =1-0.0004 \\ =0.9996 \end{gathered}$$

Therefore, the probability that the aircraft will be overloaded is equal to 0.9996.

As the probability that the aircraft will be overloaded is quite high, the pilot needs to take action for the safety of the passengers.