91影视

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

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

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

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

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.

$$\begin{gathered} P\left(\stackrel{炉}{x}>167.6\right)=1-P\left(\stackrel{炉}{x}<167.6\right) \\ =1-P\left(\frac{\stackrel{炉}{x}-渭}{\frac{蟽}{\sqrt{n}}}<\frac{167.6-渭}{\frac{蟽}{\sqrt{n}}}\right) \\ =1-P\left(z<\frac{167.6-189}{\frac{39}{\sqrt{37}}}\right) \end{gathered}$$

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

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.