Question

before every flight, the pilot must verify that the total weight of the load is less than the maximum allowable load for the aircraft. the aircraft can carry 35 passengers, and a flight has fuel and baggage that allows for a total passenger load of 5,915 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 $\frac{5,915}{35}=169$ 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 173.2 lb and a standard deviation of 38.5. the probability is approximately 0.7407 (round to four decimal places as needed.) should the pilot take any action to correct for an overloaded aircraft? a. yes. because the probability is high, the pilot should take action by somehow reducing the weight of the aircraft. b. no. because the probability is high, the aircraft is safe to fly with its current load

Explanation:

Step1: Identify the distribution parameters

Let $\mu = 173.2$ lb be the population mean weight of men, $\sigma = 38.5$ lb be the population standard - deviation, and $n = 35$ be the sample size. The maximum allowable mean weight of passengers is $\bar{x}=\frac{5915}{35}=169$ lb.

Step2: Calculate the standard error of the mean

The standard error of the mean is $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}=\frac{38.5}{\sqrt{35}}\approx6.53$.

Step3: Calculate the z - score

The z - score is calculated using the formula $z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}}$. Substituting the values, we get $z=\frac{169 - 173.2}{6.53}=\frac{- 4.2}{6.53}\approx - 0.64$.

Step4: Find the probability of overloading

We want to find $P(\bar{X}>169)$. Since $P(\bar{X}>169)=1 - P(\bar{X}\leq169)$, and from the standard normal table, $P(Z\leq - 0.64)=0.2611$. So $P(\bar{X}>169)=1 - 0.2611 = 0.7389\approx0.7407$.
Since the probability that the aircraft is overloaded is approximately $0.7407$ (a high probability), the pilot should take action to correct for an overloaded aircraft.

Answer:

A. Yes. Because the probability is high, the pilot should take action by somehow reducing the weight of the aircraft.