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 40 passengers, and a flight has fuel and baggage that allows for a total passenger load of 6,640 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 6.640 lb / 40 = 166 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 177.3 lb and a standard deviation of 35.3. the probability is approximately.9785. (round to four decimal places as needed.) should the pilot take any action to correct for an overloaded aircraft?
a. no. because the probability is high, the aircraft is safe to fly with its current load.
b. yes. because the probability is high, the pilot should take action by somehow reducing the weight of the aircraft.

Explanation:

Step1: Calculate the standard error

The standard error ($\sigma_{\bar{x}}$) is calculated by the formula $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$, where $\sigma = 35.3$ and $n = 40$.
$\sigma_{\bar{x}}=\frac{35.3}{\sqrt{40}}\approx5.58$

Step2: Calculate the z - score

The z - score is calculated by the formula $z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}}$, where $\bar{x}=\frac{6640}{40}=166$, $\mu = 177.3$, and $\sigma_{\bar{x}}\approx5.58$.
$z=\frac{166 - 177.3}{5.58}\approx - 2.025$

Step3: Find the probability

We want to find $P(\bar{X}<166)$. Using the standard normal distribution table, $P(Z < - 2.025)\approx0.0215$

Answer:

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