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 42 passengers, and a flight has fuel and baggage that allows for a total passenger load of 6,930 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 $\frac{6,930}{42}$ = 165 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 172.8 lb and a standard deviation of 39.8

the probability is approximately
(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: Identify the sampling - distribution parameters

The population mean $\mu = 172.8$ lb, the population standard deviation $\sigma=39.8$ lb, and the sample size $n = 42$. The sampling - distribution of the sample mean $\bar{X}$ has mean $\mu_{\bar{X}}=\mu = 172.8$ lb and standard deviation $\sigma_{\bar{X}}=\frac{\sigma}{\sqrt{n}}=\frac{39.8}{\sqrt{42}}\approx6.127$. The critical value for overloading is $\bar{x}=165$ lb.

Step2: Calculate the z - score

The z - score is calculated using the formula $z=\frac{\bar{x}-\mu_{\bar{X}}}{\sigma_{\bar{X}}}$. Substitute $\bar{x} = 165$, $\mu_{\bar{X}}=172.8$, and $\sigma_{\bar{X}}\approx6.127$ into the formula: $z=\frac{165 - 172.8}{6.127}=\frac{-7.8}{6.127}\approx - 1.27$.

Step3: Find the probability

We want to find $P(\bar{X}>165)$. Since $P(\bar{X}>165)=1 - P(\bar{X}\leq165)$, and from the standard normal distribution table, $P(Z\leq - 1.27)=0.1020$. So $P(\bar{X}>165)=1 - 0.1020 = 0.8980$.

Answer:

The probability is approximately $0.8980$.
Yes. Because the probability is high, the pilot should take action by somehow reducing the weight of the aircraft. So the answer is B.