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 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 6200lb/37 = 167.6lb. 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 185.47 lb and a standard deviation of 39 lb. make sure to include pdf

Answer 1

Using the normal probability distribution and the central limit theorem, it is found that the probability is of 0.9974 = 99.74%, which means that the pilot should take action.

Normal Probability Distribution

In a normal distribution with mean $\mu$ and standard deviation $\sigma$, the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

• It measures how many standard deviations the measure is from the mean.

• After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.

• By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation $s = \frac{\sigma}{\sqrt{n}}$.

In this problem, for the population, the mean and the standard deviation are given by, respectively:

$\mu = 185.47, \sigma = 39$.

For a sample of 37 passengers, we have that:

$n = 37, s = \frac{39}{\sqrt{37}} = 6.4116$

The probability that the aircraft is overloaded is one subtracted by the p-value of Z when X = 167.6, hence:

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem:

$Z = \frac{X - \mu}{s}$

$Z = \frac{167.6 - 185.47}{6.4116}$

$Z = -2.79$

$Z = -2.79$ has a p-value of 0.0026.

1 - 0.0026 = 0.9974.

There is a 0.9974 = 99.74% probability that the aircraft is overloaded. Since this is a very high probability, the pilot should take action.

To lern more about the normal probability distribution and the central limit theorem, you can check brainly.com/question/24663213