Question 1199247: 4. 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 7,098 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 7,098lb/42 =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 175.9 lb and a standard deviation of 38.4.
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
n = 42 = sample size
Assumption is that male weights are normally distributed with these parameters:
mu = 175.9 = population mean
sigma = 38.4 = population standard deviation
xbar = sample mean
The xbar distribution is the set of xbar values that randomly scatter about, but should be close to the value of mu = 175.9
The spread of this xbar distribution is: sigma/sqrt(n) = 38.4/sqrt(42) = 5.9252486 approximately.
In short, the xbar distribution we're after has these properties:
center = 175.9
spread = 5.9252486 approximately
Let's compute the z score for xbar = 169 pounds.
z = (xbar - center)/(spread)
z = (169 - 175.9)/(5.9252486)
z = -1.1645081018204
z = -1.16
This result is approximate.
The task of finding P(xbar > 169) is roughly equivalent to P(z > -1.16)
Now use a calculator such as this one
https://onlinestatbook.com/2/calculators/normal_dist.html
to find that
P(z > -1.16) = 0.877
approximately.
Using a Z table is another option you can take.
There's about an 87.7% chance of the average passenger being over 169 pounds.
This by extension leads to about an 87.7% chance of the plane being overloaded.
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Answer:
The probability of an overloaded aircraft is approximately 0.877, so the pilot should take action to correct this.
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