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 37 passengers and a flight has fuel and baggage that allows for a total passenger load of 6,031 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 tharn action to correct for an overloaded aircraft? Assume that weights of men are normally distributed with a mean of 172.4 lb and a standard deviation of 39.5 The probability is approximately 163 lb. What is the probability that the aircraft is overloaded? Should the pilot take any 37 Round to four decimal places as needed.)

Answer 1

To determine if the aircraft is overloaded, calculate the allowable weight per passenger, use the z-score to find the standardized value, and reference the standard normal distribution table. If the probability of overloading is significant, corrective actions are needed.

Step-by-step explanation:

The student is asking about the probability that an aircraft is overloaded based on the weight distribution of its passengers. According to the given information, the aircraft's maximum passenger load is 6,031 lb, and it can carry 37 passengers. If all the passengers are men, and the weight of men is normally distributed with a mean of 172.4 lb and a standard deviation of 39.5 lb, we need to calculate the probability that the mean weight of the 37 passengers exceeds the maximum allowable weight per passenger for the aircraft to be not overloaded.

First, we calculate the allowable weight per passenger:

• Total allowable passenger load = 6,031 lb
• Number of passengers = 37
• Allowable weight per passenger = Total allowable passenger load ÷ Number of passengers
• Allowable weight per passenger = 6,031 lb ÷ 37 = approximately 163 lb

To find the probability that the mean weight of the passengers is greater than 163 lb, we can use the z-score formula:

Z = (X - μ) ÷ (σ ÷ √ n)

Where:

• X = the cut-off weight (163 lb)
• μ = mean weight of men (172.4 lb)
• σ = standard deviation (39.5 lb)
• n = sample size (37)

Calculating the z-score gives us the standardized value which we can use to find the probability from the standard normal distribution table. Finally, this probability indicates the likelihood of the aircraft being overloaded, and if it is significant, the pilot should take corrective action to avoid overloading.