Question

UBackground: Airplane Salety and Average Weights: Problems 2 through 4 Alr Midwest Flight 5481, which seated 21 passengers tragicaly crashed in 2003 due to overloading and inappropriate madntenance. Rosuts from the investigation of the crash by the United States National Transportation Safety Board (NTSB) led to the revision of the average weight used in aiplane weighting calculations from 170 pounds to today's values. This accident furthemnore raised a debale within the US aviation industy about the practice of using average weights instead of real weights to caiculate ioading for small passenger aircrat. The Statistics of means, as we have studied them, is at the heart of this debate. in fact, many experts now call for using exact weights rather than average weights for small sized aircrat. WFS- 146 is exactly the bpe of airline that may fly such smaller planesi These problens will explore the Mathematics of why. The consequences of the Central Lime Theorem pleys a central role in this Understanding. Background for Probiem 2 Wo ace a small company, but we dream bigl For many years we at WFS- 146 have dreamed of owring out own, 200-seat fot aiplans. We may finaly thave tis opponinity As pat of our preparasons for making this dream a reality. carry out the following caiculations relsting to arpine sulese Problem 2 Our Company Dream: Suppose a large, jet-airplane seats 200 passengers. Assume that the airplane is completely full due to overbooking and that the passengers are chosen randomly from the customer population. Assume that the average weight of passengers in the customer population is 195 pounds in the Winter with a standard deviation of 30 pounds. Note that the number 195 pounds is drawn from a study of the total population of the United States. a)Calculate the expected total weight of the passengers. b)Applying the Central Limit Theorem: determine the standard error of the "mean passenger weight" for this airplane. c) Determine the probability that the actual total weight of the passengers is at least 5% greater than the expected total weight of the passengers. You should expect a small number. Express your answer as a percent.

Answer 1

The expected total weight of 200 passengers is calculated as 39,000 pounds. The standard error of the mean passenger weight is approximately 2.12 pounds. The probability that the actual total weight exceeds the expected total weight by at least 5% is negligible.

Step-by-step explanation:

The problem presented involves the application of statistical methods, especially the Central Limit Theorem, to address issues related to airline safety and the calculation of passenger weights. This is relevant in Mathematics, particularly in the field of statistics.

Problem Solution:

a) Expected total weight of the passengers:

Total number of passengers = 200

Average weight per passenger = 195 pounds

Expected total weight = Total number of passengers × Average weight per passenger = 200 × 195 = 39,000 pounds

b) Standard error of the mean passenger weight:

Standard deviation = 30 pounds

Standard error = Standard deviation / √(number of passengers) = 30 / √(200) = 30 / 14.14 ≈ 2.12 pounds

c) To determine the probability that the actual total weight is at least 5% greater than the expected total weight, we use the standard error to calculate the z-score and then find the corresponding probability from the standard normal distribution.

5% of expected total weight = 0.05 × 39,000 pounds = 1,950 pounds

Weight threshold for 5% increase = 39,000 pounds + 1,950 pounds = 40,950 pounds

Z-score = (Weight threshold - Expected total weight) / (Standard error × √(number of passengers)) = (40,950 - 39,000) / (2.12 × 14.14) ≈ 6.98

Probability corresponding to this z-score is extremely small and can be considered practically zero for all intents.

Therefore, the probability expressed as a percent is negligible. This reinforces the importance of careful analysis in airline safety protocol.