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Answer :
Final answer:
The probability of the aircraft being overloaded, given the weights are normally distributed, is roughly 41%. This is calculated using the Z-score and principles of normal distribution. The pilot should prioritize safety and take action if an overload possibility is identified.
Explanation:
The probability that the aircraft is overloaded can be calculated using the properties of the normal distribution. Given that the mean weight of men is 175.7lb and the standard deviation is 37.9lb, we need to find the probability that the mean weight of a random sample of 39 men would exceed 167lb, i.e., P(X > 167).
For such problems, we use the concept of standard scores or 'z-scores', which measure how many standard deviations an element is from the mean. The Z-score is calculated as Z = (X - μ) / σ, where X is the random variable (167lb), μ is the population mean (175.7lb), and σ is the population standard deviation (37.9lb).
Calculating the Z-score, we find that Z = (167 - 175.7) / 37.9 = -0.23. We can find the probability associated with this Z-score using a standard normal distribution table or a calculator with statistical functions. A Z-score of -0.23 translates to a probability of approximately 0.59 for P(X < 167). Thus, P(X > 167) = 1 - P(X < 167) = 0.41, or 41% chance that the aircraft will be overloaded.
If the pilot identifies that overload is a real possibility, action should be taken. Options could include removing some non-essential baggage, asking a passenger to take a different flight, or adding additional fuel (if possible) to offset any unpredicted overweight issues. Safety should be the utmost priority in these scenarios.
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