Answer :
To find the maximum total allowable weight when 17 males are randomly selected, we need to consider the worst case scenario. In this case, we assume that all 17 passengers are males, and we want to calculate the maximum weight that has a 0.995 probability of not being exceeded.
The weight of each male passenger is normally distributed with a mean of 168 lb and a standard deviation of 34 lb. When summing the weights of multiple passengers, the sum follows a normal distribution as well. The mean of the sum is equal to the sum of the individual means, and the standard deviation of the sum is the square root of the sum of the individual variances.
Mean of the sum = Number of passengers * Mean of each passenger = 17 * 168 = 2856 lb
Standard deviation of the sum = sqrt(Number of passengers) * Standard deviation of each passenger = sqrt(17) * 34 ≈ 82.549 lb
Now, we need to find the maximum weight that has a 0.995 probability of not being exceeded. This is equivalent to finding the z-score
corresponding to a 0.995 cumulative probability.
Using a standard normal distribution table or a calculator, we find that the z-score for a cumulative probability of 0.995 is approximately 2.575.
To find the maximum allowable weight, we can use the formula:
Maximum allowable weight = Mean of the sum + (Z-score * Standard deviation of the sum)
Maximum allowable weight = 2856 + (2.575 * 82.549) ≈ 3087.018 lb
Therefore, the maximum total allowable weight, with a 0.995 probability of not being exceeded when 17 males are randomly selected, is approximately 3087.018 lb.
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