Answer :
Final answer:
The probability that the elevator will be overloaded with 12 adult males, given their weights are normally distributed with a mean of 165 lbs and a standard deviation of 26 lbs, is approximately 0.1932 or 19.32%. This suggests that the elevator might not be entirely safe if filled with 12 adult males.
Explanation:
This problem can be solved using the principles of the normal distribution in statistics. To calculate the probability that the elevator is overloaded, we need to standardize the mean weight limit of the elevator (which is 162 pounds) to the Z score. The Z score is a measure of how many standard deviations an element is from the mean.
The formula we use is Z = (X - μ) / σ, where 'X' is the value from the dataset (162), 'μ' is the mean (165), and 'σ' is the standard deviation (26). However, because we are dealing with the mean of 12 people, we need to adjust the standard deviation by dividing it by the square root of 12. So, the adjusted standard deviation is 26/√12.
Using the given numbers, we calculate Z = (162-165) / (26/√12), leading to a Z score of about -0.8660. To find the probability of the elevator being overloaded, we need to find the area under the normal curve to the right of this Z score. Using a standard normal table or calculator, we find the area is about 0.1932.
Therefore, the probability that this elevator will be overloaded is 0.1932 or 19.32% if rounded to four decimal places. Given this relatively high probability, it can be suggested that this elevator may not be entirely safe if it is fully loaded with 12 adult males, as there's almost a 20% chance they'll exceed the weight limit.
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