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Answer :
Final answer:
To determine the probability of an elevator being overloaded, calculate the standard error of the mean for the adult male passenger weights, find the z-score for a mean weight of 156 pounds, and use the standard normal distribution to get the probability that the mean weight is greater than 156 pounds.
Explanation:
The student has asked about calculating the probability of overloading an elevator based on the weight distribution of adult male passengers.
We are given that the weights of males follows a normal distribution with a mean of 166 pounds and a standard deviation of 33 pounds. To find the probability that the elevator is overloaded because the 12 passengers have a mean weight greater than 156 pounds, we need to use the standard normal distribution, also called the Z-distribution.
First, calculate the standard error of the mean (SEM), which is the standard deviation divided by the square root of the number of samples (n). Here, SEM = 33 / sqrt(12).
Next, find the z-score that corresponds to the mean weight of 156 pounds using the formula: Z = (X - μ) / SEM, where X is 156, μ (mu) is the population mean, and SEM is the standard error of the mean just calculated.
After calculating the z-score, you can find the probability that the mean weight is above 156 pounds by looking up this z-score in the standard normal distribution table or using a statistical tool that provides cumulative distribution function (CDF) values.
This value is the probability that the elevator is not overloaded. To find the probability of the elevator being overloaded, subtract this value from 1 to get the complement.
Given this information and assuming the weights are normally distributed, it appears that the elevators' mean weight limit is safely below the mean population weight, suggesting the elevator might not be safe under these specified conditions if all passengers are of average weight or above.
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