Answer :
The probability that the elevator is overloaded because the 15 adult male passengers have a mean weight greater than 151 lb is approximately 0.8621 or 86.21%. the SEM is 32 / sqrt(15) ≈ 8.27 lb.
To find the probability that the elevator is overloaded because the 15 adult male passengers have a mean weight greater than 151 lb, we need to calculate the probability that the sample mean weight of the passengers exceeds 151 lb.
Given that the weights of males are normally distributed with a mean of 160 lb and a standard deviation of 32 lb, we can use the properties of the normal distribution to calculate this probability.
First, we need to calculate the standard error of the mean (SEM), which is the standard deviation of the sample mean weight. The SEM can be calculated using the formula SEM = standard deviation / sqrt(sample size).
In this case, the standard deviation is 32 lb, and the sample size is 15 passengers. Therefore, the SEM is 32 / sqrt(15) ≈ 8.27 lb.
Next, we need to calculate the z-score, which represents the number of standard deviations the sample mean is away from the population mean. The z-score can be calculated using the formula z = (sample mean - population mean) / SEM.
In this case, the sample mean is 151 lb, the population mean is 160 lb, and the SEM is 8.27 lb. Therefore, the z-score is (151 - 160) / 8.27 ≈ -1.09.
To find the probability corresponding to this z-score, we can refer to a standard normal distribution table or use a calculator. The probability that a standard normal distribution variable is greater than -1.09 is approximately 0.8621.
Therefore, the probability that the elevator is overloaded because the 15 adult male passengers have a mean weight greater than 151 lb is approximately 0.8621 or 86.21%.
Based on this probability, it appears that the elevator is safe. With an 86.21% probability, it is unlikely that the elevator would be overloaded with 15 adult male passengers whose mean weight exceeds 151 lb. However, it's important to note that this calculation assumes the weights are normally distributed and that all passengers are adult males. It is still necessary to consider other factors, such as the distribution of weights among different genders and age groups, to ensure the elevator's safety under diverse passenger compositions.
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