Answer :
To determine the probability that the elevator is overloaded due to the mean weight of 15 adult male passengers exceeding 156 lb, we can use the properties of the normal distribution.
Given that the weights of males are normally distributed with a mean of 164 lb and a standard deviation of 31 lb, we can calculate the probability using the z-score and the cumulative distribution function. If the probability is greater than a predetermined threshold, it suggests that the elevator may be unsafe for the given weight distribution.
To calculate the probability, we need to standardize the mean weight using the z-score. The formula for the z-score is given by z = (x - μ) / σ, where x is the mean weight, μ is the population mean weight, and σ is the population standard deviation.
In this case, the population mean weight is 164 lb, and the population standard deviation is 31 lb. Therefore, the z-score is (156 - 164) / 31 = -8 / 31.
Next, we use the standard normal distribution table or a calculator to find the probability associated with the z-score. The probability corresponds to the area under the standard normal curve to the right of the z-score. This probability represents the likelihood of the mean weight exceeding 156 lb.
If the calculated probability is greater than a predetermined threshold (e.g., 0.05 or 0.01), it suggests that the elevator is likely to be overloaded and may not be safe for the given weight distribution.
It is important to note that the safety of the elevator depends not only on the mean weight but also on the weight distribution and other factors such as elevator design, maximum capacity, and safety regulations. A thorough assessment should be conducted to ensure the elevator's safety under various conditions.
Learn more about cumulative distribution function here:
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