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Answer :
Final answer:
The probability that the elevator is overloaded because the mean weight of 15 adult male passengers is greater than 156 lb can be calculated using the properties of the normal distribution. By finding the z-score and using a standard normal distribution table or calculator, we can determine the probability associated with the z-score. If this probability is below a predetermined threshold, the elevator is considered safe. Otherwise, it may be considered unsafe.
Explanation:
To determine the probability that the elevator is overloaded, we need to calculate the probability that the mean weight of 15 adult male passengers exceeds 156 lb. We can use the properties of the normal distribution to solve this problem.
Given that the weights of males are normally distributed with a mean of 163 lb and a standard deviation of 31 lb, we can calculate the standard error of the mean using the formula:
Standard Error of the Mean (SE) = Standard Deviation / Square Root of Sample Size
For 15 passengers, the standard error of the mean is:
SE = 31 lb / sqrt(15)
Next, we can calculate the z-score, which measures the number of standard deviations a value is from the mean. The z-score can be calculated using the formula:
z = (x - mean) / SE
In this case, we want to find the probability that the mean weight exceeds 156 lb. So, the z-score is:
z = (156 - 163) / (31 / sqrt(15))
Using a standard normal distribution table or a calculator, we can find the probability associated with this z-score. Let's assume the probability is P. The probability that the mean weight exceeds 156 lb is 1 - P, since we want the probability of the mean weight being greater than 156 lb.
Finally, we can determine if the elevator appears to be safe by comparing the probability to a predetermined threshold. If the probability is below the threshold, the elevator is considered safe. Otherwise, it may be considered unsafe.
Learn more about probability and statistics here:
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