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Answer :
The probability that the elevator is overloaded because the 12 adult male passengers have a mean weight greater than 165 lb is approximately 0.0684.
To solve this problem, we need to calculate the probability that the mean weight of 12 adult male passengers exceeds 165 lb. Since the weights are normally distributed with a mean of 174 lb and a standard deviation of 35 lb, we can use the Central Limit Theorem.
First, we calculate the standard deviation of the sample mean by dividing the standard deviation of the population by the square root of the sample size:
Standard deviation of sample mean = 35 lb / sqrt(12) ≈ 10.097 lb
Next, we need to standardize the mean weight of 165 lb using the formula:
Z = (X - μ) / (σ / sqrt(n))
where X is the mean weight, μ is the population mean, σ is the population standard deviation, and n is the sample size.
Using this formula, we can calculate the Z-score:
Z = (165 - 174) / (10.097) ≈ -0.8903
Next, we find the probability of the Z-score being greater than -0.8903 using a standard normal distribution table or calculator. The probability comes out to be approximately 0.8091.
However, we are interested in the probability that the mean weight exceeds 165 lb, so we subtract the above probability from 1:
Probability (mean weight > 165 lb) = 1 - 0.8091 ≈ 0.1909
Therefore, the probability that the elevator is overloaded because the 12 adult male passengers have a mean weight greater than 165 lb is approximately 0.1909 or 19.09%.
Since this probability is higher than the acceptable threshold, the elevator does not appear to be safe for carrying 12 adult male passengers with a mean weight greater than 165 lb.
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