A roller coaster is being designed that will accommodate 60 riders. The maximum weight the coaster can hold safely is 12,000 pounds. According to the National Health Statistics Reports, the weights of adult U.S. men have a mean of 193 pounds and a standard deviation of 67 pounds, and the weights of adult U.S. women have a mean of 163 pounds and a standard deviation of 78 pounds.

a) If a random sample of 60 adult men ride the coaster, what is the probability that the maximum safe weight will be exceeded?

To find the probability that the maximum safe weight will be exceeded, we need to calculate the total weight of 60 adult men and compare it to the maximum safe weight of 12,000 pounds. The weight of adult men follows a normal distribution with a mean of 193 pounds and a standard deviation of 67 pounds.

To calculate the total weight of 60 adult men, we multiply the mean weight by 60. Thus, the total weight is 193 pounds * 60 = 11,580 pounds.

To find the probability that the maximum safe weight will be exceeded, we need to find the probability that the total weight of 60 adult men is greater than 12,000 pounds. Using the z-score formula, we calculate the z-score for the total weight:

z = (x - μ) / σ = (12,000 - 11,580) / (67 * sqrt(60)) = 0.495

Using a standard normal distribution table or a calculator, we find that the probability corresponding to a z-score of 0.495 is approximately 0.6859. Therefore, the probability that the maximum safe weight will be exceeded is approximately 0.6859 or 68.59%.