Elevator ride: Engineers are designing a large elevator that will accommodate 45 people. The maximum weight the elevator can hoid safely is 8100 pounds According to the National Health Statistics Reports, the weights of adult U.S. men have mean 176 pounds and standard deviation 65 pounds, and the weights of adult U.S. women have mean 166 pounds and standard deviaton 74 pounds. Use the TI-B4 Plus calculator. Part 1 of 3 (a) If 45 people are on the elevator, and their total weight is 8100 pounds, what is their average weight? The average welght is 180 pounds. Part 2 of 3 (b) If a random sample of 45 adult men ride the elevator, what is the probability that the maximum sole weight will be exceeded? Round the answer to at least four decimal places. The probablity that the maximum safe weight wil be exceeded is 0.3399 Pert: $2 / 3$ Part 3 of 3 (c) If a random sample of 45 adult women ride the elevator, what is the probability that the maximum sale weight will be excended? Round the ansiner to at least four decimat places. The probability that the maximum sate weght will be exceeded is $\square$

Step 1 ：The problem is asking for the probability that the total weight of 45 randomly selected adult women will exceed the maximum safe weight of the elevator, which is 8100 pounds. Given that the mean weight of an adult woman is 166 pounds and the standard deviation is 74 pounds, we can use the Central Limit Theorem to solve this problem.

Step 2 ：The Central Limit Theorem states that the sum of a large number of independent and identically distributed random variables will be approximately normally distributed. Therefore, we can model the total weight of the 45 women as a normal distribution with mean \(45*166\) and standard deviation \(\sqrt{45}*74\).

Step 3 ：We want to find the probability that this total weight exceeds 8100 pounds. The mean of the sum of weights is \(45*166 = 7470\) pounds and the standard deviation of the sum is \(\sqrt{45}*74 = 496.4070910049533\) pounds.

Step 4 ：We calculate the z-score, which is the number of standard deviations the observed value is from the mean. The z-score is \((8100 - 7470) / 496.4070910049533 = 1.269119662905286\).

Step 5 ：Finally, we find the probability that corresponds to this z-score. The probability that the maximum safe weight will be exceeded when a random sample of 45 adult women ride the elevator is approximately \(\boxed{0.1022}\).