An elevator has a placard stating that the maximum capacity is \( 4000 \mathrm{lb}-27 \) passengers. So, 27 adult male passengers can have a mean weight of up to \( 4000 / 27=143 \) pounds. Assume that weights of males are normally distributed with a mean-of 182 lb and a standard deviation of 34 lb . a. Find the poobability that 1 randomly selected adult male has a weight greater than 148 lb . b. Find the probahility that a sample of 27 randomly selected adut mates has a mean weight greater than 148 lb . c. What do you conclude about the safely of this elevator?

To tackle this fun statistical adventure, let’s zoom in on the details: For the first part, we can find the probability that one randomly selected adult male weighs more than 148 lb using the z-score formula: \( z = \frac{X - \mu}{\sigma} \). Given that the mean is 182 lb and the standard deviation is 34 lb, the z-score for 148 lb is \(-1\). Checking a standard normal distribution table shows that the probability of a single adult male weighing more than 148 lb is about \(0.8413\) or \(84.13\%\). Now, transitioning to part b, when dealing with a sample of 27 adult males, we use the Central Limit Theorem. The standard error (SE) is given by \( \sigma/\sqrt{n} = 34/\sqrt{27} \approx 6.55\). The z-score for a sample mean of 148 lb becomes much more significant, clocking in at approximately \(-5.19\). This reflects an incredibly tiny probability of that happening, typically less than \(0.0001\) or \(0.01\%\). As for the elevator's safety? With the vast majority of adult males weighing above 148 lb, the odds of hitting the threshold with 27 men are quite slim. The elevator's placard is a friendly reminder to adhere to capacity limits—safety should always be prioritized over a few extra weights!