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?

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**a. Probability that one randomly selected adult male weighs more than $$148$$ lb**

Let the weight $$X$$ be normally distributed with mean $$\mu=182$$ lb and standard deviation $$\sigma=34$$ lb, i.e.,
$$
X \sim N(182,34^2).
$$
We need to compute
$$
P(X > 148).
$$
First, standardize using
$$
Z=\frac{X-\mu}{\sigma}.
$$
For $$X=148$$,
$$
Z=\frac{148-182}{34}=\frac{-34}{34}=-1.
$$
Thus,
$$
P(X>148)=P(Z>-1).
$$
Since the standard normal distribution is symmetric,
$$
P(Z>-1)=1-P(Z\le -1)=1-(1-P(Z\le 1))=P(Z\le 1).
$$
From the standard normal table, 
$$
P(Z\le 1) \approx 0.8413.
$$
So,
$$
P(X>148) \approx 84.13\%.
$$

---

**b. Probability that the sample mean weight of 27 randomly selected adult males exceeds $$148$$ lb**

Let $$\bar{X}$$ be the sample mean of $$n=27$$ independent males. Then,
$$
\bar{X} \sim N\left(182,\left(\frac{34}{\sqrt{27}}\right)^2\right).
$$
The standard error is
$$
\sigma_{\bar{X}}=\frac{34}{\sqrt{27}}.
$$
Computing $$\sqrt{27}$$:
$$
\sqrt{27}\approx 5.196.
$$
Thus,
$$
\sigma_{\bar{X}} \approx \frac{34}{5.196} \approx 6.547.
$$

Next, standardize $$\bar{X}$$ for the threshold $$148$$ lb:
$$
Z=\frac{\bar{X}-182}{6.547}.
$$
For $$\bar{X} = 148$$,
$$
Z=\frac{148-182}{6.547}=\frac{-34}{6.547} \approx -5.20.
$$
Thus,
$$
P(\bar{X}>148)=P\left(Z > -5.20\right).
$$
Since $$-5.20$$ is very far in the left tail,
$$
P(Z>-5.20) \approx 1.
$$
In other words, the probability that the mean weight of a sample of 27 exceeds $$148$$ lb is virtually $$100\%$$.

---

**c. Conclusion about the safety of the elevator**

The elevator’s capacity placard is based on a maximum total weight of $$4000$$ lb for $$27$$ passengers, which corresponds to a mean weight of
$$
\frac{4000}{27} \approx 148 \text{ lb per passenger}.
$$
However, the actual distribution of adult male weights has a mean of $$182$$ lb and a standard deviation of $$34$$ lb. We found that:

- A single adult male has an $$\approx 84\%$$ chance of weighing more than $$148$$ lb.
- The mean weight of 27 randomly selected males is almost certain to exceed $$148$$ lb.

Thus, using a $$148$$ lb average for capacity significantly underestimates the actual weight of passengers. This indicates that the elevator is very likely to be overloaded if used by adult males, making it unsafe under typical loading conditions.
**a. Probability that one randomly selected adult male weighs more than 148 lb:** About 84%. **b. Probability that the mean weight of 27 randomly selected adult males exceeds 148 lb:** Almost certain (over 99.99%). **c. Conclusion about the elevator's safety:** The elevator is likely unsafe because the average weight of adult males exceeds the elevator's capacity limit, leading to potential overloading.

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