Elevator SafetyExercise 9 uses µ= 189 lb, which is based on Data Set 1 “Body Data” in Appendix B. Repeat Exercise 9 using µ = 174 lb (instead of 189 lb), which is the assumed mean weight that was commonly used just a few years ago. What do you conclude about the effect of using an outdated mean that is substantially lower than it should be?

Refer to exercise 9 for the following information,

Increased maximum capacity increased to 5000 lb due to 25% safety factor, which leads to a corresponding mean weight of 185 lb for 27 passengers.

Here, the mean weight of all males is 174 lb.

Let X be the weight of male passengers in the elevator.

The distribution of weights is normal.

$$\begin{gathered} X~N\left(\mu ,{\sigma }^2\right) \\ ~N\left(174,{39}^2\right) \end{gathered}$$

Define $\overline{X}$ as the sample mean distribution of 27 male passengers.

For normal population, the sample mean distribution has normal distributionwith mean and standard deviation as,

$$\begin{gathered} {\mu }_{\overline{X}}=\mu \\ {\sigma }_{\overline{X}}=\frac{\sigma }{\sqrt{n}} \\ =\frac{39}{\sqrt{27}} \\ =7.5055 \end{gathered}$$

Thus, $\overline{X}~N\left(189,{7.5055}^2\right)$ .

The z-score associated to mean weight of 185 on the sample mean distribution,

$$\begin{gathered} z=\frac{\overline{x}-{\mu }_{\overline{X}}}{{\sigma }_{\overline{X}}} \\ =\frac{185-174}{7.5055} \\ =1.4656 \end{gathered}$$

The probability that elevator is overloaded as the mean weight exceeds 185 lb when the elevator has 27 male passengers is expressed as,

$$\begin{gathered} P\left(\overline{X}>185\right)=P\left(Z>1.4656\right) \\ =1-P\left(Z<1.4656\right)...\left(1\right) \end{gathered}$$

where, the probability of 1.4656 is the right tailed area of z-score 1.4656, under the standard normal curve.

From the standard normal table, the left tailed probability for z-score 1.47 is computed as 0.9292.

Substitute value in equation (1),

$$\begin{gathered} P\left(\overline{X}>185\right)=1-0.9292 \\ =0.0708 \end{gathered}$$

Thus, the probability that the elevator is overloaded when there are 27 adult males in the elevator is 0.0708.

The chances for an elevator to be overloaded with 27 adult males when the mean weight is 174 lb is 7.08% and 70.19% when the mean weight is 189 lb.

The elevator appearsto be safe when the mean of weight is 174 lb as compared to 189 lb.

Thus, a substantially lower mean which was outdated, lowers the chances of an overloaded elevator with 27 male adults.