Question

A fitness company is building a 20-story high-rise. Architects building the high-rise know that women working for the company have weights that are normally distributed with a mean of 143 lb and a standard deviation of 29 lb, and men working for the company have weights that are normally distributed with a mean of 167 lb and a standard deviation or 25 lb. You need to design an elevator that will safely carry 18 people. Assuming a worst case scenario of 18 male passengers, find the maximum total allowable weight if we want to a 0.98 probability that this maximum will not be exceeded when 18 males are randomly selected.

The maximum weight for the elevator is ______________ pounds.

Answer 1

Solution,

Given that,

mean = $\mu$ = 167 lb (men)

standard deviation = $\sigma$ = 25 lb

n = 18

$\mu$_{$\bar x$} = $\mu$ = 167 lb

$\sigma$_{$\bar x$} = $\sigma$ / $\sqrt$ n = 25 / $\sqrt$ 18 = 5.89 lb

Using standard normal table,

P(Z > z) = 0.98

= 1 - P(Z < z) = 0.98  

= P(Z < z ) = 1 - 0.98

= P(Z < z ) = 0.02

= P(Z < -2.054) = 0.02  

z = -2.054

Using z-score formula  

$\bar x$ = z * $\sigma$ _{$\bar x$}+ $\mu$ _{$\bar x$}

$\bar x$ = -2.054 * 5.89 + 167

$\bar x$ = 154.90

The maximum weight for the elevator is 155 pounds.