Question

A ski gondola carries skiers to the top of a mountain. Assume that weights of skiers are normally distributed with a mean of 186 lb and a standard deviation of 45 lb. The gondola has a stated capacity of 25 ​passengers, and the gondola is rated for a load limit of 3500 lb. Complete parts​ (a) through​ (d) below. a. Given that the gondola is rated for a load limit of 3500 ​lb, what is the maximum mean weight of the passengers if the gondola is filled to the stated capacity of 25 ​passengers? a. The maximum mean weight is? ​(Type an integer or a decimal. Do not​ round.) b. If the gondola is filled with 25 randomly selected​ skiers, what is the probability that their mean weight exceeds the value from part​ (a)? The probability is? ​(Round to four decimal places as​ needed.) c. If the weight assumptions were revised so that the new capacity became 20 passengers and the gondola is filled with 20 randomly selected​ skiers, what is the probability that their mean weight exceeds 175 ​lb, which is the maximum mean weight that does not cause the total load to exceed 3500 ​lb? The probability is ?

Answer 1

µ=186

σ = 45

n=25

a)

The maximum mean weight is = 3500/25=140

b)

µ =    186                                      
σ =    45                                      
n=   25                                      
                                          
X =   140                                      
                                          
Z =   (X - µ )/(σ/√n) = (   140   -   186   ) / (    45   / √   25   ) =   -5.1  
                                          
P(X ≥   140   ) = P(Z ≥   -5.11   ) =   P ( Z <   5.111   ) =    1.0000           (answer)

c)

µ =    186                                      
σ =    45                                      
n=   20                                      
                                          
X =   175                                      
                                          
Z =   (X - µ )/(σ/√n) = (   175   -   186   ) / (    45   / √   20   ) =   -1.1  
                                          
P(X ≥   175   ) = P(Z ≥   -1.09   ) =   P ( Z <   1.093   ) =    0.8628           (answer)