Question

A ski gondola carries skiers to the top of a mountain. It bears a plaque stating that the maximum capacity is 11 people or 1804 lb. That capacity will be exceeded if 11 people have weights with a mean greater than (fraction) 1804 lb/11= 164lb. Assume that weights of passengers are normally distributed with a mean of 178.3 lb and a standard deviation of 40.2 lb.

a. Find the probability that if an individual passenger is randomly​ selected, their weight will be greater than 164 lb. (Round to four decimal places as​ needed.)

b. Find the probability that 11 randomly selected passengers will have a mean weight that is greater than 164 lb. (So that their total weight is greater than the gondola maximum capacity of 1804 lb. (Round to four decimal places as needed).

Answer 1

Solution :

Given that ,

mean = = 178.3

standard deviation = = 40.2

a) P(x > 164 ) = 1 - p( x< 164)

=1- p [(x - ) / < (164 - 178.3) /40.2 ]

=1- P(z < -0.36 )

= 1 - 0.3594 = 0.6406

probability =0.6406

b)

n = 11

= = 178.3  

= / n = 40.2/ 11 = 12.1208

P( >164 ) = 1 - P( < 164 )

= 1 - P[( - ) / < ( 164- 178.3) /12.1208 ]

= 1 - P(z <-1.17 )

= 1 - 0.121 = 0.8790

Probability = 0.8790