Question

A ski gondola carries skiers to the top of a mountain. Assume that weights of skiers are normally distributed with a mean of 183 lb and a standard deviation of 37 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?
The maximum mean weight is 140 lb.
​(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 11.
​(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 2020 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 nothing.
​(Round to four decimal places as​ needed.)

Answer 1

a) It is given that only 25 passengers can board a gondola at a time and the weight limit for the gondola is 3500 lbs. So, the maximum mean weight would be = 3500 / 25 = 140 lbs.

b) Now, we need to find the probability that the mean weight exceeds 140 lbs.

So, let the random variable denoting mean weight be X.

So, P(X > 140) = 1 - P(X < 140)

= 1 - P(Z < (140-183)/37)

= 1 - P(Z < -1.1621)

= 1 - 0.123

= 0.877

c) P(X > 175) = 1 - P(X< 175)

= 1 - P(Z < (175-183)/37)

= 1 - P(Z < -0.2162)

= 1 - 0.414

= 0.586

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