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 40 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 ____ 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 _____ . ​(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 _____. ​(Round to four decimal places as​ needed.)

Answer 1

a) The maximum mean weight is 3500/25=140 lb

b) Note that if Xi are iid N(183,SD=40), i=1,2,3..,25, the sample mean M~N(183,SD=40/5=8)

To find P(M>=140)

Now P(M>=140)=P((M-183)/8>(140-183)/8)=P(N(0,1)>-5.375)=1.000

The probability is 1.000

c)

Note that if Xi are iid N(183,SD=40), i=1,2,3..,20, the sample mean A~N(183,SD=40/sqrt(20)=8.944272)

To find P(A>=175)

Now P(A>=175)=P((A-183)/8.944272>(175-183)/8.944272)=P(N(0,1)> -0.8944272)=0.8144533

The probability is  0.8145

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