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Zanzabum
3 years ago
14

A ski gondola carries skiers to the top of a mountain. assume that weights of skiers are normally distributed with a mean of 199

lb and a standard deviation of 41 lb. the gondola has a stated capacity of 25 ​passengers, and the gondola is rated for a load limit of 3750 lb. complete parts​ (a) through​ (d) below.
a. given that the gondola is rated for a load limit of 3750 ​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 150 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 1. ​(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 187.5 ​lb, which is the maximum mean weight that does not cause the total load to exceed 3750 ​lb? the probability is . 8944. ​(round to four decimal places as​ needed.)
d. is the new capacity of 20 passengers​ safe? since the probability of overloading is over 50 % comma the new capacity appears to be safe enough.
Mathematics
1 answer:
Allushta [10]3 years ago
4 0
A) maximum mean weight of passengers = <span>load limit ÷ number of passengers
</span><span>
maximum mean weight of passengers = 3750 </span>÷ 25 = <span>150lb

</span>B)  First, find the z-score:
z = (value - mean) / stdev
   = (150 - 199) / 41
   = -1.20

We need to find P(z > -1.20) = 1 - P(z < -1.20)

Now, look at a standard normal table to find <span>P(z < -1.20) = 0.11507, therefore:
</span>P(z > -1.20) = 1 - <span>0.11507 = 0.8849

Hence, <span>the probability that the mean weight of 25 randomly selected skiers exceeds 150lb is about 88.5%</span> </span>

C) With only 20 passengers, the new maximum mean weight of passengers = 3750 ÷ 20 = <span>187.5lb

Let's repeat the steps of point B)

z = (187.5 - 199) / 41
   = -0.29

P(z > -0.29) = 1 - P(z < -0.29) = 1 - 0.3859 = 0.6141

</span>Hence, <span>the probability that the mean weight of 20 randomly selected skiers exceeds 187.5lb is about 61.4%

D) The mean weight of skiers is 199lb, therefore:
number</span> of passengers = <span>load limit ÷ <span>mean weight of passengers
                                     = 3750 </span></span><span>÷ 199
                                     = 18.8
The new capacity of 20 skiers is safer than 25 skiers, but we cannot consider it safe enough, since the maximum capacity should be of 18 skiers.</span>
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