b) We have to calculate the probability that a group of 25 men exceeds the average allowed weight per passenger.
As the water taxi has a load limit of 3500 lb, the maximum average weight per passenger is 3500/25 = 140 lb.
Then, we can calculate the probability that the mean of a sample of size n = 25 is greater than 140 lb.
The population distribution from where the sample is taken has a mean of 189 lb and a standard deviation of 39 lb.
We can calculate the z-score for M = 140 lb for this sample as:z
[tex]z=\frac{M-\mu}{\sigma\/\sqrt{n}}=\frac{140-189}{39\/\sqrt{25}}=\frac{-49}{39\/5}\approx-6.28[/tex]
Then, the probabilitty can be expressed as:
[tex]P(M>140)=P(Z>-6.28)\approx1[/tex]
It is almost certain that the sample mean will be greater than 140 lb.
c) We now have to calculate the probability that a sample of size n = 20 has a mean that is greater than 175 lb, the new load limit per passenger.
We can repeat the procedure calculating the z-score with this new values (M = 175 and n = 20):
[tex]z=\frac{M-\mu}{\sigma\/\sqrt{n}}=\frac{175-189}{39\/\sqrt{20}}\approx\frac{-14}{8.721}\approx-1.6[/tex]
Then, we can look up the probability for the standard normal distribution when z = -1.6 and obtain:
We can express this as:
[tex]P(M>175)=P(z>-1.6)=0.9452[/tex]
d) As the probabilty of exceeding the load limit per passenger is too high, we can consider that 20 passengers is still not safe enough.
Answer:
b) P(M > 140) = 1
c) P(M > 175) = 0.9452
d) Not safe
