Answer to Question #150606 in Statistics and Probability for Vladimyr Lubin

Question

(6.4.18) A boat capsized and sank in a lake. Based on an assumption of a mean weight of 137lb, the boat was rated to carry 70 passengers​ (so the load limit was 9590​lb). After the boat​ sank, the assumed mean weight for similar boats was changed from 137lb to 174lb. Complete parts a and b below.
a. Assume that a similar boat is loaded with 70 ​passengers, and assume that the weights of people are normally distributed with a mean of 178.7lb and a standard deviation of 40.9lb. Find the probability that the boat is overloaded because the 70 passengers have a mean weight greater than 137lb.
The probability is?

Answer 1

$P(X\ge137) = P(Z \ge \frac {137 - \mu_X} {\sigma_X}) = P(Z \ge \frac {137 - 178.7} {40.9/ \sqrt{70}}) = P(Z\ge -8.53) = \\ 1-P(Z< -8.53)=1-0.0001 = 0.9999$