Question 3 (20 points) A farmer in Iowa is planning her production of soybeans for the coming year. The farmer intends to plant two different soybean varieties: A and B. The farmer's total available land is 1600 acres. The yield in bushels per acre of soybean variety A is known to be normally distributed with a mean of 50 and a standard deviation of 6. The yield in bushels per acre of soybean variety B is known to be normally distributed with a mean of 60 and a standard deviation of 8. In addition, the yields of A and B are known to have a correlation of 0.10677. The farmer intends to allocate half of her available land to variety A, and half of her available land to variety B.

a) What is the probability that the yield of variety A is between 55 and 65 bushels per acre? (3 points)

b) What is the probability that the yield of variety A is exactly 50 bushels per acre? (2 points) 9 Question 3 (continued)

c) What is the expected value of the total soybean harvest in bushels? (3 points) d) What is the standard deviation of the total soybean harvest in bushels?

e) In order to break even, the farmer has calculated that she needs to harvest at least 78000 bushels for this coming year. What is the probability that the farmer breaks even? (5 points)

f) The farmer is considering increasing her available storage capacity to store the coming soybean harvest. How much storage capacity, in bushels, should the farmer have so that there is only a 1% probability that the farmer is not able to store the total soybean harvest?