A commodity trader believes that average volume of wheat he traded can be described by a Normal model with a mean of 32,000 metric tons and standard deviation of 2,500 metric tons.

a) If a client buys some bushels of his wheat, would it be reasonable for the client to hope that the transactions will reach 40,000 metric tons? Explain.

b) Approximately what fraction of the transactions can be expected to reach less than 30,000 metric tons?

c) Approximately what fraction of these trades can be expected to reach between 30,000 and 35,000 metric tons?

d) Estimate the IQR of the trade volumes.

e) In planning an investment strategy, the commodity trader wants to offer an opt-out guaranty (or an option to cancel a trade) to any
customer whose transaction volumes failed to deliver an agreed volume of transaction compared to the mean. However, he does not want to take too big a risk. If he is willing to give transaction refunds to no more than 1 of every 25 customers, for what volume level of transactions can he offer an opt-out guaranty?