Chapter 6.5, Problem 45E

The manager of a construction job needs to figure prices carefully before submitting a bid. He also needs to account for uncertainty (variability) in the amounts of products he might need. To oversimplify the real situation, suppose that a project manager treats the amount of sand, in yards, needed for a construction project as a random variable Y₁, which is normally distributed with mean 10 yards and standard deviation .5 yard. The amount of cement mix needed, in hundreds of pounds, is a random variable Y₂, which is normally distributed with mean 4 and standard deviation .2. The sand costs $7 per yard, and the cement mix costs $3 per hundred pounds. Adding $100 for other costs, he computes his total cost to be

$U=100+7Y_1+3Y_2.$

If Y₁ and Y₂ are independent, how much should the manager bid to ensure that the true costs will exceed the amount bid with a probability of only .01? Is the independence assumption reasonable here?