Thanks for the advice, everyone. Here is the "after" photo. I went to the local hardware store (on Small Business Saturday!), purchased a 1 1/2 x 8 inch sink tailpiece and a couple plastic washers, cut off a 2 inch section of the pipe and installed it. It appeared to me that the new pipe was the exact same diameter as the old one, but the old one was only about an inch long, which was the problem. The new pipe passed the test perfectly.