We brought the kids yesterday to Plymouth, Massachusetts to see the Plimouth Plantation, and the full-size replica of the Mayflower. My wife's family can trace itself back to John Howland, one of the original 104 passengers back in 1620 and the oldest surviving pilgrim (he died in his 80s, having fathered ten children). There were no Spragues on the Mayflower itself, but several of the passengers either later married Spragues or had daughters who did, so although as far as I know there is no similar well-documented genealogy on my side of the family, I'm almost certainly a direct male descendent. Genetic testing on my grandmother last Fall shows that I might be descended from some of the native people as well.
That's less of a big deal than you think. In fact you, my dear reader, are almost certainly a direct descendent of the Mayflower passengers too. It's obvious from simple mathematics. Four hundred years have passed since the Pilgrim days. If you assume that each of your ancestors had children somewhere between age 20 and 40, then you are something like 10 or 20 generations removed from then. Each generation has a mother and father, of course, so the total number of your direct ancestors who were alive in 1620 is somewhere between 2 to the 10th and 2 to the 20th power, or anywhere from 1,000 to 1 million people. You are a direct descendent of every single one of them, so don't be fooled by your last name or what you think of as your ethnicity.
This is true even if you think your ancestors are from China or India, or Jewish, or African. Think about it: you had as many as a million ancestors alive and breeding back in 1620. Only one supplied you with your last name, yet each of the others contributed just as much of your genetic material as he did. That's a lot of people, and a lot of years for all kinds of mixups: orphans abandoned and rescued because of war or disease, "non-paternity" events from rape or infidelity, kidnappings. All it takes is one, just one of those events and you are part of another lineage even if it's not preserved in your name or looks.
Welcome to the family!