You're right to worry about the chicken being raw in the center, since raw chicken can be dangerous. But it is easy enough to get this dish right. First of all, think about flavoring your drumsticks. You can cook them plain, just rubbing a little oil or melted butter on them first to make the skin crisp up, or you could use a spice rub or a chicken marinade recipe like this one:
When your chicken is greased, flavored or marinated, preheat the oven to 400 degrees F and bake the chicken for half an hour, turning it over once.
Remove the roasting pan or the dish and use a sharp knife to cut into the biggest drumstick. If you see any pinkness inside the chicken is not done. Return it to the oven. The chicken is done when there is no pink inside and the juices run clear. Remember that drumsticks are not as lean as breast so the meat will be darker. Chicken breast is white when it is cooked. Chicken drumsticks are brownish.
You can serve the finished chicken drumsticks hot with mashed potatoes and something like green beans or coleslaw, or you can refrigerated them for up to three days and serve them cold for lunch with a salad. The most important thing is to cut into the chicken in the thickest part and check there is no pink or blood next to the bone, then you have proof it's cooked through.