In my area I would say a 125-200 pound bear is the average. Once in a great while a 300 or 400 pound bear is killed. With that in mind my 25-06 is my bear hunting rifle out to 500ish yards. Certain places I pack my cannon, but not often. I have killed a few bears, the furthest at about 450 yards. My experience is they die real easy when chest shot.
The next story is certainly not long range hunting related, so it may not belong here, but as I am on the subject here you go. I have killed a bear with a 223 in the noggin, which is always a sure bet at close range 100ish yards. A female in my past chest shot a huge bear with 3 slugs out of her 22lr in a tree in my yard, and it was really dead when it hit the ground.
My most interesting hunting story to date is as follows. I had a young bear hunter who used his bow to shoot a bear, in the yard again. The bear was not hit good and ran into some brush and up a tree, but before the shooter could get into archery range the bear started to come out of the tree. Being as I was at the base of the tree, instinctively I utilized a large club like stick and smacked it hard on the head. I was hoping the bear would go back up the tree so the kid could shoot it again. As it turns out, I knocked the bear cold out making it fall 4-5 feet to the ground.......just long enough to be slightly self-impressed and appear cool, but scared as hell! As the bear started to come back to, whatever amount of life it had left, I used my club to smack the end of two arrows farther into the animal until they were as short as I could get them! We were in thick brush at this time and the kid was freaked out by the commotion and noises coming from the brush, and being yelled at by his father to get more arrows into the animal, and me yelling not to shoot me as I was unable to leave the scene. The bear was thrashing around raising holy hell and I knew it would expire soon from its wounds,,,,, but ****, time does not always fly when you are having fun. I just kept my club (which was quickly getting shorter) in the bears mouth, which was a trying task. By now my hollering invited my dog into the mix, thank god, and basically the bear killed itself by juggling around the arrows inside itself. We do not shoot bears with bows anymore in the yard! It can be done, but wow, I just do not want to be part of it!
Moral of the story, be prepared for super animal strength!
My only advise would be to use a 115 berger or 115 nosler bt to utilize the most of their long range capabilities in your 257 Weatherby. Yet if you like, or are comfortable with what you are shooting, then shoot the lungs, but remember bears have bigger teeth then Antelope!
Best of luck!