I would say that it is not a safe assumption. There are three main places where a wheel could be loose at, assuming the lug nuts are tight. Those are the wheel bearing(s), upper/lower ball joints, or at bushings at the base of the control arm. You might also be getting deflection out of your tie rod ends or steering rack (if equipped) as well. You could also have a combination of the three going bad. There are different things you need to do to check to see if which part is bad. Each involves looking for where the movement is occurring at. In fact, if you jack the front end up and you can wiggle the tire, it is most likely NOT your ball joints, this is because there is spring pressure keeping the ball joints tight. Like I was saying, you really need to check to see where the deflection is occurring in order to diagnose the issue.
If the wheel bearing, you should jack the vehicle up and wiggle the tire while off the ground. If you feel deflection or movement, you need to look at the back side of the tire. If the tire is moving and the rest of the suspension isn't, it's probably the bearing. It could be the inner or the outer wheel bearings, or both.
If the ball joint, jack up the front end of the vehicle so it is supported on jack stands, but not by putting them under the suspension. Use a frame rail or some other position. Now put a jack under one of the A-frames and jack it up enough to just support the spring pressure. You want the jack to be supporting the control arm and for it to be solid. After this process your tire should be about 3-4" off the ground. Next, place an 8' 2x4 under the center of the tire. You want about 6-8" to be behind the tire and the rest to be out the front. You are going to use the board as a fulcrum to help you lift the tire up a little bit. Wiggle the board back and forth and notice if there is any movement at the point of the lower ball joint. This should be easier to detect because any movement doing this method means the ball joint is bad. There should be zero deflection between the lower control arm and the spindle. To be honest with you, the upper ball joint rarely goes bad because almost all of the abuse is covered by the lower ball joint (the lower supports the weight of the truck while the upper just keeps the spindle located).
If you don't see any movement out of the ball joint, you'll need to check the bushings at the base of the control arm. You are looking for major cracking or missing rubber in the bushing. More than likely you won't see movement out of this area, but you may see that it is displaced. It will need to look as though the control arm housing is centered around where the bolts go through. If it looks like it's off to one side or the other, it may be worn out. Check both front and back of both the top and bottom control arms.
When the vehicle is back on the ground, you can check the tie rod ends. While a helper is in the cab, have them turn the steering wheel from left to right, only going far enough until the wheel stops turning easily. They won't need to move it far enough to actually move the tires. They will probably be going from the 11 to 1 o'clock position of the steering wheel. Just have them do it at a moderate pace (not fast/slow). You are going to be looking to see if there is any movement of the tie rod end but not at steering arm, both sides of the vehicle. You'll want to trace the steering linkage all the way back to the from the steering arm at the spindle to where it meets the steering unit (I don't remember if your year of F150 has a rack-n-pinion or steering box). Every place there is a joint, check to ensure there's no deflection. If there is a rack-n-pinion, you'll most likely have inner tie rod ends at the rack unit. You'll want to see if there is any movement there as well.
One of the tell tale signs that something is going bad in the steering department is if the dust cover is cracked/split. If a ball joint or tie rod end dust cover is split/cracked, that joint is most likely in the process of dieing. Pay special attention to these areas if you see it. Even if it doesn't show any deflection, it most likely will need to be changed soon. This usually occurs because non-serviceable (sealed) joints have dried out. Due to the crack, it lets dust/dirt in which helps destroy the joint even faster. Once this happens it's only a matter of time.
As far as doing the work yourself, it depends on which part is bad. If you're talking about bushings/tie rods/ball joints, this is a lot of work. If you don't have someone helping you who knows what they are doing, it would be tough to do, even with instructions. I'm not saying you couldn't get it done, but it would be a rough go of it. You would also need some specialized tools, like suspension spring compressor, a ball joint splitter, and ball joint press. Also, changing out the control arm bushings is a lot of work ... very much not easy to get it done right without knowing what you are doing (not intuitive). If you figure out it is the bearings, this is something which you can probably do, but there are a lot of parts which you have to be anal about both in cleanliness and with order/orientation. Then you'll need to know how to get the bearings/races off and the new ones on, as well as how to pack the bearings.
If you want to tackle this yourself, but are unsure of your diagnosis, you may want to take this to a shop and have them do the diagnosis. They should be able to tell you right away what's wrong with it.