I plan to build a brick wall, so I need to compact the ground first. The hardware store sells a tool called a tamper, that sells for $60 and only weighs 15 lbs, so it seems inefficient. How is that sufficient weight to pack down the ground? If a worker weighs 200+ lbs, is that more efficient to have them cut on a small square of wood and jump on that? If not, is there some alternative method of stamping the ground down?
The first question is why you think "you need to compact the ground first" - even without frost as a consideration (not knowing where you live or if it freezes there) the first step in putting up such a wall is to remove the spongy/squishy topsoil and humus and get down into the mineral subsoil, which (if undisturbed) is normally stable and fairly compact already. In a freezing climate you need to get below frost line unless you are simply making a short loose-stack (no mortar) garden wall that can shift with the ground movement and may need occasional restacking. Once down to subsoil or below frost line a reinforced concrete foundation set on undisturbed soil will support the wall, and tamping is not normally needed unless the area you are working in was recently filled and has not consolidated yet. Alternatives might include a well-drained "rubble trench" foundation or other means of preventing frost movement, which will crack your mortared brick wall.
Your typical hardware store "tamper" is of no practical use in compacting earth, as it's commonly 10x10 inches (100 square inches) and is more a "flattener" than a tamper. Good for mashing asphalt driveway patch level, perhaps.
An ordinary sledgehammer is a better hand-operated tool for tamping. Handle vertical, lift and drop, repeat. Doing a good job with one is tedious but quite possible if you prefer not to rent a powered plate compactor. For something like compacting around a fencepost, an iron digging bar can be used to compact.
With any compacting method, the soil to be compacted must be put in as shallow layers, each compacted individually, or only the top few inches/centimeters will be effectively compacted. The pressure applied to the soil surface spreads out in a cone, and is soon negligible and unable to move soil particles. Thus, compacting the bottom of a trench (which you are not building up) has very little overall effect.