What experiment can I perform to show that the higher the dissolved oxygen content of water is the better the performance of fish is?
Well, the most important thing about an experiment is to only change one variable at a time, so the variable you would change is dissolved oxygen content. I would have five tanks, one as a control tank with nothing adding or subtracting oxygen, two with supplemented oxygen, and two with reduced oxygen.
To supplement oxygen, it may be cheapest to introduce plants. Unfortunately, however, this will change a variable in the experiment. I would suggest using aeration inside the tank filter, to prevent aerator noise from disturbing the fish. Using a paddle wheel or bubble aerator in the filter would be efficient and cost effective.
To lower the portion of dissolved oxygen, you can saturate the water of the tank with carbon dioxide instead. To do this, I would run a line to a canister of pressurized carbon dioxide, feeding the gas through a bubble aerator in the filter to carbonate the water, occupying the water in the place of oxygen. Be careful with this, however, as it can be somewhat tricky to get a balance of excess carbon dioxide that fish can survive. In the event fish are unable to survive, use the tanks as control or oxygenated tanks instead.
It would be wise to keep tabs on the oxygen content of the tanks while you experiment. I suggest purchasing an oxygen meter for each tank and collecting data continuously. Fish consume resources differently depending on the time of day, activity levels, and other factors.
To test fish 'performance,' I can suggest a few methods. One is to see how active fish are through observation. Better numbers can be taken if you start keeping track of fish deaths in each tank. You can try measuring how much food the fish consume as well- fish may stop eating as much, so scooping out leftover food and weighing it can be a viable method. Certain fish can give clues about their living conditions based on color, like GloFish. Finally, reproductive activity can give clues about fish performance. Fast breeding fish like minnows would work well in that application.
To conclude, make sure you do not change anything besides oxygen between the tanks. Same feeding, same light, same number of fish. Do that, and you have the setup of a good experiment.