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O2 sensors provide input to the ECM (Engine Control Module). One thing the ECM controls is the pulse width of the signal to the fuel injector. The larger the pulse width, the more fuel is injected. The ECM takes all kinds of input to determine this pulse width. To oversimplify I'm sure, bad input to the ECM (bad O2 sensor), can cause bad output (long pulse width). This long pulse width will cause you to run rich and waste fuel.