Final answer:
Bill has probably reached NREM stage 3 sleep, known as deep sleep, characterized by low frequency, high amplitude delta waves, making it difficult for noises to wake him. While stage 4 is listed in the choices and has historically referred to deep sleep, it has been consolidated with stage 3 in modern sleep stage classifications.
Step-by-step explanation:
Bill is no longer disturbed by the sound of his leaking faucet, indicating that he has reached a deep level of sleep where external stimuli are less likely to wake him. This stage of sleep is most probably NREM stage 3, otherwise known as deep sleep or slow-wave sleep. In NREM stage 3, there is a significant reduction in heart rate and respiration, and the brain produces low frequency, high amplitude delta waves. This makes it much more difficult for sounds and other disturbances to awaken someone from sleep. Bill has therefore likely reached NREM stage 3, rather than stage 2 where sleep spindles and K-complexes appear, or REM sleep, which is associated with dreaming and muscle atonia, but not the deepest level of sleep. Stage 4 is no longer a classification used in sleep studies as stages 3 and 4 have been combined into a single stage, now known as stage 3.
Note: Based on the given choices in the original question (stage 4, stage 7, REM, stage 2), and the fact that stage 4 is an outdated term but would historically refer to deep sleep, we could surmise that the intended answer is 'a. Stage 4', which in modern terminology would be equivalent to 'c. Stage 3'.