Sleep apnea and sleep stage transitioning. Sleep apnea is a common sleep disorder characterized by collapses of the upper airway during sleep. Chance (Winter 2009)

investigated the role of sleep apnea in how people transition from one sleep stage to another. The various stages of sleep for a large group of sleep apnea patients were monitored in 30-second intervals, or “epochs.” For each epoch, sleep stage was categorized as Wake, REM, or Non-REM. The table below provides a summary of the results. Each cell of the table gives the number of epochs that occurred when transitioning from the previous sleep stage to the current sleep stage. Consider the previous and current sleep stage of a randomly selected epoch from the study.

Previous Sleep Stage Current Stage Non-REM REM Wake Non-REM 31,880 160 1,733 REM 346 7,609 175 Wake 1,588 358 6,079 Totals 33,814 8,127 7,987

a. List the sample points for this experiment.

b. Assign reasonable probabilities to the sample points.

c. What is the probability that the current sleep stage of the epoch is REM?

d. What is the probability that the previous sleep stage of the epoch is Wake?

e. What is the probability that the epoch transitioned from the REM sleep stage to the Non-REM sleep stage?