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Sleep apnea and sleep stage transitioning. Refer to the sleep apnea study published in Chance (Winter 2009), Exercise 3.54 (p. 132). Recall that the various stages of sleep for a large group of sleep apnea patients were monitored in 30-second “epochs,” and the sleep stage for the previous and current epoch was determined. A summary table is reproduced below. One of the epochs in the study is selected.

a. Given that the previous sleep stage for the epoch was the Wake state, what is the probability that the current sleep stage is REM?

b. Given that the current sleep stage for the epoch is REM, what is the probability that the previous sleep stage was not the Wake state?

c. Are the events {previous stage is REM} and {current stage is REM} mutually exclusive?

d. Are the events {previous stage is REM} and {current stage is REM} independent?

e. Are the events {previous stage is Wake} and {current stage is Wake} independent?

Exercise 3.54

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 on the next page 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.

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?

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