Studies by neuroscientists at the Massachusetts Institute of Technology (MIT) reveal that melatonin, which is secreted by the pineal gland in the brain, functions naturally as a sleep-inducing hormone. Male volunteers were given various doses of melatonin or placebos and then placed in a dark room at midday and told to close their eyes and fall asleep on demand. Of interest to the MIT researchers is the time Y (in minutes) required for each volunteer to fall asleep. With the placebo (i.e., no hormone), the researchers found that the mean time to fall asleep was 15 minutes. Assume that with the placebo treatment μ = 15 and σ = 5.

a. Consider a random sample of n = 20 men who are given the sleep-inducing hormone, melatonin. Let Y̅ represent the mean time to fall asleep for this sample. If the hormone is not effective in inducing sleep, describe the sampling distribution of Y̅.

b. Refer to part a. Find P(Y̅ ≤ 6).

c. In the actual study, the mean time to fall asleep for the 20 volunteers was Y̅ = 5. Use this result to make an inference about the true value of μ for those taking the melatonin.