To get full marks for the following questions you need to convert the question from words to a

mathematical expression (i.e. use mathematical notation), defining your random variables where

necessary, and using correct probability statements.

Suppose that the amount of sleep that adults get per night is normally distributed with a mean of

7.1 hours and a standard deviation of 1.2 hours.

(a)What amount of sleep distinguishes the longest 10%?

(b)What is the probability that a randomly selected adult sleeps between 6 and 8 hours

per night?

(c)Suppose adults who sleep less than 6 hours per night are at increased risk of diabetes.

Show that approximately 17.96% of adults are at increased risk of diabetes due to sleeping too

little.

(d) Let X be the number of people in a random sample of 20 who are at increased risk

of diabetes due to sleeping too little. What probability distribution does X follow? Justify your

answer.

(e)Using the probability distribution from part (d), find the probability that at least

3 people in the random sample of 20 are at increased risk of diabetes due to sleeping too little.

(f) Let L be the amount of time (in minutes) it takes a randomly selected adult to fall

asleep after lying in bed. Suppose L follows a uniform distribution from 5 to 60. What is the

probability that the adult will fall sleep within 15 minutes after lying in bed?