Question

More than 100 million people around the world are not getting enough sleep; the average adult needs between 7.5 and 8 hours of sleep per night. College students are particularly at risk of not getting enough shut-eye.

A recent survey of several thousand college students indicated that the total hours of sleep time per night, denoted by the random variable X, can be approximated by a normalmodel with E(X) = 6.84 hours and SD(X) = 1.24 hours.

Question 1. Find the probability that the hours of sleep per night for a random sample of 4 college students has a mean x between 6.7 and 6.93.

(use 4 decimal places in your answer)

Question 2. Find the probability that the hours of sleep per night for a random sample of 16 college students has a mean x between 6.7 and 6.93.

(use 4 decimal places in your answer)

Question 3. Find the probability that the hours of sleep per night for a random sample of 25 college students has a mean x between 6.7 and 6.93.

(use 4 decimal places in your answer)

Question 4. The Central Limit Theorem was needed to answer questions 1, 2, and 3 above.

TrueFalse    

Answer 1

1) P(6.7 < < 6.93)

= P((6.7 - )/() < (X - )/( < (6.93 - )/())

= P((6.7 - 6.84)/(1.24/) < Z < (6.93 - 6.84)/(1.24/))

= P(-0.23 < Z < 0.15)

= P(Z < 0.15) - P(Z < -0.23)

= 0.5596 - 0.4090

= 0.1506

2) P(6.7 < < 6.93)

= P((6.7 - )/() < (X - )/( < (6.93 - )/())

= P((6.7 - 6.84)/(1.24/) < Z < (6.93 - 6.84)/(1.24/))

= P(-0.45 < Z < 0.29)

= P(Z < 0.29) - P(Z < -0.45)

= 0.6141 - 0.3264

= 0.2877

3) P(6.7 < < 6.93)

= P((6.7 - )/() < (X - )/( < (6.93 - )/())

= P((6.7 - 6.84)/(1.24/) < Z < (6.93 - 6.84)/(1.24/))

= P(-0.56 < Z < 0.36)

= P(Z < 0.36) - P(Z < -0.56)

= 0.6406 - 0.2877

= 0.3529

4) True.