More than 100 million people around the world are not getting enough sleep; the

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 normal model with E(X) = 6.71 hours and SD(X) = 1.21 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 6.59 and 6.91. (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 6.59 and 6.91. (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 6.59 and 6.91. (use 4 decimal places in your answer)

Question 4. The Central Limit Theorem was needed to answer questions 1, 2, and 3 above. (true or false)

Explanation / Answer

Ans:

1)n=4

z(6.59)=(6.59-6.71)/(1.21/sqrt(4))=-0.198

z(6.91)=(6.91-6.71)/(1.21/sqrt(4))=0.331

P(-0.198<z<0.331)=P(z<0.331)-P(-0.198)=0.6295-0.4214=0.2081

2)n=16

z(6.59)=(6.59-6.71)/(1.21/sqrt(16))=-0.397

z(6.91)=(6.91-6.71)/(1.21/sqrt(16))=0.661

P(-0.397<z<0.661)=P(z<0.661)-P(-0.397)=0.7457-0.3458=0.3999

3)n=25

z(6.59)=(6.59-6.71)/(1.21/sqrt(25))=-0.496

z(6.91)=(6.91-6.71)/(1.21/sqrt(25))=0.826

P(-0.496<z<0.826)=P(z<0.826)-P(-0.496)=0.7957-0.3100=0.4857

4)True

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