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) = 6.72 hours and SD(X) = 1.16 hours.

- Find the probability that the hours of sleep per night for a random sample of 4 college students has a mean x between 6.8 and 6.91.

- Find the probability that the hours of sleep per night for a random sample of 16 college students has a mean x between 6.8 and 6.91. (use 4 decimal places in your answer)

- Find the probability that the hours of sleep per night for a random sample of 25 college students has a mean x between 6.8 and 6.91.(use 4 decimal places in your answer)

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

True

False    

Answer 1

1)

Here, μ = 6.72, σ = 0.58, x1 = 6.8 and x2 = 6.91. We need to compute P(6.8<= X <= 6.91). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (6.8 - 6.72)/0.58 = 0.14
z2 = (6.91 - 6.72)/0.58 = 0.33

Therefore, we get
P(6.8 <= X <= 6.91) = P((6.91 - 6.72)/0.58) <= z <= (6.91 - 6.72)/0.58)
= P(0.14 <= z <= 0.33) = P(z <= 0.33) - P(z <= 0.14)
= 0.6293 - 0.5557
= 0.0736

2)

Here, μ = 6.72, σ = 0.29, x1 = 6.8 and x2 = 6.91. We need to compute P(6.8<= X <= 6.91). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (6.8 - 6.72)/0.29 = 0.28
z2 = (6.91 - 6.72)/0.29 = 0.66

Therefore, we get
P(6.8 <= X <= 6.91) = P((6.91 - 6.72)/0.29) <= z <= (6.91 - 6.72)/0.29)
= P(0.28 <= z <= 0.66) = P(z <= 0.66) - P(z <= 0.28)
= 0.7454 - 0.6103
= 0.1351

3)

Here, μ = 6.72, σ = 0.232, x1 = 6.8 and x2 = 6.91. We need to compute P(6.8<= X <= 6.91). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (6.8 - 6.72)/0.232 = 0.34
z2 = (6.91 - 6.72)/0.232 = 0.82

Therefore, we get
P(6.8 <= X <= 6.91) = P((6.91 - 6.72)/0.232) <= z <= (6.91 - 6.72)/0.232)
= P(0.34 <= z <= 0.82) = P(z <= 0.82) - P(z <= 0.34)
= 0.7939 - 0.6331
= 0.1608

4)

true