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) = 6.84 hours and SD(X) = 1.2 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 6.6 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 6.6 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 6.6 and 6.91. (use 4 decimal places in your answer)

Answer 1

Sample size (n) = 4
Since we know that

P(6.6 < x < 6.91)=?

This implies that
P(6.6 < x < 6.91) = P(-0.4 < z < 0.1167) = P(Z < 0.1167) - P(Z < -0.4)
P(6.6 < x < 6.91) = 0.5464511048617128 - 0.3445782583896758
P(6.6 < x < 6.91) = 0.2019

Question 2) Sample size (n) = 16
Since we know that

P(6.6 < x < 6.91)=?

This implies that
P(6.6 < x < 6.91) = P(-0.8 < z < 0.2333) = P(Z < 0.2333) - P(Z < -0.8)
P(6.6 < x < 6.91) = 0.5922357706970902 - 0.2118553985833967
P(6.6 < x < 6.91) = 0.3804

Question 3) Sample size (n) = 25
Since we know that

This implies that
P(6.6 < x < 6.91) = P(-1.0 < z < 0.2917) = P(Z < 0.2917) - P(Z < -1.0)
P(6.6 < x < 6.91) = 0.6147419953895119 - 0.15865525393145707
P(6.6 < x < 6.91) = 0.4561
PS: you have to refer z score table to find the final probabilities.
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