Question

1. According to one study, the average hours of sleep on a weeknight for men was 6.9 hours with a standard deviation of 1.5 hours.

a. Find the proportion of men who sleep for more than 8 hours per night.

b. Find the proportion of men who sleep for less than 6 hours per night.

c. Find the proportion of men who sleep between 6-8 hours per night.  

d. Find the number of hours that represents the 90^th percentile.

Answer 1

solution:-

given that mean = 6.9 , standard deviation = 1.5

formula z = (x-mean)/standard deviation

a. P(x > 8)

=> p(z > (8-6.9)/1.5)

=> P(z > 0.73)

=> 1 - P(z < 0.73)

=> 1 - 0.7673

=> 0.2327

b. P(x < 6)

=> P(z < (6-6.9)/1.5)

=> P(z < -0.6)

=> 1 - P(z < 0.6)

=> 1 - 0.7257

=> 0.2743

c. P(6 < x < 8)

=> P((6-6.9)/1.5 < z < (8-6.9)/1.5)

=> P(-0.6 < z < 0.73)

=> P(z < 0.73) - P(z < -0.6)

=> 0.7673 - 0.2743

=> 0.4930

d. the value that corresponding to the given 90th percentile is z = 1.28

formula

=> x = z*standard deviation + mean

=> x = 1.28 * 1.5 + 6.9

=> x = 8.82

=> x = 9