Question

According to the Sleep Foundation, the average night’s sleep is 7 hours. Assume the standard deviation is 0.5 hours and that the probability distribution is normal.
a. What is the probability that a randomly selected person sleeps more than 9 hours?
b. What is the probability that a randomly selected person sleeps 5 hours or less?
c. What is the probability that a randomly selected person sleeps exactly 10 hours?
d. What is the probability that a randomly selected person sleeps more than 7 hours?
e. Doctors suggest getting between 7 and 9 hours of sleep each night. What percentage of the population gets this much sleep?

Answer 1

X=sleeping hours

$X\sim N(7,0.5)$

this can be convert to standard normal distribution by using,

$z=\frac{x-\bar{x}}{\sigma}=\frac{x-7}{0.5}\\$

a)

$P(X>9)=P(Z>\frac{9-7}{0.5})\\ P(X>9)=P(Z>4)=0\\$

probability that a randomly selected person sleeps more than 9 hours = 0

b)

$P(X\le5)=P(Z\le\frac{5-7}{0.5})\\ P(X\le9)=P(Z\le-4)=0\\$

probability that a randomly selected person sleeps 5 hours or less = 0

c) point probability of a normal distribution is 0.

Therefore,probability that a randomly selected person sleeps exactly 10 hours = 0

d) since the mean 7 hours,

probability that a randomly selected person sleeps more than 7 hours = 0.5

e)

$P(X>9)=0 \ and\ P(X>7)=0.5$

therefore,

$P(7<X<9)=0.5$

 percentage of the population gets between 7 and 9 hours of sleep=50%