According to the Sleep Foundation, the average night's sleep...

According to the Sleep Foundation, the average night's sleep is 6.8 hours (Fortune, March 20,2006)$.$ Assume the standard deviation is .6 hours and that the probability distribution is normal. a. What is the probability that a randomly selected person sleeps more than 8 hours? b. What is the probability that a randomly selected person sleeps 6 hours or less? c. Doctors suggest getting between 7 and 9 hours of sleep each night. What percentage of the population gets this much sleep?

According to the Sleep Foundation, the average night's sleep is 6.8 hours (Fortune, March
20,2006)$.$ Assume the standard deviation is .6 hours and that the probability distribution is normal.
a. What is the probability that a randomly selected person sleeps more than 8 hours?
b. What is the probability that a randomly selected person sleeps 6 hours or less?
c. Doctors suggest getting between 7 and 9 hours of sleep each night. What percentage of the population gets this much sleep?

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about its mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. It is defined by two parameters: the mean (which determines the center of the distribution) and the standard deviation (which determines the spread or dispersion of the distribution). This distribution is fundamental in statistics due to the Central Limit Theorem.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion of a set of values. In the context of a normal distribution, it determines how spread out the values are around the mean. A smaller standard deviation implies that the data points tend to be close to the mean, while a larger standard deviation indicates that the data are spread out over a wider range.

Z-Score Transformation

The z-score transformation is a method of standardizing individual data points by converting them into a number of standard deviations away from the mean. This transformation allows comparisons across different normal distributions by eliminating the effects of scale and location, making it easier to compute probabilities using the standard normal distribution.

Cumulative Probability Calculation

Cumulative probability calculation involves finding the probability that a random variable takes a value less than or equal to a particular value. When using the standard normal distribution, this probability is determined by the cumulative distribution function (CDF), which is integral in solving problems involving areas under the curve of the normal distribution.

Transcript

00:02 All right, according to the sleep foundation, the mean number of hours of sleep people get is 6 .8 hours, and the standard deviation for that, we are assuming, is 0 .6 hours.

00:15 Let x be the random variable that is the number of hours of sleep one gets.

00:25 All right, so part a asks us to find the probability that somebody gets more than 8 hours of sleep.

00:34 All right, let's z score 8 against the 3.

00:40 Statistics and that is two standard yeah just 2 .00 and that gives us according to our table 0 .972 so up here this equals the probability that z is greater than 2 .00 so we're going to take one minus the probability that z is less than or equal to 2 .00 so that's 1 minus 0 .972, which is 0 .028.

01:27 All right, part b, we need to find the probability that someone gets 6 hours of sleep or less, which is not healthy.

01:38 But, hey, it happens.

01:42 All right, let's z score 6.

01:47 All right.

01:50 This is going to be, well, this up here is going to be negative point a.

01:54 Over, that's negative 4 .06, which is negative 4 3rd, so this is 1 .33.

02:04 We check this with our table.

02:13 We get that this is equal to 0 .0 .0918.

02:21 This is the probability.

02:23 Let's see, is less than 1 .33.

02:27 Or, sorry, negative 1 .33.

02:29 So this is 0 .0918...