Ajournal published a study of the lifestyles of visually impaired students. using diaries, the students kept track of several variables, including number of hours of sleep obtained in a typical day. these visually impaired students had a mean of 8.89 hours and a standard deviation of 2.11 hours. assume that the distribution of the number of hours of sleep for this group of students is approximately normal. complete parts a through c.

a. find p(x < 6). p(x < 6) = (round to four decimal places as needed.)
b. find p(8 lessthanorequalto x lessthanorequalto 10) p(8 lessthanorequalto x lessthanorequalto 10) = (round to four decimal places as needed.)
c. find the value a for which p(x < a) = 0.2. (round to two decimal places as needed.)

a)

b)

c)

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Let X the random variable that represent interest on this case, and for this case we know the distribution for X is given by:

And let represent the sample mean, the distribution for the sample mean is given by:

a. Find P(x < 6). P(x < 6) = (Round to four decimal places as needed.)

In order to find this probability we can use the z score given by this formula

And if we use this we got this:

b. Find P(8 < x < 10)  = (Round to four decimal places as needed.)

c. Find the value a for which P(x < a) = 0.2. (Round to two decimal places as needed.)

For this case we can use the definition of z score given by:

First we need to find a z score that accumulates 0.2 of the area on the left tail, and the z score on this case is z=-0.842. and using this z score we can solve for x like this:

And if we solve for x we got:

And that would be the value required for this case.

could you provide figure.