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Korolek [52]
3 years ago
14

A journal published a study of the lifestyles of visually impaired students. Using diaries, the students kept track of several v

ariables, 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.)
Mathematics
1 answer:
Serga [27]3 years ago
6 0

Answer:

a) P(X

b) P(8 \leq X\leq 10) = 0.3640

c) x = 8.89 -0.842(2.11)=7.11

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:

X \sim N(\mu=8.89,\sigma=2.11)  

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

\bar X \sim N(\mu,\frac{\sigma}{\sqrt{n}})

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

z=\frac{x-\mu}{\sigma}

And if we use this we got this:

P(X

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

P(8 \leq X\leq 10) = P(\frac{8-8.89}{2.11} \leq Z< \frac{10-8.89}{2.11})=P(-0.422 \leq Z\leq 0.526)=P(Z

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:

z=\frac{x-\mu}{\sigma}

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:

-0.842=\frac{x-8.89}{2.11}

And if we solve for x we got:

x = 8.89 -0.842(2.11)=7.11

And that would be the value required for this case.

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