Question

A journal 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 8.888.88 hours and a standard deviation of 2.092.09 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)b. Find P(8<=x<=10)c. Find the value for which P(x

Answer 1

Answer:

a) P(X < 6) = 0.0838

b) P(8<=x<=10) = 0.3682

c) a = 7.1244

Step-by-step explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

$\mu = 8.88, \sigma = 2.09$

a. Find P(x < 6)

This is the pvalue of Z when X = 6. So:

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{6 - 8.88}{2.09}$

$Z = -1.38$

$Z = -1.38$ has a pvalue of 0.0838.

So P(X < 6) = 0.0838

b. Find P(8<=x<=10)

This is the pvalue of Z when X = 10 subtracted by the pvalue of Z when X = 8. So:

X = 10

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{10 - 8.88}{2.09}$

$Z = 0.54$

$Z = 0.54$ has a pvalue of 0.7054.

X = 8

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{8 - 8.88}{2.09}$

$Z = -0.42$

$Z = -0.42$ has a pvalue of 0.3372.

So P(8<=x<=10) = 0.7054 - 0.3372 = 0.3682

c. Find the value for which P(x < a) = 0.2

This is X = a when Z has a pvalue of 0.2. So $Z = -0.84$

$Z = \frac{X - \mu}{\sigma}$

$-0.84 = \frac{a - 8.88}{2.09}$

$a - 8.88 = 2.09*(-0.84)$

$a = 7.1244$