Homework 11: Problem 1 Prev Up Next (10 pts) Using diaries for many weeks, a stu

Question

Homework 11: Problem 1 Prev Up Next (10 pts) Using diaries for many weeks, a study on the lifestyles of visually impaired students was conducted. The students kept track of many lifestyle variables including how many hours of sleep obtained on a typical day. Researchers found that visually impaired students averaged 8.99 hours of sleep, with a standard deviation of 2.41 hours. Assume that the number of hours of sleep for these visually impaired students is normally distributed. (a) What is the probability that a visually impaired student gets at most 6.1 hours of sleep? Express your answer as a percent rounded to 2 decimal places. e.g. 1.23% Do not include the % symbol in your answer. Answer:

Explanation / Answer

NORMAL DISTRIBUTION
the PDF of normal distribution is = 1/ * 2 * e ^ -(x-u)^2/ 2^2
standard normal distribution is a normal distribution with a,
mean of 0,
standard deviation of 1
equation of the normal curve is ( Z )= x - u / sd ~ N(0,1)
mean ( u ) = 8.99
standard Deviation ( sd )= 2.41
a.
P(X > 6.1) = (6.1-8.99)/2.41
= -2.89/2.41 = -1.1992
= P ( Z >-1.1992) From Standard Normal Table
= 0.8848,
P(X < = 6.1) = (1 - P(X > 6.1)
= 1 - 0.8848 = 0.1152 ~ 11.52%
b.
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 6.3) = (6.3-8.99)/2.41
= -2.69/2.41 = -1.1162
= P ( Z <-1.1162) From Standard Normal Table
= 0.1322
P(X < 7.76) = (7.76-8.99)/2.41
= -1.23/2.41 = -0.5104
= P ( Z <-0.5104) From Standard Normal Table
= 0.3049
P(6.3 < X < 7.76) = 0.3049-0.1322 = 0.1727 ~ 17.27%
c.
P(X < 7.2) = (7.2-8.99)/2.41
= -1.79/2.41= -0.7427
= P ( Z <-0.7427) From Standard Normal Table
= 0.2288
P(X > = 7.2) = (1 - P(X < 7.2)
= 1 - 0.2288 = 0.7712 ~ 77.12%
d.
P ( Z > x ) = 0.141
Value of z to the cumulative probability of 0.141 from normal table is 1.0758
P( x-u / (s.d) > x - 8.99/2.41) = 0.141
That is, ( x - 8.99/2.41) = 1.0758
--> x = 1.0758 * 2.41+8.99 = 11.5828
e.
P(X > 7.76) = (7.76-8.99)/2.41
= -1.23/2.41 = -0.5104
= P ( Z >-0.5104) From Standard Normal Table
= 0.6951

the no.of visually impaired in 150 = 150 * 0.6951 = 104.265 ~ 105

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