Using diaries for many weeks, a study on the lifestyles of visually impaired stu

Question

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 9.52 hours of sleep, with a standard deviation of 1.06 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 6.4 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: % (b) What is the probability that a visually impaired student gets between 6.8 and 8.28 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: % (c) What is the probability that a visually impaired student gets at least 8.5 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: % (d) What is the sleep time that cuts off the top 10.3% of sleep hours? Round your answer to 2 decimal places. Answer: hours (e) If 600 visually impaired students were studied, how many students would you expect to have sleep times of more than 8.28 hours? Round to the nearest whole number. Answer: students (f) A school district wants to give additional assistance to visually impaired students with sleep times at the first quartile and lower. What would be the maximum sleep time to be recommended for additional assistance? Round your answer to 2 decimal places. Answer: hours

Explanation / Answer

Solution:

Given, Mean = 9.52 and = 1.06
Here the random variable X is a normally distribution.
a) What is the probability that a visually impaired student gets at most 6.4 hours of sleep?
P(X6.4) =P ( X6.49.52 )
=P (X/6.49.52/1.06)
= P (Z2.94)
= P (Z > 2.94)
= 1P ( Z<2.94 ) =10.9984=0.0016
b) What is the probability that a visually impaired student gets between 6.8 and 8.28 hours of sleep?
P ( 6.8<X<8.28 )=P ( 6.89.52< X<8.289.52 )
=P ( 6.89.52/1.06<X/<8.289.52/1.06)
=P ( 2.57<Z<1.17 )
=0.1159
c) What is the probability that a visually impaired student gets at least 8.5 hours of sleep?
P ( X8.5 )=P ( X8.59.52 )
=P ( X/8.59.52/1.06)
=P ( Z0.96 ))
=P ( Z<0.96 )) = 0.8315

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