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. Resparchers found that visually impaired students averaged 9.77 hours of sleep, with a standard deviation of t.22 hours. Assume that the number of hours of steep for these visually impaired students is normally distributed. What is the probabitity that a visually impaired student gets at most 6.5 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 1

To calculate the probability of a visually impaired student getting at most 6.5 hours of sleep, we first compute the z-score and then find the corresponding cumulative probability. The probability is 0.36%.

Step-by-step explanation:

To find the probability that a visually impaired student gets at most 6.5 hours of sleep, we will use the normal distribution of sleep hours given that the mean is 9.77 hours and the standard deviation is 1.22 hours.

First, we calculate the z-score for 6.5 hours using the formula:

Z = (X - μ) / σ

where X is the value of interest (6.5 hours), μ is the mean (9.77 hours), and σ is the standard deviation (1.22 hours).

Z = (6.5 - 9.77) / 1.22 = -2.68

Using standard normal distribution tables, or a calculator, we find the probability associated with a z-score of -2.68. The cumulative probability for z = -2.68 is approximately 0.0036, or 0.36%.

Therefore, the probability that a visually impaired student gets at most 6.5 hours of sleep is 0.36%.