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 9.63 hours of sleep, with a standard deviation of 2.92 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 6.3 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.

(b) What is the probability that a visually impaired student gets between 8 and 9.02 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.

(c) What is the probability that a visually impaired student gets at least 8.2 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.

(d) What is the sleep time that cuts off the top 33 % of sleep hours? Round your answer to 2 decimal places.

(e) If 400 visually impaired students were studied, how many students would you expect to have sleep times of more than 9.02 hours? Round to the nearest whole number.

(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 1

Solution: We are given:

(a) What is the probability that a visually impaired student gets at most 6.3 hours of sleep?

Answer: We have to find

Using the z-score formula, we have:

  

Now using the standard normal table, we have:

b) What is the probability that a visually impaired student gets between 8 and 9.02 hours of sleep?

Answer:

We have to find

Using the z-score formula, we have:

  

  

Now using the standard normal table, we have:

(c) What is the probability that a visually impaired student gets at least 8.2 hours of sleep?

Answer: We have to find

Using the z-score formula, we have:

  

Now using the standard normal table, we have:

(d) What is the sleep time that cuts off the top 33 % of sleep hours?

Answer: We have to first find the z-value corresponding to an area = 1- 0.33 = 0.67. Using the standard normal table, we have:

Now using the z-score formula, we have:

Therefore, the sleep time that cuts off the top 33 % of sleep hours is 10.91

(e) If 400 visually impaired students were studied, how many students would you expect to have sleep times of more than 9.02 hours?

Answer: We have to find the first

Using the z-score formula, we have:

  

Now using the standard normal table, we have:

Therefore, the number of students who would expect to have sleep times of more than 9.02 hours is:

(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?

Answer: We have to first find the z-value corresponding to an area 0.25. Using the standard normal table, we have:

Now using the z-score formula, we have:

Therefore, the maximum sleep time to be recommended for additional assistance is 7.66