I would like to know the calculator commands for these questions. Do we humans g
ID: 3063425 • Letter: I
Question
I would like to know the calculator commands for these questions.
Do we humans get enough amount of sleep we need? This is a motivational reading. You won’t use any information from this article to answer the questions in the next page.
Here is what some experts are saying...
"There is increasing evidence that there is a very strong relationship between sleep quality and physical and mental health..." A national survey of 31,044 adults revealed a link between insomnia and other sleeping problems with high blood pressure, heart failure, anxiety, and depression. Experts say that there is no evidence yet to indicate that acquiring more sleep will improve medical conditions, but some studies have shown that more sleep makes teenagers do better in school.
The US National Sleep Foundation suggests that an adult needs seven to nine hours of sleep a night. The Foundation conducted a survey in 2012 that suggested about 75% of Americans had sleeping problems, with one-third being so sleepy during the day that it affected their lives and activities.
On the other hand, when asked if everyone needed 8 hours of sleep each night, some experts stated: "It's nonsense. It's like saying everybody should have size eight shoes, or should wear Medium size clothes... However, there is a normal distribution with the average sleep length seven and a quarter hours." Many people report their sleeping hours more or less than the average, and that the determining factor for a required amount of sleep should be whether the person feels fresh and alert during the day, the same experts say.
While amount or quality of sleep undergoes a wide variety of changes during the human life span (based primarily upon age and life style), all aspects of sleep, at any given age, have a relatively normal, bell-shaped distribution.
After this motivational reading, now you will explore some sleep hours. First you need to collect a small data set in the following way.
Record, to the nearest quarter of an hour, how many hours you slept in the past 24 hours. ______4____________
Ask 3 more adults (in your family, in your classroom, at work, in the neighborhood, etc) how many hours of sleep they got in the past 24 hours and record their answers here to the nearest quarter of an hour.
Use your data set from above and the results of the following survey to answer the questions that follow:
A previous study administered to US adults showed that the amount of sleep (in hours)they get in a 24-hour period is normally distributed with a mean of 7 hours and a standard deviation of 1.25 hours.
Could you explain how to do these and the calculator commands?
What interval about the mean includes 95% of the data in the survey?
What interval about the mean includes 50% of the data in the survey?
30% of adults in the survey sleep less than how many hours of sleep?
10% of adults in the survey sleep more than how many hours of sleep?
Explanation / Answer
Let X: the amount of sleep for US adults. Then X ~ N(7,1.25)
1. What interval about the mean includes 95% of the data in the survey:
The area of interest is 0.95, So the area outside our interest area = 0.05
Area outside interest area on each side of the mean (also known as tails) = 0.05/2 = 0.025
So the area of interest lies from a point with cumulative probability of 0.025 to a point with cumulative probability of 0.975.
F(X) =0.025 corresponds to X = 4.55 (Can be found from a standard normal table, or using Excel formula NORM.INV(0.025,7,1.25).
F(X) =0.975 corresponds to X = 9.45
So the interval about mean, containing 95% of the data is [4.55,9.45].
2. For 50% of the data, the respective cumulative probabilities are 0.25 and 0.75
F(X) =0.25 corresponds to X = 6.1569
F(X) =0.75 corresponds to X = 7.8431
So the interval about mean, containing 50% of the data is [6.1569,7.8431]
3. To calculate the number of hours of sleep corresponding to the bottom 30% of adults, we note the corresponding cumulative probability is 0.3
F(X) =0.3 corresponds to X = 6.3445
So we can conclude that 30% of the adults sleep less than 6.3445 hours
4. To calculate the number of hours of sleep corresponding to the top 10% of adults, we note the corresponding cumulative probability is 0.90.
F(X) =0.9 corresponds to X = 8.6019
So, 10% of the adults sleep more than 8.6019