College students average 7.8 hours of sleep per night with a standard deviation of 45 minutes. If the amount of sleep is normally distributed, what proportion of college students sleep for more than 9 hours?
You know the value of expectation, 7.8h, you know the deviation, 0.75h and you know that the amount of students, sleeping more LESS than (or equal) 9 hours is:
/bigPhi((9h-7.8h)/0.75h)=
/bigPhi(1.6)
Hence the required probability to find such a student (sleeping MORE than 9 hours) is:
P=1-/bigPhi(1.6)
Z = (x-μ)/SD
Z = (9-7.8)/.75 = 1.6
In table in the back of your stat text called something like "areas under normal distribution," look up Z score for smaller area to get proportion.
I hope this helps a little more.
;kjl
To find the proportion of college students who sleep for more than 9 hours, we need to calculate the area under the normal distribution curve that represents the percentage of students who sleep for more than 9 hours.
First, we calculate the z-score using the formula:
z = (x - μ) / σ
where x is the value we want to find the proportion for (9 hours), μ is the mean (7.8 hours), and σ is the standard deviation (45 minutes or 0.75 hours).
So, our z-score is:
z = (9 - 7.8) / 0.75
Next, we find the area to the right of the z-score in the standard normal distribution table (also known as the z-table). This area represents the proportion of students who sleep for more than 9 hours.
Using the z-table, we find the corresponding value for our z-score. Let's assume it is denoted as P(z).
Finally, the proportion of college students who sleep for more than 9 hours can be calculated as:
Proportion = 1 - P(z)
This is because the total area under the normal distribution curve is 1, and we want the area to the right of the z-score, which is the complement of the area to the left.
Note: If you have access to statistical software or a graphing calculator, you can also directly calculate the proportion using the normal distribution function.
So, to calculate the proportion of college students who sleep for more than 9 hours, you would find the z-score, look up the corresponding value in the z-table, subtract it from 1, and that will give you the desired proportion.