University students average 7.2 hours of sleep per night with a standard deviation of 40 minutes. If the amount of sleep is normally distributed, what proportion of university students sleep for more than 8 hours
8 hr is 48 min above the mean ... this is 48/40 s.d. (1.2 z-score)
go to a z-score table to find the % of the population above 1.2
To find the proportion of university students who sleep for more than 8 hours per night, we can use the standard normal distribution.
First, we need to standardize the value of 8 hours by converting it into a z-score. To do this, we use the formula:
z = (x - μ) / σ
Where:
- x is the raw score (in this case, 8 hours)
- μ is the mean (7.2 hours)
- σ is the standard deviation (40 minutes, or 40/60 = 2/3 hours)
Converting 8 hours into z-score:
z = (8 - 7.2) / (2/3) = 0.8 / (2/3) = 1.2
Next, we need to find the proportion of values greater than this z-score in the standard normal distribution.
Using a standard normal distribution table or a calculator, we can find that the proportion of values greater than a z-score of 1.2 is approximately 0.1151.
Therefore, approximately 11.51% of university students sleep for more than 8 hours per night.